Asme Y14.5.1-2019

ASME Y14.5.
1-2019
[Revision of ASME Y14.5.1M-1994 (R2012)]
Mathematical
Definition of
Dimensioning and
Tolerancing Principles
Engineering Product Definition and

Related Documentation Practices
A N I N T E R N AT I O N A L STA N DA R D
ASME Y14.5.1-2019
[Revision of ASME Y14.5.1M-1994 (R2012)]
Mathematical
Definition of
Dimensioning and
Tolerancing Principles
Engineering Product Definition and
Related Documentation Practices
AN INTERNATIONAL STANDARD
Two Park Avenue • New York, NY • 10016 USA

x
Date of Issuance: August 31, 2020
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CONTENTS
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Committee Roster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Correspondence With the Y14 Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Section 1 Scope and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ASME Y14 Series Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Summary of Conventional Designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.7 Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Section 2 General Tolerancing and Related Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Surface Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Dimension Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Features of Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Section 3 Symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Section 4 Datum Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Datum Feature Simulators (Theoretical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.5 Constraining Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6 Tabulation of Datum Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.7 Candidate Datum Sets and Candidate Reference Frame Sets . . . . . . . . . . . . . . . . . . . . . 16
Section 5 Tolerances of Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Form Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Specifying Form Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.4 Form Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Section 6 Tolerances of Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Orientation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3 Specifying Orientation Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4 Orientation Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Section 7 Tolerances of Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Position Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.3 Projected Tolerance Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
iii
7.4 Conical Tolerance Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.5 Bidirectional Position Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.6 Position Tolerancing at MMC for Boundaries of Elongated Holes . . . . . . . . . . . . . . . . . 38
7.7 Concentricity and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Section 8 Tolerances of Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2 Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.3 Tolerance Zone Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.4 Profile Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.5 Extension of Tolerance Zone Boundaries for Sharp Corners . . . . . . . . . . . . . . . . . . . . . 48
8.6 Nonuniform Tolerance Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Section 9 Tolerances of Runout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.1 Runout Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Nonmandatory Appendices
A Principal Changes and Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B Mathematical Datum Simulators Referenced at RMB: Definitions and Properties . . . . . . 68
C Former Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
D Concepts Related to Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
E A Selection of Mathematical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
F Potential Misuse of the Swept-Sphere Definition of Size . . . . . . . . . . . . . . . . . . . . . . . . 85
Figures
1-1 Example: Distance From a Point to a True Position Axis . . . . . . . . . . . . . . . . . . . . . . . 6
1-2 Example: Cutting Surfaces to Evaluate Circular Runout . . . . . . . . . . . . . . . . . . . . . . . . 6
1-3 Examples of a Planar Surface of Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2-1 Irregular Features of Size (Type a), Collection of Features . . . . . . . . . . . . . . . . . . . . . . 11
2-2 Symbols Used in the Definition of Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2-3 Conformance to Limits of Size, Internal Feature of Size . . . . . . . . . . . . . . . . . . . . . . . . 11
2-4 Examples of Surface Attributes Leading to Variations in the Derived Median Line . . . . . 12
2-5 The Cutting Plane at a Point on a Local Size Spine, and Some of the Evaluation Lines in That
Cutting Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2-6 Profile of a Surface Tolerance Describing an External IFOSb . . . . . . . . . . . . . . . . . . . . 12
4-1 Example From Table 4-3 — Axis as Primary Datum . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4-2 Example of Testing Whether a Plane Is a Valid Datum Plane . . . . . . . . . . . . . . . . . . . . 21
4-3 Tertiary Datum Feature Simulator Is Basically Located and Oriented . . . . . . . . . . . . . . 22
4-4 Example of Translation Modifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5-1 Evaluation of Straightness of a Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5-2 Evaluation of Straightness of a Planar Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5-3 Illustration of Circularity Tolerance Zone for a Cylindrical or Conical Feature . . . . . . . . 27
6-1 Planar Orientation Zone With Primary and Secondary Datum Planes Specified . . . . . . . 31
6-2 Projection of Tolerance Vector Onto Primary Datum Plane . . . . . . . . . . . . . . . . . . . . . 31
6-3 Cylindrical Orientation Zone With Respect to a Primary Datum Plane . . . . . . . . . . . . . . 32
6-4 Linear Orientation Zone Bounded by Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7-1 First Illustration of the Difference Between Surface and Resolved Geometry Interpretations
of Position Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
iv
7-2 Second Illustration of the Difference Between Surface and Resolved Geometry
Interpretations of Position Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7-3 Tolerance Zone and Conformance: Holes at MMC or RFS, Shafts at LMC — Surface
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7-4 Tolerance Zone and Conformance: Tabs at MMC or RFS, Slots at LMC — Surface
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7-5 Tolerance Zone and Conformance: Holes at MMC or RFS, Shafts at LMC — Resolved Geometry
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7-6 Projected Tolerance Zone for a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7-7 Surface Interpretation of Conical Tolerance Zone for Holes at MMC or RFS . . . . . . . . . . 41
7-8 Resolved Geometry (Axis) Interpretation of Conical Tolerance Zone for Holes at MMC or RFS 41
7-9 Bidirectional Hole Tolerance at MMC With Cylindrical Tolerance Zones — Surface
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7-10 Bidirectional Shaft Tolerance at MMC With Parallel Plane Tolerance Zones — Surface
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7-11 Definition of the Tolerance Zone for Polar Bidirectional Tolerancing . . . . . . . . . . . . . . 44
7-12 Tolerance Zone and Conformance, Elongated Hole at MMC — Tolerance Zone is the Right
Cylinder Shown in Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7-13 Rays Are Arranged in the Lowest Order of Symmetry About an Axis or a Point . . . . . . 45
8-1 Tolerance Zone Derivation — Equally Disposed Profile . . . . . . . . . . . . . . . . . . . . . . . . 50
8-2 Tolerance Zone Derivation — Unequally Disposed Profile . . . . . . . . . . . . . . . . . . . . . . 50
8-3 Tolerance Zone Derivation — Unilaterally Disposed Profile (Outside) . . . . . . . . . . . . . . 51
8-4 Tolerance Zone Derivation — Unilaterally Disposed Profile (Inside) . . . . . . . . . . . . . . . 51
8-5 Actual Zone Definition for Equally Disposed Profile — Example of Conformance . . . . . . 52
8-6 Actual Zone Definition for Unequally Disposed Profile — Example of Conformance . . . . 52
8-7 Actual Zone for Unilateral (Outside) Profile — Example of Nonconformance . . . . . . . . . 52
8-8 Actual Zone for Unilateral (Inside) Profile — Example of Nonconformance . . . . . . . . . . 53
8-9 Profile Tolerance for a Single Feature Without a Datum Reference Frame . . . . . . . . . . . 53
8-10 Profile Tolerance Zone for a Single Feature Without a Datum Reference Frame . . . . . . . 53
8-11 Candidate Configuration #1 (System Not Optimized) . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8-12 Candidate Configuration #2 (System Optimized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8-13 Actual Values in a Multi-Feature Profile Tolerance System — Specification . . . . . . . . . . 54
8-14 Actual Values in a Multi-Feature Profile Tolerance System — Basically Related Profile
Tolerance Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8-15 Actual Values in a Multi-Feature Profile Tolerance System — Candidate Configuration #1
(System Not Optimized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8-16 Actual Values in a Multi-Feature Profile Tolerance System — Candidate Configuration #2
(System Optimized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8-17 Individual Requirements for Profile — Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8-18 Individual Requirements for Profile — Tolerance Zones . . . . . . . . . . . . . . . . . . . . . . . . 56
8-19 Individual Requirements for Profile — Individual Requirement 1 . . . . . . . . . . . . . . . . . 56
8-20 Individual Requirements for Profile — Individual Requirement 2 . . . . . . . . . . . . . . . . . 57
8-21 Datum Feature References With Profile — Specification . . . . . . . . . . . . . . . . . . . . . . . 57
8-22 Datum Feature References With Profile — Tolerance Zones and Simulators . . . . . . . . . 57
8-23 Datum Feature References With Profile — Invalid Candidate Configuration . . . . . . . . . 58
8-24 Datum Feature References With Profile — Valid Candidate Configuration . . . . . . . . . . . 58
8-25 Simultaneous Requirements for Profile — Specification . . . . . . . . . . . . . . . . . . . . . . . . 58
v
8-26 Simultaneous Requirements for Profile — Tolerance Zones . . . . . . . . . . . . . . . . . . . . . 59
8-27 Simultaneous Requirements for Profile (System Not Optimized) . . . . . . . . . . . . . . . . . . 59
8-28 Simultaneous Requirements for Profile (System Optimized) . . . . . . . . . . . . . . . . . . . . . 59
8-29 Composite Profile Lower Segment — Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8-30 Composite Profile Lower Segment —Tolerance Zones (FRTZF) . . . . . . . . . . . . . . . . . . . 60
8-31 Composite Profile Lower Segment (System Not Optimized) . . . . . . . . . . . . . . . . . . . . . 60
8-32 Composite Profile Lower Segment (System Optimized) . . . . . . . . . . . . . . . . . . . . . . . . 61
8-33 Tolerance Zone Derivation — Profile With Sharp Corner . . . . . . . . . . . . . . . . . . . . . . . 61
8-34 Actual Zone Definition — Profile With Sharp Corner . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9-1 Circular Runout Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B-1 Signed Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B-2 Least Squares Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B-3 Min-Max Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B-4 L1 Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
B-5 Progressed Least Squares Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
B-6 Mating of Actual Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
B-7 Mostly Progressed Least Squares Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B-8 Constrained L1 Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B-9 Constrained Min–Max Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B-10 Constrained Least Squares Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B-11 Constrained Simulators for a V-Shaped Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B-12 Constrained Simulators for a Peanut-Shaped Circular Feature . . . . . . . . . . . . . . . . . . . 75
B-13 Constrained Simulators for an Hourglass-Shaped Opposed Lines Feature . . . . . . . . . . . 76
B-14 Constrained Simulators for a Skew-Convex Linear Feature . . . . . . . . . . . . . . . . . . . . . . 76
B-15 Constrained Simulators for a Dented Circular Feature . . . . . . . . . . . . . . . . . . . . . . . . . 77
B-16 Constrained Simulators for a Skew-Convex Opposed Lines Feature . . . . . . . . . . . . . . . . 77
B-17 Large Voids Can Harm Constrained Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B-18 Void-Filling Can Improve Constrained Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
B-19 Void-Filling Can Degrade the Constrained Min-Max Simulator . . . . . . . . . . . . . . . . . . . 78
D-1 Size Ordering for Perfect and Imperfect Features of Size . . . . . . . . . . . . . . . . . . . . . . . 80
E-1 Examples of One-Dimensional Manifold (Left) and Not (Right) . . . . . . . . . . . . . . . . . . . 83
E-2 A Line Which Is Continuous (Left) and Not (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
E-3 A Line Which Is Tangent Continuous (Left) and Not (Right) . . . . . . . . . . . . . . . . . . . . . 83
E-4 An Object, Its Interior, and Its Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
E-5 The Result of Union Between Two Objects, and the Non-Manifold Boundary . . . . . . . . . 84
E-6 The Interior of Two Objects, and Their Union With a Manifold Boundary . . . . . . . . . . . 84
F-1 MMC and LMC Spheres, and a Part to Be Tested for Conformance . . . . . . . . . . . . . . . . 86
F-2 LMC and MMC Spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
F-3 LMC and MMC Swept Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
F-4 Conformance to the Size Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
F-8 Incorrect Determination of Conformance to the Size Tolerance . . . . . . . . . . . . . . . . . . 87
vi
Tables
1-1 Mathematical Symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4-1 Symbols for Datum Reference Frame Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4-2 Point as Primary Datum (Spherical Datum Feature) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4-3 Axis as Primary Datum (Cylindical Datum Feature) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4-4 Plane as Primary Datum (Planar or Width Datum Feature) . . . . . . . . . . . . . . . . . . . . . 20
4-5 Coincident Axis and Point as Primary Datum (Conical Datum Feature) . . . . . . . . . . . . . 20
4-6 Axis and Plane as Primary Datum (Linear Extruded Shape Datum Feature) . . . . . . . . . . 20
4-7 Axis Point and Plane as Primary Datum (Complex Datum Feature) . . . . . . . . . . . . . . . . 20
4-8 Generic Invariant Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7-1 Definition of Position Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . . . . 39
7-2 Size of Position Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . . . . . . . . 40
7-3 Size of Position Tolerance Zone — Resolved Geometry Interpretation . . . . . . . . . . . . . 40
7-4 Definition of Verifying Volume for Projected Tolerance Zone . . . . . . . . . . . . . . . . . . . . 40
7-5 Definition of Conical Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . . . . . 41
7-6 Size of Conical Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . . . . . . . . . 41
7-7 Size of Conical Tolerance Zone — Resolved Geometry (Axis) Interpretation . . . . . . . . . 42
7-8 Definition of Bidirectional Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . 42
7-9 Size of Bidirectional Tolerance Zone — Surface Interpretation . . . . . . . . . . . . . . . . . . . 42
7-10 Size of Bidirectional Tolerance Zone — Resolved Geometry (Axis) Interpretation . . . . . 43
7-11 Size of Polar Bidirectional Tolerance Zone — Resolved Geometry (Axis) Interpretation . 44
7-12 Symmetry Elements for ObtainingCorresponding Feature Elements . . . . . . . . . . . . . . . 45
8-1 Table of Profile Tolerance Dispositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
D-1 Uniqueness and Containment Relationships for Features of Size — Part 1 . . . . . . . . . . 81
D-2 Uniqueness and Containment Relationships for Features of Size — Part 2 . . . . . . . . . . 81
vii
FOREWORD
This edition is a revision of ASME Y14.5.1M-1994, Mathematical Definition of Dimensioning and Tolerancing Principles,
supporting Y14.5-2009. Efforts focused on improving usability with Y14.5 have resulted in harmonization of terms where
possible and a realignment of sections consistent with Y14.5-2009. This revision also addresses proposals submitted by
the public or members of the Y14 Standards Committee and affiliated subcommittees. See Nonmandatory Appendix A,
Principle Changes and Improvements.
Work on this revision began in May of 2012 followed by semiannual face-to-face meetings and numerous online
collaborative meetings to refine content and disposition comments. Comments on key areas are provided below to
add some context to these revisions.
This revision includes a new stabilization definition for irregularities on datum features specified at RMB, which
provides an alternative to the candidate datum set. SC5 had moved toward the concept of a single-stable solution
that minimizes the separation between the datum feature and the true geometric counterpart, and SC5.1 was
asked to study the concept and recommend a mathematical definition. A stable Constrained L2 datum definition
was selected, which applies as an alternate stabilization definition for Y14.5-2009 and the default stabilization definition
for Y14.5-2018. See Nonmandatory Appendix B for results of the study and mathematical definitions.
Profile tolerancing was a major focus of development work and the section was completely rewritten. The actual value
of profile was changed from the deviation-based two-value definition to a zone-based single-value definition compatible
with Y14.5’s tolerance zone definitions. This provides a consistent treatment of unequally disposed and unilateral profile
zones, with no change in conformance results. The updated definition allows direct comparison of the actual value with
the specified tolerance value, and consistency with actual value definitions for other geometric tolerances.
Redevelopment of the profile section also created the requirement to address applications with multiple features and a
variety of degree of freedom constraints. This was accomplished by treating profile tolerances as systems with degrees of
freedom and constraints, with the Y14.5 tolerance zone and datum reference frame definitions providing the initial
conditions. Actual values are defined for a wide variety of profile applications including single features, multifeature
groups with and without datum features, simultaneous requirements, and composite profile tolerances.
The mathematical definitions for size have not changed; they continue to use the sweeping ball concept to define the
tolerance zone volume. Two nonmandatory definitions for local size have been added: one based on opposed points and
the other based on inscribed/circumscribed circular elements.
Text and figure edits were made to improve readability and clarify content. Changes in sentence structure, organization
of content, and method of illustration are not an indication of technical changes.
This Standard is available for public review on a continuing basis. This provides an opportunity for additional public
review input from industry, academia, regulatory agencies, and the public-at-large.
ASME Y14.5.1-2019 was approved by ANSI as an American National Standard on November 7, 2019.
viii
ASME Y14 COMMITTEE
Engineering Product Definition and Related Documentation
Practices
(The following is the roster of the Committee at the time of approval of this Standard.)
STANDARDS COMMITTEE OFFICERS

W. A. Kaba, Jr., Chair
J. I. Miles, Vice Chair
F. Constantino, Secretary
STANDARDS COMMITTEE PERSONNEL

A. R. Anderson, Dimensional Dynamics, LLC S. P. Lege, U.S. Army
F. Bakos, Consultant E. F. McCarthy, E. F. McCarthy Consulting, Inc.
J. V. Burleigh, Unaffiliated P. J. McCuistion, Multimac
F. Constantino, The American Society of Mechanical Engineers J. D. Meadows, James D. Meadows & Associates, Inc.
D. O. Coon, Bell Helicopter M. E. Meloro, Northrop Grumman Corp.
R. Courson, SAE International J. Michalowicz, Stryker Corp.
K. Dobert, Siemens PLM Software J. I. Miles, Technical Consultants, Inc.
S. Hauger, Deere & Co. M. A. Murphy, Unaffiliated
J. B. Hoskins, Boeing Co. H. W. Oakes, U.S. Air Force (SAIC)
J. Houck, Woodward, Inc. B. A. Wilson, Unaffiliated
R. C. Jensen, Hexagon Manufacturing Intelligence E. F. Zwettler, Sigmetrix, LLC
W. A. Kaba, Jr., Spirit AeroSystems, Inc. K. E. Wiegandt, Contributing Member, Consultant
A. Krulikowski, Krulikowski Consulting, LLC
SUBCOMMITTEE 5.1 — MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCING

PRINCIPLES
R. C. Jensen, Chair, Hexagon Manufacturing Intelligence J. G. Salsbury, Mitutoyo America Corp.
E. Janeshewski, Vice Chair, Axymetrix Quality Engineering, Inc. C. Shakarji, NIST
R. G. Campbell, Vice Chair, Contributing Member M. Taylor, Raytheon Missile Systems
L. Bergquist, Secretary, Deere & Co. P. Thomas, Consultant
L. Aguirre, Detechnic Design Science D. Wilcox, Hexagon Manufacturing Intelligence
J. Hurt, Enventive Engineering M. Foster, Contributing Member, Applied Geometrics, Inc.
S. C. Liu, Siemens PLM Software L. Li, Contributing Member, JSL Innovations, Inc.
P. J. McCuistion, Multimac V. Srinivasan, Contributing Member, NIST
E. Morse, UNC Charlotte H. B. Voelcker, Contributing Member, Retired, Cornell University
S. Neumann, Technical Consultants, Inc.
ix
CORRESPONDENCE WITH THE Y14 COMMITTEE
General. ASME Standards are developed and maintained with the intent to represent the consensus of concerned
interests. As such, users of this Standard may interact with the Committee by requesting interpretations, proposing
revisions or a case, and attending Committee meetings. Correspondence should be addressed to:
Secretary, Y14 Standards Committee

The American Society of Mechanical Engineers
Two Park Avenue
New York, NY 10016-5990
http://go.asme.org/Inquiry
Proposing Revisions. Revisions are made periodically to the Standard to incorporate changes that appear necessary
or desirable, as demonstrated by the experience gained from the application of the Standard. Approved revisions will be
published periodically.
This Standard is always open for comment, and the Committee welcomes proposals for revisions. Such proposals
should be as specific as possible, citing the paragraph number(s), the proposed wording, and a detailed description of the
reasons for the proposal, including any pertinent documentation.
Proposing a Case. Cases may be issued to provide alternative rules when justified, to permit early implementation of
an approved revision when the need is urgent, or to provide rules not covered by existing provisions. Cases are effective
immediately upon ASME approval and shall be posted on the ASME Committee web page.
Requests for Cases shall provide a Statement of Need and Background Information. The request should identify the
Standard and the paragraph, figure, or table number(s), and be written as a Question and Reply in the same format as
existing Cases. Requests for Cases should also indicate the applicable edition(s) of the Standard to which the proposed
Case applies.
Attending Committee Meetings. The Y14 Standards Committee regularly holds meetings and/or telephone confer-
ences that are open to the public. Persons wishing to attend any meeting and/or telephone conference should contact the
Secretary of the Y14 Standards Committee. Future Committee meeting dates and locations can be found on the Committee
Page at http://go.asme.org/Y14committee.
x
ASME Y14.5.1-2019
Section 1
Scope and Definitions
1.1 SCOPE (g) The phrase “unless otherwise specified” or “UOS”

shall be used to indicate a default requirement. The phrase
This Standard presents a mathematical definition of is used when the default is a generally applied require-
geometrical dimensioning and tolerancing consistent ment and an exception may be provided by another docu-
with the principles and practices of ASME Y14.5-2009, ment or requirement.
enabling determination of actual values. While the
general format of this Standard parallels that of ASME 1.2.2 Cross-Reference of Standards
Y14.5-2009, the latter document should be consulted
for practices relating to dimensioning and tolerancing Cross-reference of standards in text with or without a
for use on engineering product definition and in date following the standard designator shall be inter-
related documentation. preted as follows:
Textual references are included throughout this Stan- (a) Reference to other ASME Y14 standards in the text
dard which are direct quotations from ASME Y14.5-2009. without a date following the standard designator indicates
All such quotations are identified by italicized type and that the edition of the standard identified in the Refer-
include paragraph reference within square brackets. ences section (subsection 1.3) shall be used to meet
Any direct references to other documents are identified the requirement.
by an immediate citation. (b) Reference to other ASME Y14 standards in the text
This Standard applies to product definition in any with a date following the standard designator indicates
representation. When reference is made in this Standard that only that edition of the standard shall be used to
to an engineering product definition, it applies to any form meet the requirement.
of product specification.
1.2.3 Invocation of Referenced Standards
1.1.1 Reference to Gaging The following examples define the invocation of a stan-
This Standard is not intended as a gaging standard. Any dard when specified in the References section
reference to gaging is included for explanatory purposes (subsection 1.3) and referenced in the text of this Stan-
only. dard:
(a) When a referenced standard is cited in the text with
1.2 ASME Y14 SERIES CONVENTIONS no limitations to a specific subject or paragraph(s) of the
The conventions in paras. 1.2.1 through 1.2.9 are used in standard, the entire standard is invoked. For example,
this and other ASME Y14 standards. “Dimensioning and tolerancing shall be in accordance
with ASME Y14.5” is invoking the complete standard
1.2.1 Mandatory, Recommended, Guidance, and because the subject of the standard is dimensioning
Optional Words and tolerancing and no specific subject or paragraph(s)
within the standard are invoked.
(a) The word “shall” establishes a requirement. (b) When a referenced standard is cited in the text with
(b) The word “will” establishes a declaration of limitations to a specific subject or paragraph(s) of the
purpose on the part of the design activity. standard, only the paragraph(s) on that subject is
(c) The word “should” establishes a recommended invoked. For example, “Assign part or identifying
practice. numbers in accordance with ASME Y14.100” is invoking
(d) The word “may” establishes an allowed practice. only the paragraph(s) on part or identifying numbers
(e) The words “typical,” “example,” “for reference,” and because the subject of the standard is engineering
the Latin abbreviation “e.g.” indicate suggestions given for drawing practices and part or identifying numbers is a
guidance only. specific subject within the standard.
(f) The word “or” used in conjunction with a require- (c) When a referenced standard is cited in the text
ment or a recommended practice indicates that there are without an invoking statement such as “in accordance
two or more options for complying with the stated with,” the standard is invoked for guidance only. For
requirement or practice. example, “For gaging principles, see ASME Y14.43” is
1
ASME Y14.5.1-2019
only for guidance and no portion of the standard is symbol proportions, select the applicable letter height
invoked. in accordance with ASME Y14.2. Multiview drawings
contained within figures are third-angle projection.
1.2.4 Parentheses Following a Definition
1.2.9 Precedence of Standards
When a definition is followed by a standard referenced
in parentheses, the standard referenced in parentheses is The following are ASME Y14 standards that are basic
the source for the definition. engineering drawing standards:
1.2.5 Notes ASME Y14.1, Decimal Inch Drawing Sheet Size and Format
ASME Y14.1M, Metric Drawing Sheet Size and Format
Notes depicted in this Standard in ALL UPPERCASE ASME Y14.2, Line Conventions and Lettering
letters are intended to reflect actual drawing entries. ASME Y14.3, Orthographic and Pictorial Views
Notes depicted in initial uppercase or lowercase letters ASME Y14.5, Dimensioning and Tolerancing
are to be considered supporting data to the contents of ASME Y14.24, Types and Applications of Engineering
this Standard and are not intended for literal entry on Drawings
drawings. A statement requiring the addition of a note ASME Y14.34, Associated Lists
with the qualifier “such as” is a requirement to add a ASME Y14.35, Revision of Engineering Drawings and
note, and the content of the note is allowed to vary to Associated Documents
suit the application. ASME Y14.36, Surface Texture Symbols
ASME Y14.38, Abbreviations and Acronyms for Use on
1.2.6 Acronyms and Abbreviations
Drawings and Related Documents
Acronyms and abbreviations are spelled out the first ASME Y14.41, Digital Product Definition Data Practices
time used in this Standard, followed by the acronym or ASME Y14.100, Engineering Drawing Practices
abbreviation in parentheses. The acronym is used there-
after throughout the text. All other ASME Y14 standards are considered specialty
types of standards and contain additional requirements or
1.2.7 Units make exceptions to the basic standards as required to
The International System of Units (SI) is featured in this support a process or type of drawing.
Standard. It should be understood that U.S. Customary
units could equally have been used without prejudice 1.3 REFERENCES
to the principles established.
The following revisions of American National Standards
1.2.8 Figures form a part of this Standard to the extent specified herein.
A more recent revision may be used provided there is no
The figures in this Standard are intended only as illus- conflict with the text of this Standard. In the event of a
trations to aid the user in understanding the practices conflict between the text of this Standard and the refer-
described in the text. In some cases, figures show a ences cited herein, the text of this Standard shall take
level of detail as needed for emphasis. In other cases, precedence.
figures are incomplete by intent so as to illustrate a
concept or facet thereof. The absence of figure(s) has ASME Y14.5-2009, Dimensioning and Tolerancing
no bearing on the applicability of the stated requirements ASME Y14.41-2012, Digital Product Definition Data Prac-
or practice. To comply with the requirements of this Stan- tices
dard, actual data sets shall meet the content requirements Publisher: The American Society of Mechanical Engineers
set forth in the text. To assist the user of this Standard, the (ASME), Two Park Avenue, New York, NY 10016-5990
designated paragraph(s) in which an illustration is refer- (www.asme.org)
enced appears in the lower right-hand corner of each
figure. This reference may not be all-inclusive. The
absence of a paragraph reference is not a reason to 1.4 MATHEMATICAL NOTATION
assume inapplicability. Some figures are illustrations of This subsection describes the mathematical notation
models in a three-dimensional environment. The used throughout this Standard, including symbology
absence of dimensioning and tolerancing annotations (typographic conventions) and algebraic notation.
in a view may indicate that the product definition is
defined in 3D. Dimensions that locate or orient and are 1.4.1 Symbology
not shown are considered basic and shall be queried
to determine the intended requirement. When the All mathematical equations in this Standard are rela-
letter “h” is used in figures for letter heights or for tionships between real numbers, three-dimensional
vectors, coordinate systems associated with datum
2
ASME Y14.5.1-2019
reference frames, and sets of these quantities. The symbol The magnitude of the cross product is equal in value to
conventions shown in Table 1-1 are used for these quan- the product of the lengths of the two vectors times the sine
tities. of the angle between them.
These symbols may be subscripted to distinguish For a given feature, the notation r( P, Γ) will denote the
between distinct quantities. Such subscripts do not
change the nature of the designated quantity. distance from a point P to true position (see para. 1.4.1) in
Technically, there is a difference between a vector and a datum reference frame Γ. When the datum reference frame
vector with location. Generally in this Standard, vectors do
is understood from the context, the notation r( P ) will be
not have location. In particular, direction vectors, which
are often defined for specific points on curves or surfaces, used. Figure 1-1 shows a case of a true position axis. If the
are functions of location on the geometry, but are not axis is represented by a point P 0 on the axis and a unit
located at those points. (Another conventional view is
that all vectors are located at the origin.) Throughout vector N , then r( P ) can be evaluated by either of the
this Standard, vectors are used to denote points in following formulas:
space. While there is a technical difference between a 2 ÄÅ ÉÑ2
ÅÅ Ñ
vector and a point in space, the equivalence used in r( P ) = P P0 (ÅÅ P P0 N ÑÑÑ
)
this Standard should not cause confusion. ÅÅÇ ÑÑÖ
1.4.2 Algebraic Notation or
A vector can be expanded into scalar components. Let î, r( P ) = (P )

P0 × N
Ĵ, and k be the unit vectors along the x, y, and z axes, respec-
The first equation is a version of the Pythagorean
tively, of a coordinate system. Then a vector V can be
Theorem. The second equation is based on the properties
uniquely expanded as of the cross product. See Figure 1-1.
V = a^ + bJ + ck
1.5 DEFINITIONS
The vector can be written V = (a, b, c). The magnitude The following terms are defined as their use applies to
this Standard. ASME Y14.5-2009 should be consulted for
(length) of vector V is denoted by |V | and can be eval-
definitions applying to dimensioning and tolerancing.
uated by
1.5.1 Actual Value
V = a2 + b2 + c 2
actual value: a unique numerical value representing a
geometric characteristic associated with one or more
A unit vector V is any vector with magnitude equal to actual features.
one. The scalar product (dot product; inner product) of NOTE: Example characteristics are flatness, perpendicularity,
two vectors V1 = (a 1 , b 1 , c 1 ) and V2 = (a 2 , b 2 , c 2 ) is position, actual mating envelope size, and actual local size.
Later Sections of this Standard provide rules for the determina-
denoted by V1 V2 . The scalar product is a real tion of actual values for specific characteristics.
number given by
1.5.2 Candidate Datum
V1 V2 = a1a2 + b1b2 + c1c2
candidate datum: one of possibly multiple datums that
and is equal in value to the product of the lengths of the two may be established from a datum feature (see
vectors times the cosine of the angle between them. The subsection 4.7).
cross product (vector product) of two vectors V1 and V2 is 1.5.3 Candidate Datum Reference Frame
denoted by V1 × V2 . The cross product is a vector V3 = (a3, candidate datum reference frame: one of possibly multiple
b3, c3)with components given by datum reference frames that may be established from one
or more datum features.
a3 = b1c2 b2c1
b3 = a2c1 a1c2 1.5.4 Candidate Datum Reference Frame Set
c3 = a1b2 a2b1 candidate datum reference frame set: the set of all candi-
date datum reference frames established from a set of
referenced datums.
3
ASME Y14.5.1-2019
1.5.5 Candidate Datum Set 1.5.13 Element, Line

candidate datum set: the set of all candidate datums that element, line: a line (either closed or open, straight or
can be established from a datum feature. See curved), e.g., a one-dimensional manifold, resulting
subsection 4.7. from the intersection of a feature and a specified
cutting surface.
1.5.6 Conformance to a Geometric Tolerance
conformance to a geometric tolerance: applied to a feature, 1.5.14 Engineering Data
that condition in which the feature does not violate the engineering data: engineering documents such as draw-
requirements defined by the specified tolerance. ings, associated lists, accompanying documents, specifica-
tions, standards, or other information prepared or used by
1.5.7 Cutting Surface a design activity and relating to the design, manufacture,
cutting surface: a theoretical surface that, when inter- procurement, testing, or inspection of items (see ASME
sected with a feature, results in a line element. Y14.100-2017).
EXAMPLE: A conical cutting surface coaxial with a datum axis
1.5.15 Envelope, Actual Mating
could be used in circular runout evaluation for a surface nomin-
ally at an angle to the datum axis, as shown in Figure 1-2. The envelope, actual mating: see ASME Y14.5-2009 [1.3.25].
intersection results in a circular line element that is evaluated.
1.5.16 Envelope, Actual Minimum Material
NOTES:
(1) A cutting plane is a cutting surface. envelope, actual minimum material: see ASME Y14.5-2009
(2) The tolerance zone is a subset of the cutting surface. [1.3.26].
1.5.8 Derived Median Line 1.5.17 Feature

derived median line: see ASME Y14.5-2009 [1.3.31]. Per ASME Y14.5-2009
feature: a physical portion of a part such as a surface, pin
1.5.9 Derived Median Plane hole, or slot or its representation on drawings, models, or
derived median plane: see ASME Y14.5-2009 [1.3.30]. digital data files. [1.3.27]
NOTE: For the purposes of this Standard, a feature is an identifi-
1.5.10 Design Geometry able subset (or finite collection of subsets) of the surface of a part.
design geometry: geometry explicitly represented in the A nominal feature is a subset of the nominal surface, while an
technical product definition (e.g., drawing or CAD actual feature is a subset of the actual surface.
model). This geometry is a representation from which A feature is required to be a two-dimensional surface,
tolerance zones can be constructed. that is, a two-dimensional manifold, possibly with
NOTE: The design geometry is not required to be at the center of boundary (see Nonmandatory Appendix E, section E-2).
the tolerance zone (e.g., the case of unequally disposed profile
tolerances). 1.5.18 Feature of Size, External
feature of size, external: a feature of size where each
1.5.11 Direction Vector surface normal is directed away from the feature’s
direction vector: a unit vector. Conventionally, directions resolved geometry.
are associated with various geometries as follows. The NOTE: Surface normals point away from the material.
direction vector of a straight line (or pair of parallel
lines) is parallel to the line(s). The direction vector of
1.5.19 Feature of Size, Internal
a plane (or a pair of parallel planes) is normal to the
plane. The direction vector of a cylinder is the direction feature of size, internal: a feature of size where each surface
vector of the cylinder axis. normal is directed toward the feature’s resolved
geometry.
1.5.12 Element, Circular NOTE: Surface normals point away from the material.
element, circular: a circular element is a closed line
element that is nominally a circle. 1.5.20 Half-Space
Examples of cutting surfaces that can create circular half-space: one of two regions separated by a theoretical
elements are surface that partitions space into exactly two regions.
(a) a cone with axis coincident with the datum axis for These regions can be finite (e.g., the interior of a spherical
circular runout
(b) a plane perpendicular to a spine for circularity
4
ASME Y14.5.1-2019
surface) or infinite (e.g., the region on either side of a actual part geometry (either surface or derived) within
planar surface). the tolerance zone.
NOTES: NOTE: The difference between tolerance zones and other zones
(1) Half-spaces are used to define tolerance zones. is that tolerance zones are inferred from the product specifica-
(2) The two half-spaces are complements of one another. tion.
1.5.21 Perfect Form 1.5.29 True Position

perfect form: a geometric shape that corresponds to the Per ASME Y14.5-2009
design geometry except for allowable variations in size,
location, or orientation. true position: the theoretically exact location of a feature of
size, as established by basic dimensions. [1.3.64]
1.5.22 Resolved Geometry NOTE: Irregular features of size type b, when specified with a
resolved geometry: the resolved geometry of a regular profile tolerance, do not have true position.
feature of size is the center point of a sphere, the axis
of a cylinder, or the center plane of a width. 1.5.30 True Profile
Per ASME Y14.5-2009
1.5.23 Size, Actual Mating
true profile: a profile defined by basic radii, basic angular
size, actual mating: a numerical value corresponding to the
dimensions, basic coordinate dimensions, basic size dimen-
actual mating envelope. This value may be a diameter for a
sions, undimensioned drawings, formulas, or mathematical
spherical or cylindrical envelope, or width for a parallel-
data, including design models. [8.2]
plane envelope, depending on the context. The radius of a
cylindrical or spherical actual mating envelope will be NOTE: The true profile is a subset of the design geometry.
designated rAM.
1.6 SUMMARY OF CONVENTIONAL
1.5.24 Size, Actual Minimum Material
DESIGNATIONS
size, actual minimum material: a numerical value corre-
sponding to the actual minimum material envelope. Throughout this Standard, conventional designations
This value may be a diameter for a spherical or cylindrical are used for various quantities. This subsection
envelope, or width for a parallel-plane envelope, summarizes these conventions.
depending on the context. The radius of a cylindrical CP = direction vector of a cutting plane
or spherical actual minimum material envelope will be D1 = direction vector of the primary datum plane
designated rAMM. D2 = direction vector of the secondary datum
plane
1.5.25 Spine = direction vector of the tertiary datum plane
D3
spine: a point, simple (non-self-intersecting) curve, or N = direction vector of the surface normal
simple surface. Spines are used in the definitions of = position vector
size and circularity. P
=
r( P , ) the distance of a point P to true position in
NOTE: Some applications of spines require them to be tangent-
datum reference frame Γ
continuous; see Nonmandatory Appendix E, section E-3. =
r( P ) the distance of a point P to true position, in
the case that the datum reference frame is
1.5.26 Spine, Local Size understood from context
spine, local size: a tangent-continuous spine from which rAM = actual mating size (radius)
cross sections are determined for actual local size deter- rAMM = actual minimum material size (radius)
mination. rTP = true position mating size (radius)
rTPMM = true position minimum material size
1.5.27 Surface of Support (radius)
surface of support: a theoretical surface that contacts an T = direction vector of a zone
actual feature at one or more points, such that the feature t = a value or size of a zone
is not on both sides of the surface. See Figure 1‐3. t0 = a specific tolerance disclosed in an engi-
neering product definition or part specifica-
1.5.28 Tolerance Zone tion
tolerance zone: a zone whose descriptive parameter t Γ = a candidate datum reference frame
corresponds to the specified tolerance value t0. Confor-
mance is determined by containment of the relevant
5
ASME Y14.5.1-2019
1.7 FORMAT (b) Conformance — mathematical definition of the

The format used in this Standard for explanation of conformance
geometric characteristics is as follows: (c) Actual value — mathematical definition of actual
value
NOTE: Definitions of conformance and actual values may contain
both narrative and mathematical descriptions; in all cases the
Table 1-1 Mathematical Symbology mathematics shall have precedence.
Quantity Symbol Presentation
Real numbers t, r, and Plain-face, italic, lowercase English
θ or lowercase Greek letters
Figure 1-1 Example: Distance From a Point to a True
Position Axis
Vectors T Boldface, italic English letters with an
arrow diacritical mark
Unit vectors N Boldface, italic English letters with a
caret diacritical mark
Functions (real or r(P ) A real number or vector symbol
vector-valued) (depending on the kind of value of
the function) followed by the
parameters of the function in
parentheses
Datum reference frames Γ Plain-face, uppercase Greek letter
(coordinate systems)
Sets S, F Plain-face, italic, uppercase English
letters
(a) Definition — narrative and mathematical descrip-

tion of the tolerance zone
Figure 1-2 Example: Cutting Surfaces to Evaluate Circular Runout
6
ASME Y14.5.1-2019
Figure 1-3 Examples of a Planar Surface of Support
7
ASME Y14.5.1-2019
Section 2
General Tolerancing and Related Principles
2.1 SURFACE POINTS Per ASME Y14.5-2009, an IFOSb is “a directly toleranced

feature or collection of features that may contain or be
This Standard describes the relationship between contained by an actual mating envelope other than a
points on the surface of a part and mathematically sphere, cylinder, or pair of parallel planes.” [1.3.32.2(b)]
defined tolerance zones. In order to effectively apply See Figure 2-6.
mathematics to these relationships, it is assumed that This subsection establishes definitions for the size
the location of the points on the surface are known limits, conformance, and actual size value for features
exactly and it is also known to which feature each of size. Subject to Rule #1 of ASME Y14.5-2009 [2.7.1],
point belongs. size limits of regular features of size also control form
NOTES: variation. The method by which Rule #1 is applied is
(1) The definition of what constitutes an “actual surface,” discussed in para. 2.3.2.1. For the definition of form
whether on an atomic scale or some other scale, is not controls, refer to Section 5.
addressed in this Standard. What constitutes an "actual
surface" may vary by industry and application.
2.3.1 Limits of Size
(2) The assignment of points to specific features is known as
partitioning. This process is not addressed in this Standard. Size is the value used to characterize a feature of size. A
size tolerance describes both upper and lower limits for
2.2 DIMENSION ORIGIN this parameter. A feature of size satisfies its size tolerance
if the feature is contained within a tolerance zone corre-
When a dimension origin symbol is used to specify the sponding to these limits. Paragraphs 2.3.1 through 2.3.4
distance between two features, the feature from which the are applicable for regular features of size and IFOSa; para.
dimension originates defines an origin plane for defining 2.3.5 applies to IFOSb.
the tolerance zone. In such cases, the origin plane shall be (a) Definition. A size tolerance zone is the volume
established using the same rules as are provided for between two half-space boundaries, to be described
primary datum features (although a datum is not below. The tolerance zone does not have a unique
created). See Section 4, Datum Reference Frames. form. Each half-space boundary is formed by sweeping
a ball of appropriate radius along an acceptable spine,
2.3 FEATURES OF SIZE as discussed below. The radii of the balls are determined
Features of size are classified as regular, irregular (type by the size limits: one ball radius is the least material
a), or irregular (type b). These are abbreviated FOS, IFOSa, condition limit (rLMC) and one is the maximum material
or IFOSb for ease of notation. Per ASME Y14.5-2009 condition limit (rMMC).
Regular feature of size: one cylindrical or spherical A zero-dimensional spine is a point, and applies to
surface, a circular element, and a set of two opposed parallel spherical features. A one-dimensional spine is a simple
elements or opposed parallel surfaces, each of which is asso- (non-self-intersecting) curve in space, and applies to cy-
ciated with a directly toleranced dimension. [1.3.32.2] lindrical features. A two-dimensional spine is a simple
Circular elements and opposed parallel elements can be (non-self-intersecting) surface, and applies to parallel-
considered [regular] “elements of size” when defined with plane features. These three types of spines can be
appropriate cutting surface (see para. 1.5.7). more rigorously defined, respectively, as connected
Per ASME Y14.5-2009, an IFOSa is “a directly toleranced regular (in the relative topology) subsets of d-manifolds,
feature or collection of features that may contain or be for d = 0, 1, and 2. A d-dimensional spine will be denoted as
contained by an actual mating envelope that is a sphere, Sd. Also, a (solid) ball of radius r will be denoted as Br.
cylinder, or pair of parallel planes.” [1.3.32.2(a)] See A solid G(Sd, Br) is obtained by sweeping the ball Br so
Figure 2-1. that its center lies in Sd. G(S0, Br) is a single ball bounded by
a sphere. If S1 is a line segment, G(S1, Br) is a solid bounded
NOTE: For an IFOSa, the actual mating envelope is explicitly
by a cylindrical surface and two spherical end caps. If S2 is a
defined by the Engineering Data. For example, supplemental
geometry defining a cylinder or parallel planes is needed to iden-
planar patch, G(S2, Br) is a solid bounded by two planar
tify the feature of size in Figure 2-1. patches and some canal surfaces. (Canal surfaces are
8
ASME Y14.5.1-2019
obtained by sweeping spheres, or balls, so that their 2.3.2.2 MMC Limit, Where Rule #1 of Y14.5 Does Not
centers lie on a curve in space.) Figure 2-2 shows a Apply
one-dimensional spine and its associated solid for a
(a) Definition. If Rule #1 does not apply, the MMC limit
ball of radius r.
of the size tolerance is a half-space formed by sweeping a
S1 and S2 need not be portions of lines or planes, respec-
ball having the diameter of the size limit along a spine of
tively. If necessary, S1 or S2 can be extended to infinity, or
appropriate dimension for the feature (zero-dimensional
closed upon itself, so that the resulting solid G is a half-
point for a sphere feature of size, one-dimensional curve
space. The spine, along with the balls, also defines the
for a cylindrical feature of size, or a two-dimensional
symmetric axis transformation of such solids.
surface for a parallel plane feature of size). This is referred
(b) Conformance. A regular feature of size or an IFOSa,
to as the MMC half-space.
F, conforms to the limits of size rLMC and rMMC if there exist
(b) Conformance. For an external features, the toler-
two spines, S ℓ corresponding to r LMC and S m corre-
ance limit is satisfied if the feature is contained within
sponding to rMMC, and two associated solids, Gℓ = G(Sℓ,
the MMC half-space for some spine. For internal features,
BrLMC) and Gm = G(Sm, BrMMC), that satisfy two conditions
the tolerance limit is satisfied if the feature is contained
described below.
within the complement of the MMC half-space for some
(1) H Hm
spine.
(2) F H m H (c) Actual Value. The actual mating size is the diameter
In the above conditions, "⊂" indicates set contain- of the smallest ball that can be swept along a spine so that
ment and " – " indicates set difference. the feature is contained within the resulting half-space
If F is an external feature, then let Hℓ = Gℓ and Hm = Gm. If (for external features) or largest ball that can be swept
F is an internal feature, then let Hℓ be the complement of Gℓ along a spine so that the feature is contained within
and Hm be the complement of Gm (see Figure 2-3). the complement of the resulting half-space (for internal
NOTE: Additional guidance on the spines is detailed in features).
Nonmandatory Appendix F.
(c) Actual Value. Two actual values are defined. The 2.3.2.3 LMC Size Limit
actual external (to the material) size of an external (a) Definition. The LMC size limit describes a half-space
(respectively, internal) feature is the smallest (respec- formed by sweeping a ball having the diameter of the LMC
tively, largest) size of the ball to which the feature size limit along a spine of appropriate dimension for the
conforms. The actual internal size is the largest (respec- feature (zero-dimensional point for a sphere feature of
tively, smallest) size of the ball to which the feature size, one-dimensional curve for a cylindrical feature of
conforms. The size may be expressed as a radius or size, or a two-dimensional surface for a parallel plane
diameter, as appropriate to the application. feature of size). The resulting volume is referred to as
the LMC half-space.
2.3.2 Variation of Size (b) Conformance. For an external feature, the tolerance
Regular features of size and IFOSa are naturally para- limit is satisfied if the feature is contained in the comple-
meterized by the size of the balls (spheres) used to ment of the LMC half-space for some spine. For an internal
construct the size tolerance zone. IFOSb must have an feature, the tolerance limit is satisfied if the feature is
explicit parameterization where each size value corre- contained within the LMC half-space for some spine.
sponds to a boundary. (c) Actual Value. The actual minimum material size is
the largest ball that can be swept along a spine so that the
2.3.2.1 MMC Limit, Where Rule #1 of Y14.5 Applies feature is contained within the complement of the
(a) Definition. Where Rule #1 applies, the MMC limit of resulting half-space (for external features) or smallest
the size tolerance is a bound on the allowed size of the ball that can be swept along a spine so that the feature
perfect-form actual mating envelope of the feature of is contained within the resulting half-space (for internal
size. This can be modeled as a half-space formed by a cyl- features).
inder (sphere/width) with size equal to the MMC size
limit. The interior and boundary of this cylinder 2.3.3 Actual Local Size Limits
(sphere/width) is referred to as the MMC half-space. In addition to the tolerance zone containment require-
(b) Conformance. If the feature of size is contained ments for satisfying a size tolerance, requirements on the
within MMC half-spaces (for external features) or actual local size of a feature of size may be induced from
contained within the complement of the MMC half- the specification. ASME Y14.5 is not overly prescriptive in
space (for internal features), the tolerance limit is satis- its definition of actual local size, stating merely that actual
fied. local size is “any individual distance at any cross section of a
(c) Actual Value. The actual mating size is the size of the feature of size.” In some sections of the ASME Y14.5 stan-
unrelated actual mating envelope for the feature of size. dard, it appears this is a point-to-point distance, while in
9
ASME Y14.5.1-2019
other sections there is reference to the actual local size of (b) Conformance. If the distance between the two
circular elements. opposed points satisfies the size limits, this actual local
Both of these notions rely on cross sections, which are size conforms to the tolerance.
determined relative to a local size spine; this spine is one‐ (c) Actual Value. The actual value for an individual
dimensional (cylindrical features), two‐dimensional actual local size is the Euclidean distance between the
(parallel plane features), or a point (spherical features). two opposed points where the evaluation line intersects
that actual feature.
2.3.3.1 Establishing the Local Size Spine. A local size
spine must be tangent-continuous in order for cross NOTE: Conformance to the actual local size requirements is
sections to be taken along the entire feature. Per ASME neither a necessary nor a sufficient condition for conformance
to the size requirement based on swept spheres.
Y14.5-2009
The derived median line/plane is made up of the center
2.3.3.3 Evaluation of Actual Local Size (Circular
points of all cross sections/line segments of the feature
Elements). If the actual local size is to be evaluated by
normal (perpendicular) to the axis/center plane of the unre-
the circular elements method, two actual local sizes
lated actual mating envelope. [1.3.30, 1.3.31]
exist for every point on the local size spine in the
As surface flaws may cause these center points to form a
cross section perpendicular to the local size spine at
discontinuous line or plane, evaluation of actual local size
that point (cylindrical features of size), or every plane
requires constructing a tangent-continuous line or plane.
passing through the center point (spherical features of
Examples of non-unique center points or a tangent-
size).
discontinuous center line are shown in Figure 2-4.
– Where Rule #1 does not apply, both “maximum mate-
Because of these properties of the derived median line
rial” and “least material” local sizes are of interest. If Rule
and plane, a local size spine is used that approximates
#1 applies, only the “least material” local size is of interest.
the derived median line or plane, but is tangent-contin-
– Cross sections are formed by using a cutting plane for
uous.
cylindrical and spherical features of size.
NOTE: The local size spine can be thought of as a “well-behaved” – Circular element actual local size is not defined for
derived median line or plane, possibly obtained through some parallel planes features of size.
smoothing of the derived median line or plane constructed using (a) Definition. The actual least material local size in a
the ASME Y14.5 definition.
particular cross section is the diameter of the minimum
When straightness of the derived median line is eval- circumscribed circle (internal features) or the maximum
uated (see para. 5.4.1.1), the nominally circular cross inscribed circle (external features) corresponding to the
sections obtained from cylindrical features of size are actual feature in that cross section.
evaluated to find the centers. In the absence of other spec- (b) Conformance. If the diameter of the defined circle(s)
ification, these centers will be the center of a “mating satisfies the limits of size, this actual local size conforms to
circle” in that cross section, i.e., an inscribed circle for the tolerance.
an internal feature, and a circumscribed circle for an (c) Actual Value. The actual value for an actual local size
external feature. (s) is the diameter of the defined circle(s).
2.3.3.2 Evaluation of Actual Local Size (Opposed NOTE: Conformance to the actual local size requirements is only
Points). If actual local size is to be evaluated by the intended to be an estimate of conformance to a size specification
opposed points method, an actual local size exists: by the swept-sphere interpretation.
– for every line perpendicular to the two‐dimensional
local size spine at the point this line intersects the spine 2.3.4 Continuous Features of Size
(parallel plane features of size);
Where multiple features of size are indicated to be a
– for every line passing through the one-dimensional
continuous feature ( CF ), the MMC (perfect form)
local size spine in any cross section perpendicular to
the spine (cylindrical features of size); or size definition remains the same for the portions(s) of
– for every line passing through the center point (spher- the surface which form the continuous feature.
ical features of size). (a) For the opposed points definition of the actual local
These lines are referred to as evaluation lines. See size, surface control only exists for those portions of the
Figure 2-5 for a graphical representation of actual local continuous feature that have opposed points as defined by
sizes on a cylindrical feature. two intersections of an evaluation line with the feature
(a) Definition. The actual local size for a particular surface. For portions of the continuous feature with no
evaluation line is the Euclidean distance between the opposed elements, no actual local size exists.
opposed points where the evaluation line intersects (b) For the circular element definition of the actual
the actual feature. If there are more than two intersecting local size, surface control only exists where the inscribed
points, no size exists for this evaluation line. or circumscribed circle is sufficiently constrained by the
continuous feature in the cross section of evaluation.
10
ASME Y14.5.1-2019
2.3.5 Limits for Irregular Feature of Size (Type b) 2.3.6 Variation of Size Under Rule #1 for Irregular
As shown in Figure 2-6, IFOSb may be specified using Features of Size
profile tolerance. When the limits of “size” are defined by a Rule #1 applies only to regular features of size. No stan-
profile tolerance, there is no unique size actual value for dard syntax exists for requiring Rule #1 for irregular
IFOSb. Nonmandatory Appendix D contains additional features of size.
discussion.
Figure 2-1 Irregular Features of Size (Type a), Collection of Features
Figure 2-2 Symbols Used in the Definition of Size Figure 2-3 Conformance to Limits of Size, Internal
Feature of Size
11
ASME Y14.5.1-2019
Figure 2-4 Examples of Surface Attributes Leading to

Variations in the Derived Median Line
Figure 2-5 The Cutting Plane at a Point on a Local Size Spine, and Some of the Evaluation Lines in That Cutting Plane
Figure 2-6 Profile of a Surface Tolerance Describing an External IFOSb
12
ASME Y14.5.1-2019
Section 3
Symbology
There are no concepts in Section 3 of ASME Y14.5-2009

that require mathematical definition.
13
ASME Y14.5.1-2019
Section 4
Datum Reference Frames
4.1 GENERAL 4.4.1 Datum Feature Simulator Requirements

This Section contains mathematical methods for estab- Per ASME Y14.5-2009
lishing datums and datum reference frames from datum Datum feature simulators shall have the following re-
features on a part. Datum reference frames are coordinate quirements:
systems used to locate and orient part features. (a) perfect form
(b) basic orientation relative to one another for all the
4.2 CONCEPTS datum references in a feature control frame
(c) basic location relative to other datum feature simu-
In this Standard, sometimes it is explained that the part lators for all the datum references in a feature control
is assumed to be fixed in space and the datum reference frame, unless a translation modifier or movable datum
frame is established in relation to the part. This approach target symbol is specified
can be contrasted to the explanation in ASME Y14.5-2009, (d) movable location when the translation modifier or
where the datum reference frame is assumed to be fixed in the movable datum target symbol is specified
space and the part is moved into the datum reference (e) fixed at the designated size, when MMB or LMB is
frame. The two approaches are different, but the end specified
results are identical. (f) adjustable in size, when the datum feature applies at
RMB [4.5.2]
4.3 DEGREES OF FREEDOM Referencing datum features in an order of precedence in
Per ASME Y14.5-2009 a feature control frame establishes a series of datum
All parts have six degrees of freedom, three translational feature simulators. The size, form, orientation, location,
and three rotational, which may be constrained by datum growth, or movement of these simulators is based on
feature references in a feature control frame. [4.2] the datum feature modifier specified. This Standard
The three translational degrees of freedom are termed x expands the description of the datum feature simulator
(translation along X axis), y (translation along Y axis), and z requirements as follows:
(translation along Z axis). The three rotational degrees of (a) They shall be the perfect inverse shape (opposite
freedom are termed u (rotation about the X axis), v (rota- material counterpart) of the datum feature unless other-
tion about the Y axis), and w (rotation about the Z axis). wise specified.
(b) They shall have perfect basic orientation relative to
one another as referenced in a feature control frame.
4.4 DATUM FEATURE SIMULATORS
(c) They shall have perfect basic location relative to one
(THEORETICAL)
another as referenced in a feature control frame unless a
Datum features are identified on a drawing and refer- translation modifier or movable datum target symbol is
enced in an order of precedence in the feature control specified.
frame. Since datum features have manufactured varia- (d) They shall have movable location when the transla-
tions, perfect datums cannot be directly derived from tion modifier or the movable datum target symbol is speci-
these imperfect features. Theoretical datum feature simu- fied.
lators are perfect, and unless otherwise specified (see (e) They shall be fixed at the designated MMB size,
para. 4.7.11), contact the imperfect datum feature at when the datum feature is modified at MMB or fixed
its extremities or high points. These theoretical datum at the designated LMB size, when the datum feature is
feature simulators derive datums from the datum modified at LMB.
features, constrain the part’s degrees of freedom, and (f) They shall expand, contract, or otherwise progress,
create a datum reference frame. In ASME Y14.5, there in their order of precedence, to make maximum contact
are two types of datum feature simulators: physical with the datum feature when the datum feature is speci-
and theoretical. This Standard will only reference the fied at RMB.
theoretical datum feature simulators.
14
ASME Y14.5.1-2019
4.5 CONSTRAINING DEGREES OF FREEDOM datum feature constrains three or more of the six
degrees of freedom. A lower precedence datum feature
Per ASME Y14.5-2009 shall not constrain any degrees of freedom already estab-
Where datum features are referenced in a feature control lished by any higher precedence datum feature. Therefore,
frame, the part is constrained in rotation and translation the secondary datum feature, if specified, only constrains
relative to the applicable datum feature simulators in the additional degrees of freedom that were not previously
specified order of precedence with applicable modifiers that constrained by the primary datum feature. The tertiary
establish the datum reference frame. [4.4] datum feature, if specified, only constrains the remaining
This establishes the relationship between the part and degree of freedom that was not previously constrained by
the datum reference frame. the primary and secondary.
4.5.1 Primary Datum Feature Types and NOTE: It is possible lower precedence datum features are not
needed because the higher precedence datum feature(s) have
Constrained Degrees of Freedom
constrained all six degrees of freedom.
The following primary datums are derived from the
associated datum feature simulator: 4.5.3 Partially Constrained Datum Reference
(a) A planar datum feature (nominally flat) establishes Frame
a datum feature simulator that creates a datum plane and
constrains three degrees of freedom (one translation and A fully constrained datum reference frame is
two rotations). Refer to Cases 3.1 through 3.18 in constrained in all three translational directions (x, y,
Table 4-4. and z) and all three rotations (u, v, and w). Depending
(b) A width as a datum feature (two opposed parallel on how a designated feature is toleranced from a
surfaces) establishes a datum feature simulator that datum reference frame, a partially constrained datum
creates a datum center plane and constrains three reference frame may be sufficient to define the tolerance
degrees of freedom (one translation and two rotations). zone for the designated feature. For example, if the datum
Refer to Cases 3.1 through 3.18 in Table 4-4. reference frame is defined by only a plane (Case 3.1 in
(c) A spherical datum feature establishes a datum Table 4-4), then it is sufficiently constrained to define
feature simulator that creates a datum center point the parallelism tolerance zone for the designated
and constrains three translational degrees of freedom. feature with respect to the datum plane.
Refer to Cases 1.1 through 1.14 in Table 4-2.
(d) A cylindrical datum feature establishes a datum 4.5.4 Datum Reference Frames for Composite
feature simulator that creates a datum axis and constrains Tolerances
four degrees of freedom (two translations and two rota- In composite tolerancing, the feature-relating toler-
tions). Refer to Cases 2.1 through 2.20 in Table 4-3. ances (lower segments of the feature control frame)
(e) A conical datum feature establishes a datum feature control only the rotational degrees of freedom for the
simulator that creates a datum axis and a datum point and pattern. The datum reference frame for such a tolerance
constrains five degrees of freedom (three translations and releases the translational degrees of freedom normally
two rotations). Refer to Cases 4.1 through 4.4 in Table 4-5. constrained by the procedures given above.
(f) A datum feature of linear extruded shape estab-
lishes a datum feature simulator that creates a datum 4.5.5 Customized Datum Reference Frames
plane and a datum axis and constrains five degrees of
freedom (two translations and three rotations). Refer ASME Y14.5-2009, subsections 4.22 and 4.23 address
to Cases 5.1 through 5.4 in Table 4-6. customized datum reference frames. Customized datum
(g) A complex datum feature establishes a datum reference frames are used to override the degrees of
feature simulator that creates a datum plane, datum freedom constrained by datum features in an order of
point, and a datum axis and constrains six degrees of precedence. The degrees of freedom to be constrained
freedom (three translations and three rotations). Refer by each datum feature shall be explicitly stated by
to Case 6.1 in Table 4-7. placing the designated degree of freedom (x, y, z, u, v,
w) brackets behind each datum feature reference. This
4.5.2 Datum Feature Order of Precedence in a Standard does not show the possible combinations of
Datum Reference Frame customized datum reference frames.
Datum features are specified in an order of precedence 4.6 TABULATION OF DATUM REFERENCE FRAMES
to relate the part to the datum reference frame. This order
of precedence (e.g., primary, secondary, and tertiary) This subsection presents tables of datum reference
specified in the feature control frame determines frames organized by the geometry of the primary datum.
which degrees of freedom each datum feature shall Table 4-1 presents the symbols used in the rest of the
constrain. Unless otherwise specified, the primary Section.
15
ASME Y14.5.1-2019
Tables 4-2 through 4-7 show all valid combinations of 4.7.1 Establishing a Candidate Datum Set
noncustomized datum reference frames and, for each case,
the free degrees of freedom (DOF), the invariants, and the Paragraphs 4.7.2 through 4.7.6 establish the process for
conditions under which the datum reference frame is associating candidate datum sets with individual datum
valid. An invariant in a datum reference frame is a quantity features. While a candidate datum set is associated with an
(distance or angle) that does not change under uncon- individual datum feature, datum precedence is used in the
strained degrees of freedom allowed by that datum refer- definition. The candidate datums from these sets are used
ence frame. (See Table 4-8.) to construct the candidate datum reference frames as
Coordinate system labels are somewhat arbitrary described in para. 4.7.8.
unless the coordinate system is labeled on the
4.7.2 Types of Datum Features
drawing. However, generally, the following conventions
apply. If the primary datum is a point, it establishes The method of establishing datums depends on the type
the origin. If the primary datum is an axis, it establishes of datum feature (e.g., flat surface, cylinder, width, sphere),
the z coordinate axis. If it is a plane, it establishes the x-y the datum feature precedence (primary, secondary, or
coordinate plane (and hence the direction of the z axis). tertiary), and sometimes the material condition of the
Secondary and tertiary datums establish additional datum reference (RMB, MMB, or LMB).
elements of the coordinate system. The following classification of datum features is used in
The example in Figure 4-1 is a specific row from Table paras. 4.7.3 through 4.7.6:
4-3. (a) planar datum features
Figure 4-1 shows Case 2.8: a datum reference frame (b) datum features of size that are cylinders, widths,
consisting of a primary datum axis, a secondary datum and spheres
axis, and no tertiary datum. The validity conditions indi- (1) referenced at RMB
cate that this case applies only if the two axes (A and B) are (2) referenced at MMB
not equal and are parallel. The only free degree of freedom (3) referenced at LMB
is translation along the z axis. As a result, the invariants This Standard does not specify how to establish candi-
include x and y coordinates, and all angle relationships date datum sets for screw threads, gears, or splines.
between features and the datum reference frame.
4.7.3 Planar Datum Features
4.7 CANDIDATE DATUM SETS AND CANDIDATE Because of the imperfections in the datum feature’s
REFERENCE FRAME SETS form, the candidate datum set may consist of more
Sometimes multiple valid datum reference frames may than one datum. This is equivalent to “rocking” the
be established from a set of datum features. This may datum feature on a perfect plane. The definition below
happen for multiple reasons: limits the amount that the datum feature can “rock” in
(a) A datum feature is referenced at MMB or LMB and a manner that is roughly proportional to the form varia-
the datum feature is allowed displacement due to its tion of the datum feature.
departure from MMB or LMB. (a) Primary Planar Datum Features. The candidate
(b) A datum reference frame does not constrain all six datum set for a nominally flat primary datum feature
degrees of freedom (see para. 4.5.3). is defined by the following procedure:
(c) The datum feature simulator does not make unique (1) Consider a plane P which is an external surface of
contact with the datum feature (if the datum feature support for the datum feature. Let C be the set of contact
“rocks”). Some examples of this are a nominally specified points of the datum feature and P.
flat surface manufactured convex, a cylinder specified at (2) Consider an arbitrary line L in P. Orthogonally
RMB produced tapered or barreled, or a width specified at project each point on the boundary of the datum
RMB produced tapered or diamond shaped. feature onto L, giving the line segment L′. Consider
The set of all possible candidate datums that can be regions of L′ that are within some fraction x of the
established from a datum feature is called a candidate endpoints of L′. That is, if the length of L′ is n, consider
datum set. Since a datum feature may generate more regions of L′ within a distance xn of the endpoints of L′.
than one datum, multiple datum reference frames may Unless otherwise specified on the drawing, the value of x
exist for a single feature control frame. The set of all shall be 1/3. If all of the orthogonal projections of the
possible candidate datum reference frames established points in C are within either single region, then plane
from one or more datum features is called a candidate P is rejected as a valid datum plane.
datum reference frame set. (3) Do this for all lines in P.
NOTE: Parallel lines will yield identical results.
If no line rejects P, then P is a candidate datum for the
datum feature.
16
ASME Y14.5.1-2019
The procedure is illustrated in Figure 4-2, which shows or tertiary datum features, the simulators are constrained
one line direction L. The line segment L′ is bounded by the to be basically oriented and, if applicable, basically located
projection of the datum feature onto L. The particular line to the higher precedence datums.
direction illustrated in the figure does not reject P as a (b) Width (Both Internal and External). The candidate
valid datum plane since the projections of the contact datum set for a width is the set of all center planes of all
points onto L are not all in region 1 or all in region 2 datum feature simulators of the datum feature. For
of L′. secondary or tertiary datum features, the simulators
NOTE: Only the direction of L in the plane P is important. P is a are constrained to be basically oriented and, if applicable,
candidate datum for the datum feature if it is not rejected by any basically located to the higher precedence datums.
line direction in P. Figure 4-3 shows an example of a tertiary datum feature
simulator specified at RMB oriented and located to higher
(b) Secondary Planar Datum Features Referenced at
precedence datums. Datum center plane C in the figure
RMB. The candidate datum set for a secondary planar
must be established from the datum feature under the
datum feature is determined by one of the following:
constraint that the simulator is oriented to datum
(1) If the primary datum is a point, use the procedure
plane A and oriented and located to datum axis B.
for a primary planar datum feature to establish the
(c) Sphere (Both Internal and External). The candidate
secondary datum.
datum set for a sphere is the set of center points of all
(2) If the primary datum is an axis nominally perpen-
datum feature simulators of the datum feature.
dicular to the secondary datum, then, for each candidate
datum in the candidate datum set for the primary datum
4.7.5 Datum Features of Size Referenced at MMB
feature, the secondary datum is the unique plane which is
perpendicular to that primary datum and which forms a (a) Cylinder [External] {Internal}. The candidate datum
surface of support for the secondary datum feature. set for a cylinder is the set of axes of all datum feature
(3) If neither (1) nor (2) applies, use the procedure simulators of MMB size that [enclose] {are enclosed
for a primary planar datum feature modified in the within} the datum feature. For secondary or tertiary
following ways: Given a primary datum from the datum features, the simulators are constrained to be basi-
primary candidate datum set, each plane P being consid- cally oriented and, if applicable, basically located to the
ered as a secondary datum must be basically oriented to higher precedence datums.
the primary datum. Also, each line L in P being considered (b) Width [External] {Internal}. The candidate datum
must be perpendicular to the direction vector of the set for a width is the set of center planes of all datum
primary datum. (Only one line in P must be considered.) feature simulators of MMB size that [enclose] {are
NOTE: If the secondary datum feature is referenced at MMB, enclosed within} the datum feature. For secondary or
LMB, or BASIC, the secondary datum feature simulator is tertiary datum features, the simulators are constrained
located at the specified boundary or distance. to be basically oriented and, if applicable, basically
located to the higher precedence datums.
(c) Tertiary Planar Datum Features Referenced at RMB.
(c) Sphere [External] {Internal}. The candidate datum
If the first two datums leave a rotational degree of
set for a sphere is the set of center points of all datum
freedom, then the candidate datum set is formed by
feature simulators of MMB size that [enclose] {are
the procedure for a primary planar datum feature modi-
enclosed within} the datum feature. For secondary or
fied such that each plane P being considered must be basi-
tertiary datum features, the simulators are constrained
cally oriented relative to the datums of higher precedence,
to be basically located to the higher precedence datums.
and one line L is to be considered, which must be perpen-
dicular to the axis established by the higher precedence 4.7.6 Datum Features of Size Referenced at LMB
datums. If the first two datums do not leave a rotational
degree of freedom, the candidate datum set consists of the (a) Cylinder [External] {Internal}. The candidate datum
plane which is basically oriented relative to the datums of set for a cylinder is the set of axes of all datum feature
higher precedence and which forms a surface of support simulators of LMB size that [are enclosed within]
for the datum feature. {enclose} the datum feature. For secondary or tertiary
NOTE: If the tertiary datum feature is referenced at MMB, LMB, or datum features, the simulators are constrained to be basi-
BASIC, the tertiary datum feature simulator is located at the cally oriented and, if applicable, basically located to the
specified boundary or distance. higher precedence datums.
(b) Width [External] {Internal}. The candidate datum
4.7.4 Datum Features of Size Referenced at RMB set for a width is the set of center planes of all datum
feature simulators of LMB size that [are enclosed
(a) Cylinder (Both Internal and External). The candidate within] {enclose} the datum feature. For secondary or
datum set for a cylinder is the set of axes of all datum tertiary datum features, the simulators are constrained
feature simulators of the datum feature. For secondary
17
ASME Y14.5.1-2019
to be basically oriented and, if applicable, basically located tion and profile tolerances that are located by basic dimen-
to the higher precedence datums. sions, related to common datum features referenced in the
(c) Sphere [External] {Internal}. The candidate datum same order of precedence at the same boundary conditions.
set for a sphere is the set of center points of all datum [4.19]
feature simulators of LMB size that [are enclosed All features with simultaneous requirements must use
within] {enclose} the datum feature. For secondary or the same candidate datum reference frame in the candi-
tertiary datum features, the simulators are constrained date datum reference frame set.
to be basically located to the higher precedence datums.
4.7.11 Alternate Stabilization Procedures
4.7.7 Translation Modifier
In accordance with ASME Y14.5-2009, if irregularities
Per ASME Y14.5-2009 on a datum feature are such that the part is unstable when
Where it is necessary to indicate that the basic location of brought into contact with the corresponding datum
the datum feature simulator is unlocked and the datum feature simulator, the default stabilization procedure is
feature simulator is able to translate to fully engage the per the candidate datum set as outlined in this Standard.
feature, the translation modifier is added to the feature ASME Y14.5-2009 does allow for different stabilization
control frame following the datum feature reference and procedures to be specified. When a single solution that
any other applicable modifiers. [4.11.10] minimizes the separation between the feature and the
The datum feature simulator is constrained to be basi- simulator is specified the default procedure is a
cally oriented but not basically located to the higher prece- constrained L2 for datum features of size and the
dence datums. See Figure 4-4. constrained L2 applied to the external envelope for
planar datum features.
4.7.8 Establishing a Candidate Datum Reference The background and details of this procedure can be
Frame Set found in Nonmandatory Appendix B.
The construction of a particular candidate datum refer-
ence frame shall proceed as follows:
(a) A primary datum is selected from the candidate
datum set associated with the primary datum feature.
(b) If a secondary datum is called out, the choice of Table 4-1 Symbols for Datum Reference Frame Tables
primary datum establishes, by the rules of the former
subsections, a candidate datum set for the secondary Symbol Description
datum feature. A secondary datum is chosen from this set. A Primary datum
(c) Similarly, if a tertiary datum is called out, the choice B Secondary datum
of primary and secondary datums establishes a candidate C Tertiary datum
datum set for the tertiary datum feature. A tertiary datum PT Point
is chosen from this last set. AX Axis
All the candidate datum reference frames established in PL Plane
this manner constitute the candidate datum reference {LI …} Line through …
frame set. {LI … : … } Line through … such that … is true
≠ Not coincidental with
4.7.9 Conformance and Actual Value
⊂ Contained within
For tolerances that reference a datum reference frame, ⊄ Not contained within
if the feature does not violate the constraints defined by || Parallel with
the tolerance for at least one candidate datum reference
⊥ Perpendicular to
frame in the candidate datum reference frame set, then the
∧ Logical AND
feature is in conformance to the tolerance. There is a
∨ Logical OR (one or the other, or both)
candidate actual value associated with each candidate
datum reference frame in the candidate datum reference ¬ Logical NOT
frame set. The actual value associated with the tolerance is ∩ Intersection
the minimum candidate actual value. x, y, z Position in a Cartesian coordinate system
u, v, w Rotation about x, y, z axis, respectively
4.7.10 Simultaneous Requirements γz Angle relative to datum axis z
r
Per ASME Y14.5-2009 Spherical radius: x 2 + y 2 + z2
A simultaneous requirement is where two or more ρz Cylindrical radius: x2 + y2
geometric tolerances apply as a single pattern or part
requirement. A simultaneous requirement applies to posi- — No entry (e.g., not applicable, none)
18
ASME Y14.5.1-2019
Table 4-2 Point as Primary Datum (Spherical Datum Feature)

Datums
Case A B C Free DOF Invariants Validity Conditions
1.1 PT — — u, v, w r —
1.2 PT PT — w ρz, z, γz A≠B
1.3 PT PT PT — All (A ≠ B) ∧ (C ⊄ {LI AB})
1.4 PT PT AX — All (A ≠ B) ∧ (C ≠ {LI AB})
1.5 PT PT PL — All (A ≠ B) ∧ ¬(C ⊥ {LI AB})
1.6 PT AX — — All A⊄B
1.7 PT AX — w ρz, z, γz A⊂B
1.8 PT AX PT — All (A ⊂ B) ∧ (C ⊄ B)
1.9 PT AX AX — All (A ⊂ B) ∧ (B ≠ C)
1.10 PT AX PL — All (A ⊂ B) ∧ ¬ (B ⊥ C)
1.11 PT PL — w ρz, z, γz —
1.12 PT PL PT — All C ⊄ {LI A: LI ⊥ B}
1.13 PT PL AX — All C ≠ {LI A: LI ⊥ B}
1.14 PT PL PL — All ¬ (C || B)
Table 4-3 Axis as Primary Datum (Cylindical Datum Feature)

Datums
2.1 AX — — z, w ρz, γz —
2.2 AX PT — — All B⊄A
2.3 AX PT — w ρz, z, γz B⊂A
2.4 AX PT PT — All (B ⊂ A) ∧ (C ⊄ A)
2.5 AX PT AX — All (B ⊂ A) ∧ (A ≠ C)
2.6 AX PT PL — All (B ⊂ A) ∧ ¬ (A ⊥ C)
2.7 AX AX — — All (A ≠ B) ∧ ¬ (A || B)
2.8 AX AX — z x, y, u, v, w (A ≠ B) ∧ (A || B)
2.9 AX AX PT — All (A ≠ B) ∧ (A || B)
2.10 AX AX AX — All (A ≠ B) ∧ (A || B) ∧ ¬ (A || C)
2.11 AX AX PL — All (A ≠ B) ∧ (A || B) ∧ ¬ (A || C)
2.12 AX PL — — All ¬ ((A || B) ∨ (A ⊥ B)
2.13 AX PL — z x, y, u, v, w A || B (including A ⊂ B)
2.14 AX PL — w ρz, z, γz A⊥B
2.15 AX PL PT — All A || B
2.16 AX PL PT — All (A ⊥ B) ∧ (C ⊄ A)
2.17 AX PL AX — All (A || B) ∧ ¬ (A || C)
2.18 AX PL AX — All (A ⊥ B) ∧ (A ≠ C)
2.19 AX PL PL — All (A || B) ∧ ¬ (A || C)
2.20 AX PL PL — All (A ⊥ B) ∧ ¬(A ⊥ C)
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ASME Y14.5.1-2019
Table 4-4 Plane as Primary Datum (Planar or Width Datum Feature)

Datums
3.1 PL — — x, y, w z, γz —
3.2 PL PT — w ρz, z, γz —
3.3 PL PT PT — All C ⊄ {LI B: LI ⊥ A}
3.4 PL PT AX — All C ≠ {LI B: LI ⊥ A}
3.5 PL PT PL — All ¬ (A || C)
3.6 PL AX — — All ¬ ((A || B) ∨ (A ⊥ B)
3.7 PL AX — w ρz, z, γz A⊥B
3.8 PL AX — x y, z, u, v, w A || B
3.9 PL AX PT — All (A ⊥ B) ∧ (C ⊄ B)
3.10 PL AX PT — All A || B
3.11 PL AX AX — All (A ⊥ B) ∧ (B ≠ C)
3.12 PL AX AX — All (A || B) ∧ ¬ (B || C)
3.13 PL AX PL — All (A ⊥ B) ∧ ¬ (B ⊥ C)
3.14 PL AX PL — All (A || B) ∧ ¬ (B || C)
3.15 PL PL — x y, z, u, v, w ¬ (A || B)
3.16 PL PL PT — All ¬ (A || B)
3.17 PL PL AX — All ¬(A || B) ∧ ¬ (C || {LI (A ∩ B)})
3.18 PL PL PL — All ¬(A || B) ∧ ¬ (C || {LI (A ∩ B)})
Table 4-5 Coincident Axis and Point as Primary Datum Table 4-8 Generic Invariant Cases
(Conical Datum Feature) Index Invariant Cases Case Number(s)
Datums Free Validity 1 r 1.1
Case A B C DOF Invariants Conditions 2 ρz, γz 2.1
4.1 AX & PT — — w ρz, z, γz — 3 ρz, z, γz 1.2, 1.7, 1.11, 2.3, 2.14, 3.2, 3.7, 4.1
4.2 AX & PT PT — — All B⊄A 4 z, γz 3.1
4.3 AX & PT AX — — All A≠B 5 x, y, u, v, w or y, z, u, v, w 2.8, 2.13, 3.8, 3.15, 5.1
4.4 AX & PT PL — — All ¬ (A ⊥ B) 6 All All others
Table 4-6 Axis and Plane as Primary Datum (Linear

Extruded Shape Datum Feature)
Datums Free Validity
Case A B C DOF Invariants Conditions
5.1 AX & PL — — z x, y, u, v, w —
5.2 AX & PL PT — — All —
5.3 AX & PL AX — — All ¬ (A || B)
5.4 AX & PL PL — — All ¬ (A || B)
Table 4-7 Axis Point and Plane as Primary Datum

(Complex Datum Feature)
Datums Free Validity
Case A B C DOF Invariants Conditions
6.1 AX, PT & — — — All —
PL
20
ASME Y14.5.1-2019
Figure 4-1 Example From Table 4-3 — Axis as Primary Datum
Datums
2.8 AX AX — z x, y, u, v, w (A ≠ B) ∧ (A || B)
Figure 4-2 Example of Testing Whether a Plane Is a Valid Datum Plane
21
ASME Y14.5.1-2019
Figure 4-3 Tertiary Datum Feature Simulator Is Basically Located and Oriented
Figure 4-4 Example of Translation Modifier
22
ASME Y14.5.1-2019
Section 5
Tolerances of Form
5.1 GENERAL Form tolerances are applicable to single (individual)

features or elements of single features; therefore, form toler-
This Section establishes the principles and methods for ances are not related to datums. The following subpara-
the mathematical evaluation of ASME Y14.5-2009 dimen- graphs cover the particulars of the form tolerances:
sioning and tolerancing to control the form of features. straightness, flatness, circularity, and cylindricity. [5.4]
5.2 FORM CONTROL 5.4.1 Straightness

Per ASME Y14.5-2009 Per ASME Y14.5-2009
Form tolerances control straightness, flatness, circu- Straightness is a condition where an element of a surface,
larity, and cylindricity. When specifying a form tolerance, or derived median line, is a straight line. A straightness
consideration must be given to the control of form already tolerance specifies a tolerance zone within which the consid-
established through other tolerances such as size (Rule #1), ered element of a surface or derived median line must lie. A
orientation, runout, and profile controls. [5.2] straightness tolerance is applied in the view where the
elements to be controlled are represented by a straight
5.3 SPECIFYING FORM TOLERANCES line. [5.4.1]
Per ASME Y14.5-2009 5.4.1.1 Straightness at RFS of a Cylinder (Applied to a
Form tolerances critical to function and interchange- Derived Median Line)
ability are specified where the tolerances of size do not (a) Definition. A straightness tolerance at RFS of a cyl-
provide sufficient control. A tolerance of form may be speci- inder specifies that the derived median line must lie within
fied where no tolerance of size is given (e.g., in the control of some cylindrical zone whose diameter is the specified
flatness after assembly of the parts). A form tolerance speci-
tolerance. A straightness zone for a derived median
fies a zone within which the considered feature, its line
elements, its derived median line, or its derived median line is a cylindrical volume consisting of all points P satis-
plane must be contained. [5.3] fying the condition
While the shape of the tolerance zone is defined (a cyl- t
inder, a zone bounded by two concentric cylinders, a zone T× P ( A ) 2
bounded by two concentric circles, a zone bounded by two
parallel planes, or a zone bounded by two parallel lines), where
the extent of the tolerance zone (e.g., the length of the = any point on the center line of the straightness
A
cylinder) must also be considered. The following two zone
cases are to be considered: T = the direction vector of the straightness axis
(a) The extent of the tolerance zone is restricted to t = the diameter of the straightness zone
control a limited area or length of the surface shown
by a chain line drawn parallel to the surface profile dimen- (b) Conformance. A cylindrical feature conforms to a
sioned for length and location. straightness tolerance at RFS t 0 if all points of the
(b) In all other cases, the extent of the tolerance zone is derived median line lie within a straightness zone as
limited to the actual feature surface. For a derived median
defined above with t = t0. That is, there exist T and A
line or a derived median plane the extent is defined by
projecting the actual surface points onto the derived such that with t = t0, all points of the derived median
median line or derived median plane. line are within the straightness tolerance zone.
(c) Actual Value. The actual value of straightness at RFS
5.4 FORM TOLERANCES of a cylinder is the smallest straightness tolerance to
which the derived median line will conform.
Per ASME Y14.5-2009
23
ASME Y14.5.1-2019
5.4.1.2 Straightness at MMC/LMC of a Cylinder = a point on the surface, contained by the cutting
PS
(Applied to a Derived Median Line) plane
T = the direction vector of the center line of the
(a) Definition. A straightness tolerance at MMC/LMC of
straightness zone
a cylinder specifies that the actual surface must not violate
t = the size of the straightness zone (the separation
the virtual condition boundary. Per ASME Y14.5-2009
between the parallel lines)
The collective effect of size and form variation can
produce a virtual condition or outer or inner boundary
Figure 5-1 illustrates a straightness zone for surface line
equal to the MMC/LMC size plus the straightness tolerance.
elements of a cylindrical feature. Figure 5-2 illustrates a
[5.4.1.2]
straightness zone for surface line elements of a planar
No resolved geometry interpretation is provided for
feature.
MMC or LMC tolerances. The surface interpretation for
(b) Conformance. A surface line element conforms to
straightness at MMC/LMC of a cylinder is that the
the straightness tolerance t0 for a cutting plane if all
surface of an actual cylindrical feature must lie within
points of the surface line element lie within some straight-
(or without for a hole) the virtual condition boundary.
(b) Conformance. A cylindrical feature conforms to a ness zone as defined above with t = t0. That is, there exist T
straightness tolerance at MMC/LMC if all surface and A such that with t = t0, all points of the surface line
points lie within the virtual condition boundary (or
element are within the straightness tolerance zone.
without for a hole).
A surface conforms to the straightness tolerance t0 if it
(c) Actual Value. The actual value of straightness at
conforms simultaneously for all toleranced surface line
MMC/LMC of a cylinder is the smallest straightness toler-
elements corresponding to the actual mating envelope.
ance to which feature will conform.
(c) Actual Value. The actual value of straightness for a
5.4.1.3 Straightness of Surface Line Elements surface is the smallest straightness tolerance to which the
surface will conform.
(a) Definition. A straightness tolerance for the line
elements of a feature specifies that each line element 5.4.2 Flatness
must lie in a zone bounded by two parallel lines which
are separated by the specified tolerance and which are Per ASME Y14.5-2009
in the cutting plane defining the line element. Flatness is the condition of a surface or derived median
When a straightness tolerance is applied to the longi- plane having all elements in one plane. A flatness tolerance
tudinal elements of a cylindrical feature, the cutting plane specifies a tolerance zone defined by two parallel planes
contains the axis of the unrelated actual mating envelope within which the surface or derived median plane must
of the cylinder. For a planar surface, a straightness toler- lie. [5.4.2]
ance is applied in the view where the elements to be 5.4.2.1 Flatness of a Planar Surface
controlled are represented by a straight line.
A straightness zone for a surface line element is an area (a) Definition. A flatness tolerance of a plane specifies
that all points of the surface must lie in some zone bounded
between parallel lines consisting of all points P satisfying
by two parallel planes which are separated by the speci-
the condition fied tolerance.
t
T× P ( A ) 2
A flatness zone is a volume consisting of all points P
satisfying the condition
and t
(
T P A ) 2
(
CP P )
PS = 0
where
CP ( A PS ) = 0 = any point on the mid-plane of the flatness zone
A
CP T = 0 T = the direction vector of the parallel planes
defining the flatness zone
where t = the size of the flatness zone (the separation of the
= any point on the center line of the straightness parallel planes)
A
zone
CP = the normal to the cutting plane; perpendicular to (b) Conformance. A feature conforms to a flatness toler-
N , and perpendicular to the mating axis for cy- ance t0 if all points of the feature lie within some flatness
lindrical surfaces or parallel to the view surface zone as defined above, with t = t0. That is, there exist T and
normal for planar surfaces
N = the mating envelope surface normal
24
ASME Y14.5.1-2019
A such that with t = t0, all points of the feature are within 5.4.3 Circularity (Roundness)
the flatness tolerance zone. Per ASME Y14.5-2009
(c) Actual Value. The actual value of flatness for a planar Circularity is a condition of a surface where
surface is the smallest flatness tolerance to which the (a) for a feature other than a sphere, all points of the
surface will conform. surface intersected by any plane perpendicular to an
axis or spine (curved line) are equidistant from that axis
5.4.2.2 Flatness at RFS of a Width
or spine;
(a) Definition. A flatness tolerance at RFS of a width (b) for a sphere, all points of the surface intersected by
specifies that all points of the derived median plane any plane passing through a common center are equidistant
must lie in some zone bounded by two parallel planes from that center.
which are separated by the specified tolerance. A circularity tolerance specifies a tolerance zone bounded
A flatness zone is a volume consisting of all points P by two concentric circles within which each circular element
of the surface must lie, and applies independently at any
satisfying the condition plane described in (a) and (b) above. [5.4.3]
t (a) Definition. A circularity tolerance specifies that all
(
T P A ) 2 points of each circular element of the surface must lie in
some zone bounded by two concentric circles whose radii
where differ by the specified tolerance. Circular elements are
= any point on the mid-plane of the flatness zone
A obtained by taking cross sections perpendicular to
some spine. For a sphere, the spine is zero-dimensional
T = the direction vector of the parallel planes
defining the flatness zone (a point), and for a cylinder, cone, torus, or other swept-
t = the size of the flatness zone (the separation of the sphere shape, the spine is one-dimensional (a simple, non-
parallel planes) self-intersecting, tangent-continuous curve). The
concentric circles defining the circularity zone are
(b) Conformance. A width conforms to a flatness toler- centered on, and in a plane perpendicular to, the spine.
ance t0 at RFS if all points of the derived median plane lie A circularity zone at a given cross section is an annular
within some flatness zone as defined above, with t = t0. area consisting of all points P satisfying the following con-
That is, there exist T and A such that with t = t0, all points ditions:
of the derived median plane are within the flatness toler-
ance zone.
T P ( A )=0
(c) Actual Value. The actual value of flatness at RFS for a
and
width is the smallest flatness tolerance to which the
derived median plane will conform. t
P A r
2
5.4.2.3 Flatness at MMC/LMC of a Width Feature
where
(a) Definition. A flatness tolerance at MMC/LMC of a = any point on the spine
width specifies that actual surface must not violate the A
r = a radial distance (which may vary between
virtual condition boundary. The collective effect of size
circular elements) from the spine to the center
and form variation can produce a virtual condition
of the circularity zone (r > 0 for all circular
boundary equal to the MMC/LMC size plus/minus the flat-
elements)
ness tolerance.
No resolved geometry interpretation is provided for T = for a cylinder or cone, a unit vector that is tangent
MMC or LMC tolerances. The surface interpretation defi- to the spine at A . For a sphere, T is a unit vector
nition for flatness at MMC/LMC of a width is that the actual
that points radially in all directions from A
width feature must lie within (or without for a slot) the
virtual condition boundary. t = the size of the circularity zone
(b) Conformance. A width conforms to a flatness toler-
ance at MMC/LMC if all surface points of the feature lie Figure 5-3 illustrates a circularity zone for a circular
within the virtual/resultant condition boundary (or element of a cylindrical or conical feature.
without for a slot). (b) Conformance. A cylindrical or conical feature
(c) Actual Value. The actual value of flatness at MMC/ conforms to a circularity tolerance t0 if there exists a
LMC for a width is the smallest flatness tolerance to which one-dimensional spine such that at each point A of
the derived median plane will conform. the spine the circular element perpendicular to the
25
ASME Y14.5.1-2019
(a) Definition. A cylindricity tolerance specifies that all

tangent vector T at A conforms to the circularity toler- points of the surface must lie in some zone bounded by two
ance t0. That is, for each circular element there exist A and coaxial cylinders whose radii differ by the specified toler-
ance.
r such that with t = t0, all points of the circular element are
A cylindricity zone is a volume between two coaxial cyl-
within the circularity tolerance zone.
A spherical feature conforms to a circularity tolerance t0 inders consisting of all points P satisfying the condition
if there exists a point (a zero‐dimensional spine) such that t
each circular element in each cutting plane containing the T× P ( A ) r
2
point conforms to the circularity tolerance t0. That is, for
each circular element there exist T , r, and a common A where
= any point on the cylindricity axis
such that with t = t0, all points of the circular element are A
within the circularity tolerance zone. r = the radial distance from the cylindricity axis to
(c) Actual Value. The actual value of circularity for a the center of the zone
feature is the smallest circularity tolerance to which T = the direction vector of the cylindricity axis
the feature will conform. t = the size of the cylindricity zone
5.4.4 Cylindricity (b) Conformance. A feature conforms to a cylindricity

tolerance t0 if all points of the feature lie within some cylin-
Per ASME Y14.5-2009 dricity zone as defined above with t = t0. That is, there exist
Cylindricity is a condition of a surface of revolution in
which all points of the surface are equidistant from a T , A , and r such that with t = t0, all points of the feature are
common axis. A cylindricity tolerance specifies a tolerance within the cylindricity tolerance zone.
zone bounded by two concentric cylinders within which the (c) Actual Value. The actual value of cylindricity for a
surface must lie. In the case of cylindricity, unlike that of surface is the smallest cylindricity tolerance to which it
circularity, the tolerance applies simultaneously to both will conform.
circular and longitudinal elements of the surface (the
entire surface). [5.4.4]
NOTE: The cylindricity tolerance is a composite control of form
which includes circularity, straightness, and taper of a cylindrical
feature.
26
ASME Y14.5.1-2019
Figure 5-1 Evaluation of Straightness of a Cylindrical Surface
Figure 5-2 Evaluation of Straightness of a Planar Surface
Figure 5-3 Illustration of Circularity Tolerance Zone for a Cylindrical or Conical Feature
27
ASME Y14.5.1-2019
Section 6
Tolerances of Orientation
6.1 GENERAL (b) a tolerance zone defined by two parallel planes at the
specified basic angle from, parallel to, or perpendicular to
This Section establishes the principles and methods for one or more datum planes or a datum axis, within which the
mathematical evaluation of ASME Y14.5-2009 dimen- axis of the considered feature must lie.
sioning and tolerancing to control the orientation of (c) a cylindrical tolerance zone at the specified basic
features. angle from, parallel to, or perpendicular to one or more
datum planes or a datum axis, within which the axis of
6.2 ORIENTATION CONTROL the considered feature must lie.
Per ASME Y14.5-2009 (d) a tolerance zone defined by two parallel lines at the
An orientation tolerance controls parallel, perpendicular, specified basic angle from, parallel to, or perpendicular to a
and all other angular relationships. Note that an orienta- datum plane or axis, within which the line element of the
tion tolerance, when applied to a plane surface, controls surface must lie. [6.4.2]
flatness to the extent of the orientation tolerance. When While the shape of the tolerance zone is defined (a cyl-
the flatness control in the orientation tolerance is not suffi- inder, a zone bounded by two parallel planes, or a zone
cient, a separate flatness tolerance should be considered. An bounded by two parallel lines), the extent of the tolerance
orientation tolerance does not control the location of zone (e.g., the length of the cylinder) must also be consid-
features. When specifying an orientation tolerance, considered. There are two cases to be considered:
eration must be given to the control of orientation already (a) The extent of the tolerance zone is restricted to
established through other tolerances such as location, control a limited area or length of the surface shown
runout, and profile controls. [6.2] by a chain line drawn parallel to the surface profile dimen-
sioned for length and location.
6.3 SPECIFYING ORIENTATION TOLERANCES (b) In all other cases, the extent of the tolerance zone is
limited to the actual feature surface. For a feature axis,
Per ASME Y14.5-2009 tangent plane, or center plane the extent is defined by
When specifying an orientation tolerance, the considered projecting the actual surface points onto the axis,
feature shall be related to one or more datums. Orientation tangent plane, or center plane.
tolerances are constrained only in rotational degrees of
freedom relative to the referenced datums; they are not 6.4 ORIENTATION TOLERANCE
constrained in the translational degrees of freedom.
Thus, with orientation tolerances, even in those instances Per ASME Y14.5-2009
where datum features may constrain all degrees of freedom, Tolerance zones apply to the full extent of the feature,
the tolerance zone only orients to that datum reference unless otherwise indicated. Where it is a requirement to
frame. Sufficient datum features should be referenced to control only individual line elements of a surface, a quali-
constrain the required rotational degrees of the fying notation, such as EACH ELEMENT or EACH RADIAL
freedom. If the primary datum feature alone does not ELEMENT, is added to the drawing. This permits control of
constrain sufficient degrees of freedom, additional individual elements of the surface independently in relation
datum features may be specified. [6.4] to the datum and does not limit the total surface to an
An orientation tolerance specifies a zone within which the encompassing zone. Although orientation tolerances are
considered feature, its line elements, its axis, or its center only constrained in rotational degrees of freedom relative
plane must be contained. [6.4.1] to the referenced datums, the notation of EACH RADIAL
An orientation tolerance specifies one of the following: ELEMENT adds a requirement for the tolerance zone(s)
(a) a tolerance zone defined by two parallel planes at the to be constrained in location relative to the axis from
specified basic angle from, parallel to, or perpendicular to which the radial elements emanate. Tolerances for individ-
one or more datum planes or a datum axis, within which the ual elements may also be specified using a line profile toler-
surface or center plane of the considered feature must lie. ance. [6.4.3]
28
ASME Y14.5.1-2019
Where it is desired to control a tangent plane established – parallelism of a plane, center plane, or tangent plane to
by the contacting points of a surface, the tangent plane a datum plane
symbol is added in the feature control frame after the – perpendicularity of an axis to a datum axis with a
stated tolerance. Where a tangent plane symbol is specified planar orientation tolerance zone
with a geometric tolerance, the flatness of the tolerance – parallelism of an axis to a datum plane with a planar
feature is not controlled by the geometric tolerance. [6.5] orientation tolerance zone
Mathematically, the equations describing angularity, In these cases, the orientation of the planar orientation
parallelism, and perpendicularity are identical when zone is fully constrained by the primary datum. In the
generalized in terms of the angle(s) between the tolerance following two cases, if a secondary datum is not specified,
zone and the related datum(s). Accordingly, the generic then the orientation tolerance may not provide adequate
term orientation is used in place of angularity, parallelism, control:
and perpendicularity in the definitions. – angularity and parallelism of an axis to a datum axis
An orientation zone is bounded by a pair of parallel with a planar orientation tolerance zone
planes, a cylindrical surface, or a pair of parallel lines. – angularity and perpendicularity of an axis to a datum
Each of these cases is defined separately below. If the plane with a planar orientation tolerance zone
tolerance value is preceded by the diameter symbol, When controlling perpendicularity of an axis within a
then the tolerance zone is a cylindrical volume; if the nota- planar orientation tolerance zone with just a primary
tion EACH ELEMENT or EACH RADIAL ELEMENT appears datum, the planar orientation zone can rotate to
then the tolerance zone is an area between parallel lines; contain the axis with an actual value of zero.
in all other cases the tolerance zone is a volume between If a secondary datum is specified, the planar orientation
parallel planes by default. zone is further restricted to be oriented relative to the
direction vector, D2 , of the secondary datum by
6.4.1 Planar Orientation Tolerance Zone
l
o cos for a secondary datum axis
(a) Definition. An orientation tolerance which is not T D2 = m o
o
n sin for a secondary datum plane
preceded by the diameter symbol and which does not
include the notation EACH ELEMENT or EACH RADIAL
where T ʹ is the normalized projection of T onto a plane
ELEMENT specifies that the considered surface, center
plane, tangent plane, or axis must lie in a planar orienta- normal to D1, and α is the basic angle between the
tion zone bounded by two parallel planes separated by the secondary datum and T ʹ. T ʹ is given by
specified tolerance and basically oriented to the primary
datum and, if specified, to the secondary datum as well. T (T D1)D1
T =
A planar orientation zone is a volume consisting of all T (T D1)D1
points P satisfying the condition
t Figure 6-1 shows the relationship of the planar orienta-
(
T P A ) 2 tion zone direction vector to the primary and secondary
datums. Figure 6-2 illustrates the projection of T onto the
where
= any point on the mid-plane of the planar orienta- primary datum plane to form T ʹ.
A
tion zone (b) Conformance. A surface, center plane, tangent plane,
T = the direction vector of the planar orientation or axis S conforms to an orientation tolerance t0 if all
zone points of S lie within some planar orientation zone as
t = the size of the planar orientation zone (the sep-
defined above with t = t0. That is, there exist T and A
aration of the parallel planes)
such that with t = t 0 , all points of S are within the
The planar orientation zone is oriented such that, if D1 is planar orientation tolerance zone.
the direction vector of the primary datum, then NOTE: If the orientation tolerance refers to both a primary datum
l cos for a primary datum axis and a secondary datum, then T is fully determined. If the consid-
o
T D1 = m o sin for a primary datum plane
ered feature is a feature of size (cylinder or width), then the
o
n orientation tolerance may be applied on a MMC or LMC material
condition basis. This increases the width of the planar orienta-
where Θ is the basic angle between the primary datum and tion tolerance zone by the additional tolerance.
the direction vector of the planar orientation zone. (c) Actual Value. The actual value of orientation for S is
In the following four cases, a secondary datum has no the smallest orientation tolerance to which S will conform.
effect:
– perpendicularity of a plane, center plane, or tangent
plane to a datum axis
29
ASME Y14.5.1-2019
6.4.2 Cylindrical Orientation Tolerance Zone there exists T and A such that with t = t0, all points
(a) Definition. An orientation tolerance which is of S are within the cylindrical orientation tolerance zone.
preceded by the diameter symbol specifies that the toler- NOTE: If the orientation tolerance refers to both a primary datum
anced axis must lie in a cylindrical orientation zone and a secondary datum, then T is fully determined. If the orienta-
bounded by a cylinder with a diameter equal to the speci- tion tolerance is applied to a cylinder, then a MMC or LMC mate-
fied tolerance and whose axis is basically oriented to the rial condition may be applied. This increases the diameter of the
primary datum and, if specified, to the secondary datum as cylindrical orientation tolerance zone by the additional toler-
well. ance.
A cylindrical orientation zone is a volume consisting of (c) Actual Value. The actual value of orientation for S is
all points P satisfying the condition the smallest orientation tolerance to which S will conform.
t
T× P ( A ) 2
6.4.3 Linear Orientation Tolerance Zone
(a) Definition. An orientation tolerance which includes
where the notation EACH ELEMENT or EACH RADIAL ELEMENT
= any point on the axis of the cylindrical orientation
A specifies that each line element of the toleranced surface
zone must lie in a linear orientation zone bounded by two
T = the direction vector of the axis of the cylindrical parallel lines which are
orientation zone (1) in the cutting plane defining the line element
t = the diameter of the cylindrical orientation zone (2) separated by the specified tolerance
(3) basically oriented to the primary datum and, if
The axis of the cylindrical orientation zone is oriented specified, to the secondary datum as well.
such that, if D1 is the direction vector of the primary For a surface point Ps , a linear orientation zone is an
datum, then
area consisting of all points P in a cutting plane of direc-
l
o cos for a primary datum axis
T D1 = m tion vector Cp that contains Ps . The points P satisfy the
o sin for a primary datum plane
o
n following conditions:
where Θ is the basic angle between the primary datum and
the direction vector of the axis of the cylindrical orienta-
(
Cp P )
Ps = 0
tion zone. and

In the following cases, a secondary datum has no effect:
t
– parallelism of an axis to a datum axis, and
– perpendicularity of an axis to a datum plane.
(
T× P × A ) 2
In these cases, the orientation of the cylindrical orienta-
where
tion zone is fully constrained by the primary datum. If a = any point on the center line of the linear orienta-
secondary datum is specified, the cylindrical orientation A
tion zone
zone is further restricted to be oriented relative to the =
Cp the normal to the cutting plane and basically
direction vector, D2, of the secondary datum by oriented to the datum reference frame
l
o cos for a secondary datum axis = a point on S
T D2 = m Ps
o
o
n sin for a secondary datum plane T = the direction vector of the center line of the linear
orientation zone
where T ʹ is the normalized projection of T onto a plane t = the size of the linear orientation zone (the sep-
aration between the parallel lines)
normal to D1 , and α is the basic angle between the
secondary datum and T ʹ. T ʹ is given by
The position vector A , which locates the center line of
T (T D1)D1 the linear orientation zone, also locates the cutting plane
T =
T (T D1)D1 through the following constraint:
(
Cp Ps A )=0
Figure 6-3 illustrates a cylindrical orientation zone.
(b) Conformance. An axis S conforms to an orientation In some cases, if a secondary datum is not specified then
tolerance t0 if all points of S lie within some cylindrical the orientation tolerance may not provide adequate
orientation zone as defined above with t = t0. That is, control: angularity of a surface line element to a datum
axis or a datum plane. If a secondary datum axis is
30
ASME Y14.5.1-2019
specified, and the toleranced feature in its nominal condi- where Θ is the basic angle between the primary datum and
tion is rotationally symmetric about that datum axis, then the direction vector of the linear orientation zone.
the cutting planes are further restricted to contain the NOTE: If a secondary datum is not specified, at least one of T and
datum axis as follows:
D1 will not be defined, and so the actual value will not be defined.
(
Cp Ps B )=0 Figure 6-4 illustrates a linear orientation zone bounded
by parallel lines on a cutting plane for the controlled
where B is a position vector that locates the datum axis. surface.
(b) Conformance. A surface, center plane, or tangent
Otherwise, the cutting planes are required to be parallel to
plane S conforms to an orientation tolerance t0 for a
one another.
The direction vector of the center line of the linear cutting plane Cp if all points of the intersection of S
orientation zone, T , is constrained to lie in the cutting with Cp lie within some linear orientation zone as
plane by defined above with t = t0. That is, there exist T and A
Cp T = 0 such that with t = t 0 , all points of S are within the
linear orientation tolerance zone.
The center line of the linear orientation zone is oriented A surface S conforms to the orientation tolerance t0 if it
such that, if D1 is the direction vector of the primary datum, conforms simultaneously for all surface points and cutting
then planes Cp.
NOTE: If the orientation tolerance refers to both a primary datum
l cos for a primary datum axis
o and a secondary datum, then T is fully determined.
T D1 = m
o
o
n sin for a primary datum plane (c) Actual Value. The actual value of orientation for S is
the smallest orientation tolerance to which S will conform.
Figure 6-1 Planar Orientation Zone With Primary and Figure 6-2 Projection of Tolerance Vector Onto Primary
Secondary Datum Planes Specified Datum Plane
31
ASME Y14.5.1-2019
Figure 6-3 Cylindrical Orientation Zone With Respect to a

Primary Datum Plane
Figure 6-4 Linear Orientation Zone Bounded by Parallel Lines
32
ASME Y14.5.1-2019
Section 7
Tolerances of Location
7.1 GENERAL Figure 7-2 shows the converse situation. The shaft
shown in the figure is controlled by a position tolerance
This Section establishes the principles of tolerances of t0 at MMC. The shaft was manufactured with perfect form.
location; included are position, concentricity, and If the radius of the shaft is rAM and the MMC radius is rMMC,
symmetry used to control the following relationships: the radius of the tolerance zone for the axis is rMMC − rAM +
(a) center distance between (features of size, such as) t0/2. If the height of the shaft is h, and the axis of the actual
holes, slots, bosses, and tabs shaft is tilted to an extreme orientation within the toler-
(b) location of features of size [such as in (a) above] as a
group with respect to datum reference frames ance zone, a simple geometric analysis shows that point P
(c) coaxiality and coplanarity of features of size lies outside the theoretical boundary (virtual condition)
(d) concentricity or symmetry of features of size — by a distance
center distances of correspondingly-located feature ÄÅ ÉÑ
Å Ñ
elements equally disposed about a datum axis or plane rAMÅÅÅÅ 1 + [(t + 2(r MMC rAM)) / h]2 1ÑÑÑÑ
ÅÇ ÑÖ
7.1.1 Material Condition Basis The feature illustrated in Figure 7-2 is acceptable per
Position tolerances are applied on an MMC, RFS, or LMC the resolved geometry interpretation but unacceptable
basis. A position tolerance may be explained either in per the surface interpretation.
terms of the surface method of the actual feature or in Throughout most of this Section, both a surface inter-
terms of size and the resolved geometry (center point, pretation and a resolved geometry interpretation are
axis, or center plane) of the applicable (mating or supplied. In a few cases (e.g., projected tolerance
minimum material) actual envelope. These two interpre- zones) only a surface interpretation is provided. When-
tations will be called the “surface interpretation” and the ever the two interpretations do not produce equivalent
“resolved geometry interpretation,” respectively. (See results, the surface interpretation shall take precedence.
subsection 7.2 and the following for the precise definitions
of position tolerancing interpretations.) These interpre- 7.1.2 Patterns of Features
tations are not equivalent. They differ in part because For the purposes of this Standard, all tolerances of loca-
the resolved geometry interpretation relies on an assump- tion are considered to apply to patterns of features, where
tion that the feature is of perfect form and in part because a pattern may consist of only a single feature. The control
the derivation of the surface interpretation assumes of the location of the pattern as a group is called the
perfect orientation. pattern-locating tolerance zone framework (PLTZF).
Two examples will illustrate these interpretations. When the pattern consists of two or more features,
Consider the illustration in Figure 7-1. The hole shown there is the possibility, through the use of composite tol-
in the figure is controlled by a zero position tolerance erancing, to control the relative location of features within
at MMC. The theoretical boundary (virtual condition) the pattern. This is done by specifying a secondary location
has a diameter equal to the MMC diameter of the hole. tolerance, called the feature-relating tolerance zone
The actual hole was manufactured with poor form, but framework (FRTZF), in conjunction with the PLTZF.
is within the limits of size (see para. 2.3.1). The hole There may be more than one FRTZF for a pattern. All
does not violate the theoretical boundary (virtual condi- features within a single pattern are controlled simulta-
tion), and would be acceptable per the surface interpreta- neously. That is, all features must be evaluated with
tion. The resolved geometry (the axis of the hole) is further respect to a single datum reference frame from the candi-
away from true position than allowed by the combined date datum reference frame set for the control.
effects of the position tolerance (zero) and the bonus toler-
ance resulting from the unrelated actual mating envelope 7.2 POSITION TOLERANCING
of the hole. The hole would not be acceptable per the
resolved geometry interpretation. This subsection presents a general explanation of posi-
tion tolerancing for features of size. A position tolerance
can be explained in terms of a zone within which the
33
ASME Y14.5.1-2019
resolved geometry (center point, axis, or center plane) of a For MMC or LMC material condition basis, the boundary
feature of size is permitted to vary from true (theoretically
exact) position. Basic dimensions establish the theoretical ( ) = b, with b as given here, is the theoretical
defined by r P
true position from specified datum features and between boundary (virtual condition).
interrelated features. A position tolerance can be (c) Actual Value. The actual value of position deviation
explained in terms of a surface boundary. is the smallest value of t0 to which the feature conforms.
Throughout this subsection, whenever the true position NOTE: For LMC and MMC controls the actual value of deviation
can be negative. A negative value can be interpreted as the
( )
is understood from context, the notation r P will denote
unused portion of the bonus tolerance resulting from the depar-
the distance from a point P to the true position. For ture of the feature from the applicable limit of size.
( ) is the distance to the true position

spheres, r P 7.2.2 In Terms of the Resolved Geometry of a
Feature
center point. For cylinders, r(P ) is the distance
(a) Definition. For a pattern of features of size, a posi-
between P and the true position axis. For parallel tion tolerance specifies that the resolved geometry (center
point, axis, or center plane, as applicable) of each unre-
( )
plane features, r P is the distance between P and the
lated actual mating envelope (for features at MMC or RFS)
true position center plane. These definitions should or unrelated actual minimum material envelope (for
also be understood to be for a particular choice of features at LMC) must lie within a corresponding position
datum reference frame from the candidate datum refer- tolerance zone. Each tolerance zone is bounded by a
ence frame set. Throughout this Section, all spherical and sphere, cylinder, or pair of parallel planes of size equal
cylindrical sizes are in terms of radius unless otherwise to the total allowable tolerance for the corresponding
specified. All tolerance values are assumed to be feature. Each tolerance zone is located and oriented by
diameters for spheres and cylinders, and full widths the basic dimensions of the pattern relative to a datum
for parallel planes, in accord with common practice. reference frame. A position tolerance zone is a spherical,
cylindrical, or parallel-plane volume defined by all points
7.2.1 In Terms of the Surface of a Feature
(a) Definition. For a pattern of features of size, a posi-
( )
P that satisfy the equation r P b, where b is the radius
tion tolerance specifies that the surface of each actual or half-width of the tolerance zone.
feature must not violate the boundary of a corresponding Figure 7-5 illustrates the tolerance zone for holes at
position tolerance zone. Each boundary is a sphere, cyl- MMC and RFS and for shafts at LMC. The figure shows
inder, or pair of parallel planes of size equal to the collec- the position of a point on the axis of the actual mating
tive effect of the limits of size, material condition basis, and envelope that is outside the tolerance zone. A similar
applicable position tolerance. Each boundary is located figure for holes at LMC or shafts at MMC or RFS would
and oriented by the basic dimensions of the pattern show the actual mating envelope surrounding the
and datum(s) if specified. Each position tolerance zone feature surface. The feature axis extends for the full
length of the feature.
is a volume defined by all points P that satisfy the appro- (b) Conformance. A feature conforms to a position
priate equation from Table 7-1, where b is a position toler- tolerance t0 at a specified material condition basis if all
ance zone size parameter (radius or half-width). points of the resolved geometry of the applicable envelope
Figure 7-3 illustrates the tolerance zone for a cylindrical (as determined by the material condition basis) lie within
hole at MMC or RFS, or a shaft at LMC. The tolerance zone is some position tolerance zone as defined above with b
a cylindrical volume. determined by the appropriate formula from Table 7-3.
Figure 7-4 illustrates the tolerance zone for a tab at Furthermore, the surface must conform to the applicable
MMC or RFS, or a slot at LMC. The tolerance zone is a size limits.
pair of parallel planar volumes. (c) Actual Value. The position deviation of a feature is
(b) Conformance. A feature conforms to a position the diameter of the smallest zone (smallest value of b)
tolerance t0 at a specified material condition basis if all which contains the center point or all points on the
points of the feature lie outside some position zone as axis or center plane (within the extent of the feature)
defined above with b determined by the appropriate of the applicable actual mating envelope of the feature.
value from Table 7-2.
The surface must conform to the applicable size limits. 7.3 PROJECTED TOLERANCE ZONE
In the case of an internal feature at MMC or RFS or an
external feature at LMC, there is a further condition (a) Definition. For a cylindrical or parallel-plane
that the feature must surround the tolerance zone. feature, a projected tolerance specifies that a volume,
called a verifying volume, with a boundary of perfect
34
ASME Y14.5.1-2019
form, called a verifying boundary, can be defined such that 7.4 CONICAL TOLERANCE ZONE
the following two conditions hold. First, the axis or center
plane of the verifying boundary is contained within a A conical position tolerance zone is specified by
projected position tolerance zone, itself a boundary of different position tolerance values at each end of a cylin-
perfect form extending outward from the feature by drical feature. A conical tolerance can be interpreted
the specified projection length. Second, the surface of either in terms of the surface of the feature or in
the feature does not violate the verifying volume. terms of the axis of the feature.
A projected position tolerance zone is a cylindrical or
7.4.1 In Terms of the Surface of the Feature
parallel-plane volume defined by all points P that satisfy
(a) Definition. For a pattern of cylindrical features, a
( )
the equation r P b, where b is the radius or half-width position control tighter at one end of the features than
of the tolerance zone. A verifying volume is a cylindrical or the other specifies that the surface of each actual
feature must not violate a corresponding perfect-form
parallel-plane volume defined by all points P that satisfy conical boundary. This boundary is a frustum of height
the appropriate equation from Table 7-4, where w is a size and diameters equal to the collective effects of the
parameter for the verifying volume. limits of size, material condition basis, and applicable posi-
Figure 7-6 illustrates a typical case. The projected toler- tion tolerances at each end of the feature. The boundary is
ance zone is positioned and oriented by the choice of located and oriented by the basic dimensions of the
datum reference frame. A plane perpendicular to the feature. A position tolerance zone is a conical volume
true position axis is located to contact the part surface defined by all points P that satisfy the appropriate equa-
that defines the end of the cylindrical feature. The
height of the tolerance zone is the specified projection tion from Table 7-5, where b(P ) is the radius of the toler-
length and starts at the point where the true position
ance zone at height P . The radius b(P ) is related to the
axis intersects the contacting plane. The verifying
volume is shown for a hole at MMC or RFS. (A similar position tolerance zone size parameters r1 and r2 by
picture would apply for a shaft at LMC. For a shaft at
MMC or RFS, or a hole at LMC, the verifying envelope
b(P ) = r1 1( )
( P ) + r2 (P )
would surround the feature.)
(b) Conformance. A feature conforms to a position where
tolerance t0, projected a distance h, and at a specified (P P1) (P2 P1)
material condition basis, if there exists at least one veri- (P ) = 2
fying volume for which the following conditions hold. All P2 P1
points of the feature lie outside the verifying volume as
defined above with w determined from the material basis
as follows: is the position of P along the axis between P1 and P 2 ,
(1) for MMC, w = rMMC scaled so that ( P1) = 0 and ( P2) = 1 .
(2) for RFS, w = rAM Figure 7-7 illustrates the tolerance zone for holes at
(3) for LMC, w = rLMC MMC and RFS. Shafts at LMC would have a similar
The verifying envelope satisfies r P ( ) t0/2 for all figure. A similar figure for holes at LMC or shafts at
MMC or RFS would show the envelope surrounding
points P on the resolved geometry starting at the inter- the feature surface. The tolerance zone axis extends
section of the resolved geometry with the contacting plane between points P1 and P 2 , which are the intersection
and ending at the intersection of the resolved geometry
of the true position axis with two planes, one at each
with a second plane parallel to the contacting plane and
end of the basic feature, at nominal distance and nominally
separated from it by a distance h.
located and oriented relative to the datum reference
NOTE: For RFS features, this definition can also be considered the frame.
resolved geometry interpretation. No resolved geometry inter- (b) Conformance. A cylindrical feature conforms to
pretation is provided for MMC or LMC tolerances.
position tolerances t1 and t2 at a specified material condi-
(c) Actual Value. The position deviation of a feature is tion basis if all points of the feature lie outside the position
the size of the smallest projected zone such that the tolerance zone as defined above, with ri (i ∊ {1,2}) deter-
resolved geometry of the actual mating envelope lies mined according to Table 7-6. The surface must also
within the zone for the full projection height. conform to the applicable size limits. In the case of a
hole, there is a further condition that the hole must
surround the tolerance zone.
35
ASME Y14.5.1-2019
(c) Actual Value. No definition for actual value of posi- 7.5 BIDIRECTIONAL POSITION TOLERANCING
tion deviation is provided for the surface interpretation.
Refer to para. 7.4.2 to the resolved geometry (axis) inter- A bidirectional position tolerance zone for a cylindrical
pretation for a definition of actual value. feature is specified by different position tolerance values
in different directions perpendicular to the basic feature
7.4.2 In Terms of the Resolved Geometry (Axis) of axis. Bidirectional position tolerancing results in two
the Feature distinct tolerance zones for locating each cylindrical
feature. Each tolerance zone is considered separately
(a) Definition. For the axes of cylindrical features in the following. As with other tolerances, however,
within a pattern, a position tolerance tighter at one rules for simultaneous or separate requirements apply
end specifies that the axes of the actual mating envelopes to the components of a bidirectional position tolerance.
(for features at MMC or RFS) or of the actual minimum (See para. 4.7.10.) Bidirectional position tolerancing
material envelopes (for features at LMC) must lie may be applied in either a rectangular or a polar (cylin-
within corresponding position tolerance zones. Each of drical) coordinate system. Rectangular bidirectional tol-
these tolerance zones is bounded by a frustum of erancing can be explained in terms of either the surface or
height and diameters equal to the collective effects of the axis of the feature. A resolved geometry (axis) inter-
the limits of size, material condition basis, and applicable pretation only is provided for polar bidirectional toleranc-
position tolerances at each end of the feature. The axis of ing.
the frustum is located and oriented by the basic dimen-
sions of the feature. The frustum is located along the axis 7.5.1 In Terms of the Surface of the Feature
by the nominal surfaces bounding the feature. The posi-
This paragraph establishes the surface interpretation of
tion tolerance zone is a conical volume defined by all
bidirectional position tolerancing when applied in a
points P that satisfy the equation r P ( ) b(P ), where rectangular coordinate system.
(a) Definition. For a pattern of cylindrical features, each
( )
b P is the radius of the tolerance zone at the height bidirectional position tolerance specifies that each surface
must not violate a tolerance boundary. For holes at MMC
along the axis of P . (See the surface interpretation,
or RFS and shafts at LMC, each tolerance boundary is a
para. 7.4.1, for details.) cylinder of diameter equal to the collective effects of
Figure 7-8 illustrates the axis definition for holes. The the limits of size, material condition basis, and applicable
tolerance zone axis extends between points P1 and P 2 , position tolerance. Each boundary is located and oriented,
which are the intersection of the true position axis by the basic dimensions of the pattern relative to the speci-
with the nominal surfaces bounding the feature. fied datum reference frame and by the applicable direction
(b) Conformance. A cylindrical feature conforms to of tolerance control, such that the axis of each boundary
position tolerances t1 and t2 at a specified material condi- lies in the plane containing the true position axis of the
tion basis if all points on the axis of the applicable envelope corresponding feature and normal to the direction in
(as determined by the material condition basis) lie within which the tolerance applies. The orientation and position
the position tolerance zone as defined above, with ri (i ∊ of the boundary axis within this plane is unconstrained.
{1,2}) determined by the appropriate formula from For holes at LMC and shafts at MMC or RFS, each toler-
Table 7-7. Furthermore, the surface must conform to ance boundary is a pair of parallel planes separated by a
the applicable size limits. distance equal to the collective effects of the limits of size,
(c) Actual Value. A cylindrical feature controlled by a material condition basis, and applicable position toler-
conical tolerance zone has two actual values for position ance. The center plane of each boundary is that plane
deviation, one at each end of the feature. The actual value containing the axis of the corresponding feature and
at each end is the smallest diameter circle that contains the normal to the direction in which the tolerance applies.
axis of the actual mating envelope at that end. Each circle is A position tolerance zone is a volume defined by all
in the plane perpendicular to the true position axis at the points P that satisfy the appropriate equation from
end point of the feature axis, and is centered on the true Table 7-8, where b is a position tolerance zone size pa-
position axis. In the case that the actual mating envelope rameter (radius or half-width).
can rock, it may be possible to decrease the actual value of (b) Conformance. A cylindrical feature conforms to a
position deviation at one end at the expense of the devia- bidirectional position tolerance t0 at a specified material
tion at the other end. No rule is defined for selecting among condition basis if all points of the feature lie outside some
possible pairs of actual values. position tolerance as defined above with b determined by
the appropriate value from Table 7-9.
Figure 7-9 shows an example of bidirectional toleranc-
ing of a hole at MMC. Each callout creates its own cylin-
drical position tolerance zone. The tolerance zone
36
ASME Y14.5.1-2019
corresponding to the 0.4 mm tolerance, shown in the 7.5.3 Polar Bidirectional Tolerancing in Terms of
bottom left, is free to be located and oriented only in the Resolved Geometry (Axis) of the Feature
the plane indicated by the vertical dashed line. Similarly,
the tolerance zone corresponding to the 0.2 mm tolerance, This paragraph establishes the resolved geometry
shown in the bottom right, is only free to be located and (axis) interpretation of bidirectional position tolerancing
oriented left-to-right in the view shown. Each of these when applied in a cylindrical coordinate system. (While
planes of motion are determined by the basic dimensions the term “polar” is used in ASME Y14.5-2009, and used
from the indicated datums. herein for consistency, a cylindrical coordinate system
A similar example is illustrated for shafts at MMC in is being used. The tolerances are specified in the plane
Figure 7-10. In this case, each callout creates a tolerance normal to the axis of the cylindrical coordinate system.)
zone bounded by parallel planes. The tolerance zone (a) Definition. For axes of cylindrical features within a
corresponding to the 0.4 mm tolerance is bounded by pattern, polar bidirectional position tolerances specify
the vertical planes separated by 16.6 mm. The tolerance that the axes of the actual mating envelopes (for features
zone corresponding to the 0.2 mm tolerance is bounded by at MMC or RFS) or of the minimum material envelopes (for
the horizontal planes separated by 16.4 mm. features at LMC) must lie within corresponding position
(c) Actual Value. No definition for actual value of bidir- tolerance zones. Each tolerance zone is bounded radially
ectional position deviation is provided in terms of the by two concentric cylindrical arcs and tangentially by two
surface of the feature. Refer to para. 7.5.2 to the resolved planes symmetrically disposed about the true position of
geometry (axis) interpretation for a definition of actual the feature and oriented at the basic polar angle of the
value. feature. The plane separation and the difference in cylin-
drical arc radii are each equal to the total allowable toler-
7.5.2 In Terms of the Resolved Geometry (Axis) of ance for the corresponding feature, including any effects of
the Feature feature size. Each tolerance zone is located and oriented by
the basic dimensions of the pattern. A polar bidirectional
This paragraph establishes the resolved geometry position tolerance zone is a (cylindrical shell) volume
(axis) interpretation of bidirectional position tolerancing
defined by all points P that satisfy the following two equa-
when applied in a rectangular coordinate system.
(a) Definition. For axes of cylindrical features within a tions:
pattern, bidirectional position tolerances specify that the
axis of each unrelated actual mating envelope (for features P 0 < br
at MMC or RFS) or minimum material envelope (for
features at LMC) must lie within two corresponding posi- and
tion tolerance zones. Each position tolerance zone is
bounded by two parallel planes separated by a distance
equal to the total allowable tolerance for the corre-
(P A ) Nt bt
sponding feature, including any effects of feature size.

where
Each position tolerance zone is located and oriented = a point on the true position axis of the feature
by the basic dimensions of the pattern relative to the speci- A
fied datum reference frame. A bidirectional position toler- br = the tolerance zone size parameter for the cylin-
drical boundaries of the tolerance zone, equal in
ance zone is a (slab) volume defined by all points P that value to half the difference in radii of the bound-
aries
satisfy the equation r P( ) b, where b is half the thickness
bt = the tolerance zone size parameter for the planar
of the tolerance zone. boundaries of the tolerance zone, equal in value
(b) Conformance. A cylindrical feature conforms to a to half of the distance between the boundaries
position tolerance t0 at a specified material condition Nt = the direction vector of the plane containing the
basis if all points on the axis of the applicable envelope axis of the polar (cylindrical) coordinate system
(as determined by the material condition basis) lie within and the true position axis of the feature
the position tolerance zone as defined above with b deter- =
P the distance of P from the axis of the polar (cy-
mined by the appropriate formula from Table 7-10.
Furthermore, the surface must conform to the applicable lindrical) coordinate system
size limits. ρ0 = the distance of the true position axis from the
(c) Actual Value. The position deviation of a feature is axis of the polar (cylindrical) coordinate system
the thickness of the smallest tolerance zone to which the
axis conforms. The relationship between these quantities is illustrated
in Figure 7-11.
NOTE: There is a distinct actual value for each tolerance callout in
a bidirectional position tolerance.
37
ASME Y14.5.1-2019
(b) Conformance. A cylindrical feature conforms to a elongated-hole cylinders (unconstrained in location or

polar, bidirectional position tolerance with radial compo- orientation), such that the following conditions hold.
nent tr and tangential component tt, each applied at a One cylinder, with w and ℓ equal to the MMC limits of
specified material condition basis, if all points on the size, is surrounded by the hole surface. The other cylinder,
axis of the applicable envelope (as determined by the with w and ℓ equal to the LMC limits of size, surrounds the
material condition basis) lie within the position tolerance hole surface.
zone as defined above with br and bt determined by the (c) Actual Value. No actual value of position deviation
appropriate formula from Table 7-11, with t = tr and t = tt, for elongated holes is defined.
respectively. Furthermore, the surface must conform to
the applicable size limits. 7.7 CONCENTRICITY AND SYMMETRY
(c) Actual Value. As with rectangular, bidirectional
position tolerancing, two actual values of position devia- This subsection provides definitions of concentricity
tion are defined. The actual value of position deviation in and symmetry tolerances that control concentricity
either the radial or tangential direction is the thickness of and symmetry of features. Concentricity and symmetry
the smallest tolerance zone to which the applicable axis controls are similar concepts and are treated together
conforms. in this Section. Concentricity is that condition where
the median points (centroids) of all diametrically
opposed elements of a figure of revolution (or correspond-
7.6 POSITION TOLERANCING AT MMC FOR
ingly located elements of two or more radially disposed
BOUNDARIES OF ELONGATED HOLES features) are congruent with a datum axis or center point.
An elongated hole is an internal feature consisting of Symmetry is that condition where one or more features is
two parallel, opposed, planar faces terminated by cylin- equally disposed about a datum plane. A symmetry toler-
drical end caps, tangent to the planar faces, with axes ance is used for the mathematical concept of symmetry
inside the hole. For purposes of position tolerancing, about a plane and a concentricity tolerance is used for
an elongated hole is considered a feature of size, charac- the mathematical concept of symmetry about a point
terized by two size parameters, its length and width. Such or symmetry about an axis. Concentricity and symmetry
tolerancing is always considered to be bidirectional in controls are applied to features on an RFS basis only.
nature, even if a single tolerance value is applied. Only Datum references must also be RMB.
a surface interpretation is provided. (a) Definition. A concentricity or symmetry tolerance
(a) Definition. For a pattern of elongated holes, a posi- specifies that the centroid of corresponding point
tion tolerance at MMC specifies that the surface of each elements on the surfaces of the actual features must
actual hole must not violate the boundary of a corre- lie in the symmetry tolerance zone. The tolerance zone
sponding tolerance zone. Each boundary is a right cylinder is bounded by a sphere, cylinder, or pair of parallel
with an elongated cross section of perfect form as shown planes of size equal to the total allowable tolerance for
in Figure 7-12. Each boundary is located and oriented by the features. The tolerance zone is located and oriented
the basic dimensions of the pattern relative to the speci- by the basic dimensions of the feature(s) relative to the
fied datum reference frame. Each position tolerance zone specified datum reference frame. The tolerance zone is a
is the volume interior to the corresponding boundary (the spherical, cylindrical, or parallel-plane volume defined by
shaded area in Figure 7-12). The boundary size is char-
acterized by two size parameters, ℓ and w, representing,
( )
all points P that satisfy the equation r P b, where b is
respectively, the half-length and half-width of the toler- the radius or half-width of the tolerance zone.
ance zone. Corresponding point elements are obtained by inter-
(b) Conformance. A position tolerance for an elongated secting a pattern of symmetry rays with the actual
hole specifies two values feature. The rays of symmetry are determined per
(1) tw, controlling position deviation in the direction Table 7-12. If the feature is symmetric about a plane, a
of the hole width two-fold symmetry pattern is always used. For point
(2) tℓ, controlling position deviation along the length and axis symmetry, the symmetry pattern is constructed
of the hole using the lowest order of symmetry of the basic feature.
An elongated hole conforms to position tolerances tw One consequence of this is that surfaces of revolution use
and tℓ if all points of the hole surface lie outside the posi- two-fold patterns of symmetry rays about the axis or
tion tolerance zone as defined above, with w = wMMC − tw/2 center of symmetry. The feature elements are located
and ℓ = ℓMMC − tℓ/2, where wMMC is the MMC width of the at the intersection of the symmetry rays and the actual
elongated hole and ℓMMC is the MMC length of the elon- feature surface.
gated hole. This principle is illustrated in Figure 7-13. A feature that
Furthermore, the hole must surround the tolerance has basic three-fold symmetry about a point or (as shown
zone and must conform to the limits of size. An elongated in the figure) an axis results in a three-fold symmetry for
hole conforms to the limits of size if there exist two right, the symmetry rays. If the symmetry of the feature is six-
38
ASME Y14.5.1-2019
fold, however, the symmetry rays are arranged in a two- Table 7-1 Definition of Position Tolerance Zone —
fold pattern. Surface Interpretation
(b) Conformance. A feature conforms to its symmetry or Material Condition Basis
concentricity tolerance if the centroids of corresponding
MMC or RFS LMC
points of intersection of the rays with the feature all lie
Feature Type Internal
within the tolerance zone as defined above with b = t0/2. ( ) <b
r P ( )>b
r P
(c) Actual Value. The actual value of concentricity or
External r( P ) > b r( P ) < b
symmetry deviation is the smallest tolerance value to
which the feature will conform.
Figure 7-1 First Illustration of the Difference Between

Surface and Resolved Geometry Interpretations of
Position Tolerancing Figure 7-3 Tolerance Zone and Conformance: Holes at
MMC or RFS, Shafts at LMC — Surface Interpretation
Figure 7-2 Second Illustration of the Difference Between Figure 7-4 Tolerance Zone and Conformance: Tabs at
Surface and Resolved Geometry Interpretations of MMC or RFS, Slots at LMC — Surface Interpretation
Position Tolerancing
39
ASME Y14.5.1-2019
Table 7-2 Size of Position Tolerance Zone — Surface Interpretation

Material Condition Basis
b MMC RFS LMC
Feature Type Internal t0 t0 t
rMMC rAM r LMC + 0
2 2 2
External t0 t0 t0
rMMC + rAM + r LMC
2 2 2
Figure 7-5 Tolerance Zone and Conformance: Holes at

MMC or RFS, Shafts at LMC — Resolved Geometry
Interpretation
Table 7-3 Size of Position Tolerance Zone — Resolved Geometry Interpretation

b MMC RFS LMC
Feature Type Internal t0 t0 t0
+ (rAM rMMC) + (r LMC rAMM)
2 2 2
External t0 t0 t0
+ (rMMC rAM) + (rAMM r LMC)
2 2 2
Table 7-4 Definition of Verifying Volume for Projected

Tolerance Zone
MMC or RFS LMC
( )<w
r P ( )>w
r P
External r( P ) > w r( P ) < w
40
ASME Y14.5.1-2019
Figure 7-6 Projected Tolerance Zone for a Hole Figure 7-7 Surface Interpretation of Conical Tolerance
Zone for Holes at MMC or RFS
Table 7-6 Size of Conical Tolerance Zone — Surface

Interpretation
Table 7-5 Definition of Conical Tolerance Zone — Surface Material Condition Basis
Interpretation ri MMC RFS LMC
Material Condition Basis Feature Internal ti ti t
rMMC rAM r LMC + i
Type 2 2 2
MMC or RFS LMC
External ti ti ti
Feature Type Internal rMMC + rAM + r LMC
r( P ) < b( P ) r( P ) > b( P ) 2 2 2
External r( P ) > b( P ) r( P ) < b( P )
Figure 7-8 Resolved Geometry (Axis) Interpretation of
Conical Tolerance Zone for Holes at MMC or RFS
41
ASME Y14.5.1-2019
Table 7-7 Size of Conical Tolerance Zone — Resolved Geometry (Axis) Interpretation
ri MMC RFS LMC
Feature Type Internal ti ti ti
2 2 2
External ti ti ti
2 2 2
Table 7-8 Definition of Bidirectional Tolerance Zone —

Surface Interpretation
MMC or RFS LMC
( )<b
r P ( )>b
r P
External r( P ) > b r( P ) < b
( )
GENERAL NOTE: In this table, r P is the distance from P to the
resolved geometry of the tolerance zone boundary. The tolerance
zone boundary is a cylinder for a hole at MMC or RFS and for
shafts at LMC; it is a pair of parallel planes for shafts at MMC or
RFS and for holes at LMC.
Table 7-9 Size of Bidirectional Tolerance Zone — Surface Interpretation

b MMC RFS LMC
Feature Type Internal t0 t0 t
rMMC rAM r LMC + 0
2 2 2
External t0 t0 t0
rMMC + rAM + r LMC
2 2 2
42
ASME Y14.5.1-2019
Figure 7-9 Bidirectional Hole Tolerance at MMC With Cylindrical Tolerance Zones — Surface Interpretation
Figure 7-10 Bidirectional Shaft Tolerance at MMC With Parallel Plane Tolerance Zones — Surface Interpretation
Table 7-10 Size of Bidirectional Tolerance Zone — Resolved Geometry (Axis) Interpretation
b MMC RFS LMC
Feature Type Internal t0 t0 t0
2 2 2
External t0 t0 t0
2 2 2
43
ASME Y14.5.1-2019
Figure 7-11 Definition of the Tolerance Zone for Polar

Bidirectional Tolerancing
Table 7-11 Size of Polar Bidirectional Tolerance Zone — Resolved Geometry (Axis) Interpretation
br or bt MMC RFS LMC
Feature Type Internal t t t
+ (rAM r MMC) + (r LMC rAMM)
2 2 2
External t t t
+ (r MMC rAM) + (rAMM r LMC)
2 2 2
Figure 7-12 Tolerance Zone and Conformance, Elongated

Hole at MMC — Tolerance Zone is the Right Cylinder
Shown in Cross Section
44
ASME Y14.5.1-2019
Table 7-12 Symmetry Elements for Obtaining

Corresponding Feature Elements
Symmetry Tolerance
Type Type Patterns of Symmetry Rays
Point Concentricity Rays from the datum point
Axis Concentricity Rays from, and perpendicular to, the
datum axis
Plane Symmetry Rays from, and perpendicular to, the
datum plane
Figure 7-13 Rays Are Arranged in the Lowest Order of

Symmetry About an Axis or a Point
45
ASME Y14.5.1-2019
Section 8
Tolerances of Profile
8.1 GENERAL to the true profile at all points along the profile. The bound-
aries of the tolerance zone follow the geometric shape of the
This Section establishes the principles and methods for true profile. [8.3.1]
mathematical evaluation of ASME Y14.5-2009 dimen- (a) Definition. A profile tolerance zone for a continuous
sioning and tolerancing applicable to the control of profile. true profile is constructed by sweeping a line segment of
length t0, where t0 is the specified tolerance, along each
8.2 PROFILE point on the true profile. The line segment is kept normal
Profile tolerances define tolerance zones to control to the true profile at each point. If the line segment inter-
surfaces relative to a true profile. Unless otherwise speci- sects the true profile at another point, the line segment
fied, profile tolerance zones maintain a basic relationship shall be truncated at that intersection. The profile toler-
to the true profile and any datums referenced. This ance zone is the union of line segments obtained from all
enables profile tolerance zones to always control form points on the basic surface. A maximum material
and, where applicable, size (curvature), orientation, boundary (MMB) and a least material boundary (LMB)
and location of the considered feature. are created, each one a locus of endpoints of the line
segments.
8.2.1 Types of Profile Tolerances The disposition of the generating line segment follows
the disposition indicated by the tolerance specification.
A profile tolerance may be applied to the entire surface See Table 8-1 and Figures 8-1 through 8-4.
of a part, multiple features, individual surfaces, or to in- (1) For equally disposed tolerances, the line segment
dividual profiles taken at various cross sections through a is centered on the basic surface with equal extension in
part. The two types of profile tolerances are profile of a both directions.
surface and profile of a line, and they are explained in (2) For unilateral and unequally-disposed toler-
paras. 8.2.1.1 and 8.2.1.2. ances, the line segment is disposed in the manner indi-
8.2.1.1 Profile of a Surface. Per ASME Y14.5-2009 cated by the unequally-disposed symbol Ⓤ or
The tolerance zone established by the profile of a surface graphical indication. Several different dispositions are
tolerance is three-dimensional (a volume), extending along defined for a profile tolerance t0 with a value tu following
the length and width (or circumference) of the considered the unequally-disposed symbol.
feature or features. [8.2.1.1] (b) Conformance. A feature conforms to a profile toler-
For profile of a surface, the true profile is comprised of ance t0 if all points on the actual surface are within the
surface geometry. The controlled component is the entire tolerance zone described above.
surface of the actual feature(s). (c) Actual Value. The actual value of a profile tolerance
is based on an enveloping zone called the actual zone that
8.2.1.2 Profile of a Line. No mathematization of the is generated in the same way as the tolerance zone. This
definition, conformance, or actual value of profile of a concept applies in the same way to equally disposed,
line is provided in this Standard. unequally disposed, and unilateral profile specifications.
The generating line segment of the tolerance value t0 is
8.3 TOLERANCE ZONE BOUNDARIES lengthened or shortened by an equal amount at each end.
Uniform, bilateral, unequally disposed, or nonuniform The length change at each end is called the growth param-
tolerance zones may be defined by profile tolerances. eter g. The line segment for the actual zone has a length
equal to t0 + 2g. The actual zone has the minimum g nec-
8.3.1 Uniform Tolerance Zones essary to contain the actual surface.
(1) If the actual zone is contained within the toler-
Per ASME Y14.5-2009 ance zone, the growth parameter g is negative.
A uniform tolerance zone is the distance between two (2) If the actual zone is not contained within the
boundaries equally or unequally disposed about the true tolerance zone, g is positive.
profile or entirely disposed on one side of the true
profile. Profile tolerances apply normal (perpendicular)
46
ASME Y14.5.1-2019
(3) If the actual zone is equivalent to the tolerance alizes the concept of datum feature shift, to apply to profile
zone, g = 0. applications in which a datum reference frame is not
NOTE: Any necessary truncation is performed after the length- present (i.e., no datum features are referenced). See
ening or shortening of the line segments. Figures 8-11, 8-12, 8-15, and 8-16.
The actual value of a profile tolerance for a single feature 8.4.1.3 Candidate Tolerance Zone Transformations.
is the value of t0 + 2g for the enveloping zone to which the By default, profile tolerance zones remain static relative
actual surface will conform. See Figures 8-5 through 8-8. to the true profile. With composite profile tolerancing, the
NOTE: This Section defines the actual value of a uniform profile feature-relating tolerance zones are constrained in rota-
tolerance in the context of a single feature in a given candidate tion only and are thus permitted to translate.
datum reference frame. Applications involving multiple features This is an example of a tolerance zone transformation, in
and optimization within degrees of freedom are addressed in which the tolerance zone framework is permitted to trans-
para. 8.4.1.5 of this Standard. late relative to the datums (and datum reference frame). In
The value after the Ⓤ symbol for the actual zone is not this Standard, the term candidate tolerance zone transfor-
part of the actual value definition, but can be calculated mation is used to describe a particular transformation of
using the expression (tu + g). the tolerance zone relative to the basic reference-body.
See para. 8.4.7 of this Standard for details on composite
profile.
8.4 PROFILE APPLICATIONS
8.4.1.4 Candidate Configurations. In this Standard,
Subsection 8.3 explained the profile tolerance zone and
the generic term candidate configuration is used to
actual value in the context of a single feature in a fully
describe a particular state of a profile tolerance
constrained datum reference frame. Profile tolerances
system. This can include a candidate spatial relationship,
can also be applied to multiple features, with a variety
candidate tolerance zone transformations, or both.
of possible datum feature constraints and tolerance
zone transformations. These are most easily explained 8.4.1.5 Default Optimization Criteria for Calculation
using the system concept, in which the profile tolerance of Actual Values. For most geometric characteristics
is treated as a system with degrees of freedom and applied to single feature (profile or otherwise), the
constraints. This approach allows optimization and actual value is the smallest tolerance to which the
actual value calculation for applications with mixed toler- feature will conform. In terms of optimization, the objec-
ance values and combined controls. tive function is to minimize the size of the zone that will
just envelop the feature. For profile tolerance systems in-
8.4.1 Optimization of Profile Tolerance Systems volving multiple features and mixed tolerance values, this
Conformance and actual value calculation for a profile concept cannot be uniquely applied and must be general-
tolerance system is based on the concept of optimization ized. In each candidate configuration of such systems,
within constraints. The actual surface geometry and the candidate actual values for each characteristic can be
tolerance zones for the profile tolerances in the system are calculated.
fitted to each other within the applicable constraints. In this Standard, the default optimization method is to
Profile tolerance systems include the following: minimize the maximum value of the growth parameter g
(a) basic reference-body with true profile(s) for the geometric characteristics in the system. This repre-
(b) tolerance zone(s) for considered feature(s) sents a generalization of the single-feature concept of
(c) actual part geometry with actual considered feature minimizing the size of the enveloping zone. Actual
surface(s) values are defined as the candidate actual values in the
(d) actual datum feature surfaces (if applicable) candidate configuration in which the largest value of g
(e) datum feature simulators or datum target simula- for any feature in the system is minimized.
tors (if applicable)
(f) datums and a datum reference frame (if applicable)
8.4.2 Constraint Properties of Profile Tolerance
Zones
8.4.1.1 Basic Reference-Body. This is the design
geometry (para. 1.1.1). A profile tolerance zone is defined in relation to a theo-
retically exact true profile that is considered rigid. The
8.4.1.2 Candidate Spatial Relationships. ASME Y14.5- tolerance zone has perfect form and by default is not
2009 uses the term “datum feature shift” to describe rela- permitted to transform (translate, rotate, or progress)
tive movement between the datum feature(s) and a datum relative to the true profile. This enables the profile
reference frame. In this Section, the term “candidate zone to control form and, where applicable, size of the
spatial relationship” is used to describe a particular considered feature. See Figures 8-9 through 8.12.
instance of rotation and translation between the actual
part geometry and the basic reference-body. This gener-
47
ASME Y14.5.1-2019
8.4.3 Effect of Pattern Creation (Grouping) 8.4.6 Effect of Simultaneous Requirements

Mechanisms Where simultaneous requirements apply, all profile
Pattern creation mechanisms (ALL AROUND, ALL tolerances in the system must be optimized simulta-
OVER, between points, nX, n SURFACES, INDICATED) neously in a common candidate spatial relationship, as
have the following system-level effects: described in para. 8.4.1.5. See Figures 8-25 through 8-28.
(a) All profile characteristics in the pattern are treated
as one profile tolerance system 8.4.7 Composite Profile
(b) All profile tolerance zones in the pattern must be Per ASME Y14.5-2009
evaluated simultaneously, in a common candidate config- This provides a composite application of profile toleranc-
uration. ing for the location and constraint (rotation and transla-
The basic reference-body provides a rigid framework tion) of a feature pattern (PLTZF) as well as the
for mutual constraint of the profile tolerance zones. Each interrelation (location, size, form, orientation) or profiled
feature’s true profile is a subset of the basic reference- features within these patterns (FRTZF). [8.6.1.3]
body, and all of the true profiles in the profile tolerance The toleranced feature shall lie within both the PLTZF
system are therefore basically related. The tolerance and the FRTZF.
zones are not permitted to transform relative to the Each complete horizontal segment of a composite profile
true profile. See Figures 8-13 through 8-16. The combina- feature control frame constitutes a separately verifiable
tion of basically related tolerance zones and simultaneous component of multiple interrelated requirements.
evaluation enables the control of mutual orientation and In terms of the system concept, the profile tolerance in
mutual location of the considered features, even in the each segment of a composite feature control frame may be
absence of a datum reference frame. evaluated as a separate profile tolerance system.
8.4.4 Effect of Individual Profile Specifications 8.4.7.1 Pattern Locating Tolerance Zone Framework
(PLTZF). Conformance and actual value are the same as a
Multiple profile tolerances for which a grouping single segment feature control frame.
mechanism does not apply have the following properties:
(a) Each profile tolerance is each treated as a distinct 8.4.7.2 Feature Relating Tolerance Zone Framework
requirement. See Figures 8-17 and 8-18. (FRTZF). The feature relating tolerance zone framework
(b) Each profile tolerance may be optimized individu- has a slightly different behavior, in that only its rotation is
ally, i.e., actual values may be calculated in different candi- controlled by the datums. Per ASME Y14.5-2009
date configurations. See Figures 8-19 and 8-20. If datums are specified in the lower segment(s), they
govern the rotation of the FRTZF relative to the datums
8.4.5 Datum Feature References and within the boundaries established and governed by
the PLTZF. [8.6.1.3(b)(2)]
Referencing a datum feature in a profile feature control
In terms of the profile tolerance system, the FRTZF has
frame adds the following to the profile tolerance system:
an additional transformation:
(a) basic datum feature surfaces (subsets of the basic
– The FRTZF may freely translate relative to the basic
reference-body).
reference-body, in order to achieve conformance or
(b) datum feature simulators. Unless otherwise speci-
optimal actual value. The tolerance zone shall not progress
fied, a datum feature simulator is a surface coincident with
or rotate relative to the basic reference-body. These prop-
or derived from the basic datum feature with the surface
erties make the FRTZF capable of refining form, size, and
normal pointing in the opposite direction.
orientation without refining location. See Figures 8-29
(c) datums and a datum reference frame.
through 8-32.
(d) actual datum feature surfaces.
(e) requirements for contact between the actual datum
features and the simulators, as defined in ASME Y14.5- 8.5 EXTENSION OF TOLERANCE ZONE
2009 and further explained in Section 4 of this Standard. BOUNDARIES FOR SHARP CORNERS
These may involve maximum contact (RMB references) or Per ASME Y14.5-2009
envelopment (MMB or LMB references). Where a profile tolerance encompasses a sharp corner,
The requirements for contact impose a constraint on the the tolerance zone extends to the intersection of the
system that reduces the set of valid candidate spatial rela- boundary lines. [8.3.1]
tionships. See Figure 8-21 through 8-24. In this Standard, a sharp corner is considered to be a
point at which the true profile is not tangent continuous. In
such cases, the boundaries are extended until they inter-
sect. This defines an extension to the tolerance zone. For
planar boundaries, a planar extension is created for each
boundary and trimmed at the intersections. For curved
48
ASME Y14.5.1-2019
boundaries, a default method of creating the extension Table 8-1 Table of Profile Tolerance Dispositions
geometry (linear, constant curvature, other) is not Tolerance Value Disposition tu
defined. Enveloping zones for actual value calculation
t0 Equally disposed tu = t0/2 (implied)
should use the same extension method used for creating
t0 Unequally disposed 0 <tu < t0
the tolerance zone. See Figures 8-33 and 8-34.
t0 Unilateral (outside) tu = t0
t0 Unilateral (inside) tu = 0
8.6 NONUNIFORM TOLERANCE ZONE
(a) Definition. Per ASME Y14.5-2009
A nonuniform tolerance zone is a maximum material
boundary and a least material boundary, of unique
shape, that encompasses the true profile. These boundaries
are defined in a CAD file or by basic dimensions on a drawing
with phantom lines to indicate the tolerance zone. The term
“NONUNIFORM” replaces the tolerance value within the
feature control frame.” [8.3.2]
(b) Conformance. A feature conforms to a nonuniform
profile tolerance if all points on the actual surface are
within the specified tolerance zone.
(c) Actual Value. There is no unique tolerance value
associated with a nonuniform tolerance zone, therefore
a unique actual value is not defined.
49
ASME Y14.5.1-2019
Figure 8-1 Tolerance Zone Derivation — Equally Disposed Profile
Figure 8-2 Tolerance Zone Derivation — Unequally Disposed Profile
50
ASME Y14.5.1-2019
Figure 8-3 Tolerance Zone Derivation — Unilaterally Disposed Profile (Outside)
Figure 8-4 Tolerance Zone Derivation — Unilaterally Disposed Profile (Inside)
51
ASME Y14.5.1-2019
Figure 8-5 Actual Zone Definition for Equally Disposed Profile — Example of Conformance
Figure 8-6 Actual Zone Definition for Unequally Disposed Profile — Example of Conformance
Figure 8-7 Actual Zone for Unilateral (Outside) Profile — Example of Nonconformance
52
ASME Y14.5.1-2019
Figure 8-8 Actual Zone for Unilateral (Inside) Profile — Example of Nonconformance
Figure 8-9 Profile Tolerance for a Single Feature Without a Datum Reference Frame
Figure 8-10 Profile Tolerance Zone for a Single Feature Without a Datum Reference Frame
53
ASME Y14.5.1-2019
Figure 8-11 Candidate Configuration #1 (System Not Optimized)
Figure 8-12 Candidate Configuration #2 (System Optimized)
Figure 8-13 Actual Values in a Multi-Feature Profile Tolerance System — Specification
54
ASME Y14.5.1-2019
Figure 8-14 Actual Values in a Multi-Feature Profile Tolerance System — Basically Related Profile Tolerance Zones
Figure 8-15 Actual Values in a Multi-Feature Profile Tolerance System —

Candidate Configuration #1 (System Not Optimized)
Figure 8-16 Actual Values in a Multi-Feature Profile Tolerance System —

Candidate Configuration #2 (System Optimized)
55
ASME Y14.5.1-2019
Figure 8-17 Individual Requirements for Profile — Specification
Figure 8-18 Individual Requirements for Profile — Tolerance Zones
Figure 8-19 Individual Requirements for Profile — Individual Requirement 1
56
ASME Y14.5.1-2019
Figure 8-20 Individual Requirements for Profile — Individual Requirement 2
Figure 8-21 Datum Feature References With Profile — Specification
Figure 8-22 Datum Feature References With Profile — Tolerance Zones and Simulators
57
ASME Y14.5.1-2019
Figure 8-23 Datum Feature References With Profile — Invalid Candidate Configuration
Figure 8-24 Datum Feature References With Profile — Valid Candidate Configuration
Figure 8-25 Simultaneous Requirements for Profile — Specification
58
ASME Y14.5.1-2019
Figure 8-26 Simultaneous Requirements for Profile — Tolerance Zones
Figure 8-27 Simultaneous Requirements for Profile (System Not Optimized)
Figure 8-28 Simultaneous Requirements for Profile (System Optimized)
59
ASME Y14.5.1-2019
Figure 8-29 Composite Profile Lower Segment — Specification
Figure 8-30 Composite Profile Lower Segment —Tolerance Zones (FRTZF)
Figure 8-31 Composite Profile Lower Segment (System Not Optimized)
60
ASME Y14.5.1-2019
Figure 8-32 Composite Profile Lower Segment (System Optimized)
Figure 8-33 Tolerance Zone Derivation — Profile With Sharp Corner
Figure 8-34 Actual Zone Definition — Profile With Sharp Corner
61
ASME Y14.5.1-2019
Section 9
Tolerances of Runout
9.1 RUNOUT TOLERANCE D1 × P ( A ) =r

Per ASME Y14.5-2009
Runout is a tolerance used to control the functional rela- and
tionship of one or more features to a datum axis established t
from a datum feature specified at RMB. [9.2] (
D1 P B ) 2
The types of features controlled by runout tolerances
include those surfaces constructed around a datum axis where
and those constructed at right angles to a datum axis. [9.3] = a position vector locating the datum axis
A
Surfaces constructed around a datum axis are those = a position vector locating the center of the zone
B
surfaces that are either parallel to the datum axis or
D1 = the direction vector of the datum axis
are at some angle other than 90 deg to the datum axis.
The mathematical definition of runout is separated r = the radial distance from P to the axis
S
into two definitions: one for surfaces constructed t = the size of the zone (i.e., the height of the cylin-
around the datum axis, and one for surfaces constructed drical surface for each surface point)
at right angles to the datum axis. A feature may consist of
surfaces constructed both around and at right angles to (b) Conformance. The circular element through a
the datum axis. Separate mathematical definitions
surface point P S conforms to the circular runout tolerance
describe the controls imposed by a single runout tolerance
on the distinct surfaces that comprise such a feature. t0 if all points of the element lie within some circular
Circular and total runout are addressed in paras. 9.1.1 runout zone as defined above with t = t0. That is, there
and 9.1.2, respectively. exists B such that with t = t0, all points of the surface
Nominal diameters, lengths, radii, and angles establish a element are within the circular runout tolerance zone.
cross-sectional design geometry having perfect form and
A surface conforms to the circular runout tolerance if all
orientation. The design geometry may be translated
circular surface elements conform.
axially or radially, but may not be tilted or scaled with (c) Actual Value. The actual value of circular runout for
respect to the datum axis. The tolerance band is a surface that is constructed at a right angle to a datum axis
equally disposed about this geometry and then revolved is the smallest circular runout tolerance to which it will
around the datum axis, a volumetric tolerance zone or an
conform.
annulus-like surface of revolution is generated.
9.1.1.2 Surfaces Constructed Around a Datum Axis
9.1.1 Circular Runout
(a) Definition. The tolerance zone for each circular
9.1.1.1 Surfaces Constructed Normal to a Datum Axis element on a surface constructed around a datum axis
(a) Definition. The tolerance zone for each circular is generated by revolving a line segment about the
element on (contained within) a surface that is datum axis. The line segment is normal to the desired
constructed at a right angle to a datum axis is generated surface (i.e., the true geometric shape) and is of length
by revolving a line segment about the datum axis. The line t0, where t0 is the specified tolerance. Depending on
segment is parallel to the datum axis and is of length t0, the orientation of the feature, the resulting tolerance
where t0 is the specified tolerance. The resulting tolerance zone will be either a flat annular area, or the surface
zone is the surface of a cylinder of height t0. of a truncated cone. Mapping these two types to
feature geometry, the former is associated with a right
For a surface point P S , a circular runout zone is the
circular cylinder and the latter with a general surface
surface of a cylinder consisting of the set of points P satis- of revolution.
fying the conditions
62
ASME Y14.5.1-2019
ÄÅ ÉÑ
For a surface point P S , a datum axis ÅÅÅÅ A , D1 ÑÑÑÑ, and a given
zone bounded by two parallel planes perpendicular to
ÅÇ ÑÖ the datum axis and separated by the specified tolerance.
mating surface as defined by the true geometric shape, a For a surface constructed at right angles to a datum axis,
circular runout zone for a surface constructed around a
a total runout zone is a volume consisting of the points P
datum axis consists of the set of points P satisfying the
satisfying
conditions t
D1 (P B)
D1 P( B ) 2
= D1 N
P B where
= a position vector locating the mid-plane of the
B
and zone
t D1 = the direction vector of the datum axis
P B d t = the size of the zone (the separation of the parallel
2
planes)
N PS ( B )>0
(b) Conformance. A surface conforms to the total runout
where tolerance t0 if all points of the surface lie within some total
= a position vector locating the datum axis runout zone as defined above with t = t0. That is, there
A
= the point of intersection of the datum axis and the exists B such that with t = t0, all points of the surface
B
line through P S parallel to the direction vector N are within the total runout tolerance zone.
D = the direction vector of the datum axis (c) Actual Value. The actual value of total runout for a
1
surface constructed at right angles to a datum axis is the
d = the distance from B to the center of the zone as
smallest total runout tolerance to which it will conform.
measured parallel to N , where d ≥ t/2
9.1.2.2 Surfaces Constructed Around a Datum Axis
N =
the surface normal at P S determined from the
(a) Definition. A total runout tolerance zone for a
mating surface as defined by the true geometric surface constructed around a datum axis is a volume
shape of revolution generated by revolving an area about the
t = the size of the zone as measured parallel to N datum axis. This area is generated by moving a line
segment of length t0, where t0 is the specified tolerance,
Figure 9-1 illustrates a circular runout zone on a noncy- along the desired contour as defined by the true geometric
lindrical surface of revolution. shape with the line segment kept normal to, and centered
(b) Conformance. The circular element through a on, the desired contour (i.e., the true geometric shape) at
each point. The area that is generated by sweeping the line
surface point P S conforms to the circular runout tolerance segment along the true geometric shape is revolved about
t0 for a given mating surface if all points of the circular the datum axis. The resulting tolerance zone is a volume
element lie within some circular runout zone as defined between two surfaces of revolution separated by the
above with t = t0. That is, there exists d such that with t = t0, specified tolerance, t0.
all points of the circular element are within the circular Given a datum axis defined by the position vector A and
runout tolerance zone.
A surface conforms to a circular runout tolerance t0 if all the direction vector D1, let B be a point on the datum axis
circular elements of the surface conform to the circular locating one end of the desired contour as defined by the
runout tolerance for the same mating surface as true geometric shape, and let r be the distance from the
defined by the true geometric shape. datum axis to the desired contour as defined by the true
(c) Actual Value. The actual value of circular runout for
geometric shape at point B . Then, for a given B and r, let
a surface constructed around a datum axis is the smallest
circular runout tolerance to which it will conform. C(B , r ) denote the desired contour.
9.1.2 Total Runout NOTE: Points on the desired contour as defined by the true
geometric shape can be represented by [d, r + f(d)], where d
9.1.2.1 Surfaces Constructed Normal to a Datum Axis
is the distance along the datum axis from B .
(a) Definition. A total runout tolerance for a surface
constructed at right angles to a datum axis specifies For each possible C(B , r ), a total runout zone is defined
that all points of the surface must lie in a tolerance as the set of points P satisfying the condition
63
ASME Y14.5.1-2019
Figure 9-1 Circular Runout Zone
t
P P (b) Conformance. A surface conforms to a total runout
2 tolerance t0 if all points of the surface lie within some total
where runout zone as defined above with t = t0. That is, there exist
P ʹ
= the projection of onto the surface generated by
P B and r such that with t = t0, all points of the surface are
within the total runout tolerance zone.
rotating C(B , r ) about the datum axis (c) Actual Value. The actual value of total runout for a
t = the size of the zone, measured normal to the surface constructed around a datum axis is the smallest
desired contour as described by the true total runout tolerance to which it will conform.
geometric shape
64
ASME Y14.5.1-2019
NONMANDATORY APPENDIX A
PRINCIPAL CHANGES AND IMPROVEMENTS
A-1 GENERAL A-5 SECTION 3, SYMBOLOGY

The purpose of this Appendix is to provide users a list of A-5.1 No Changes to Section 3
the principal changes and improvements in this revision of
the Standard as compared to the previous issue. The
A-6 SECTION 4, DATUM REFERENCE FRAMES
changes are summarized for each Section or Appendix
in the form of additions, clarifications, extensions of prin- A-6.1 All Figures Redrawn
ciples, or resolution of differences.
All figures were redrawn to show improvements for
A-2 STANDARD Y14 FORMAT clarity and readability.
The format of the Standard has been revised to present A-6.2 Terminology
the material in the sequence presented in ASME Y14.5-
2009. Revised in accordance with ASME Y14.5-2009.
A-6.3 Translation Modifier

A-3 SECTION 1, SCOPE AND DEFINITIONS
A-6.4 Tertiary Datum Simulator
A-3.1 Added Para. 1.2, ASME Y14 Series
Conventions Located and oriented to higher precedence datum.
Explains the ASME Y14 series conventions used in the A-6.5 Added New Datum Feature Types to the DOF
Y14 series of standards and the conventions were applied Tables
throughout this Standard.
Cone and linear extrusion.
A-3.2 Added References
A-6.6 Added Para. 4.7.11, Alternate Stabilization
Added reference to ASME Y14.41-2012, Digital Product Procedures
Definition Data Practices was added in para. 1.3.
The definition was added for a single unique datum with
A-3.3 New or Revised Terms and Definitions a constrained L2 as default.
Definitions and terms have been enhanced by expan-
sion, addition, clarification, and reorganization. A-7 SECTION 5, TOLERANCES OF FORM
A-7.1 Renumbered and Made Into Separate Section

A-4 SECTION 2, GENERAL TOLERANCING AND
RELATED PRINCIPLES Section 5 was previously Section 6 and included Orien-
tation, Profile, and Runout
A-4.1 All Figures Redrawn
A-7.2 All Figures Redrawn
All figures were redrawn to show improvements for
clarity and readability. All figures were redrawn to show improvements for
clarity and readability.
A-4.2 Terminology
A-7.3 Terminology
Revised in accordance with ASME Y14.5-2009
Revised in accordance with ASME Y14.5-2009.
A-4.3 Added Definitions for Actual Local Size
A-7.4 Added Definition
Opposing points and circular element.
Derived Median Plane and Flatness
65
ASME Y14.5.1-2019
A-8 SECTION 6, TOLERANCES OF ORIENTATION A-10.6 Growth Parameter
A-8.1 Renumbered and Made Into Separate Section The growth parameter g was introduced to represent
the amount of length adjustment at each end of the line
Section 5 was previously Section 6 and included Orien- segment.
tation, Profile, and Runout
A-10.7 Relationship Between Actual Zone,
A-8.2 All Figures Redrawn Tolerance Zone, Growth Parameter, and
All figures were redrawn to show improvements for Conformance
clarity and readability. Ranges for the growth parameter g were summarized
for conforming, nonconforming, and barely conforming
A-8.3 Terminology conditions.
Revised in accordance with ASME Y14.5-2009.
A-10.8 Actual Value Formula
A-9 SECTION 7, TOLERANCES OF LOCATION A general formula was defined for the actual value of a
profile tolerance for a single feature, a = t0 + 2g. This
A-9.1 All Figures Redrawn formula applies to uniform profile tolerances of any dispo-
All figures were redrawn to show improvements for sition.
clarity and readability.
A-10.9 Application of Actual Zones
A-9.2 Terminology Examples were added illustrating actual zone definition
Revised in accordance with ASME Y14.5-2009. for equally disposed, unequally disposed, unilateral
outside, and unilateral inside dispositions. Conforming
A-10 SECTION 8, TOLERANCES OF PROFILE and nonconforming examples are shown.
A-10.1 Renumbered and Made Into Separate A-10.10 New Subsection for Profile Applications
Section A new subsection was added to address profile applica-
tions involving multiple features, introducing the concept
Section 8 was previously Section 6 and included Form,
of a profile tolerance specification as a system that is opti-
Orientation, and Runout
mized within constraints.
A-10.2 Reorganization of Section
A-10.11 Basic Reference-Body
The Profile Control section from ASME Y14.5.1M-1994
The new term “basic reference-body” was introduced to
was moved to Nonmandatory Appendix C, Former Prac-
represent the design geometry used as the basis for profile
tices, and replaced with new text and figures.
tolerance zones.
A-10.3 Profile Tolerance Zone Definition
A-10.12 Spatial Relationship
A definition of a uniform profile tolerance zone was
The new term “candidate spatial relationship” was
added, describing a generating line segment swept
introduced to generalize the Y14.5 concept of datum
along the true profile with the disposition indicated by
feature shift to apply to situations in which datum
the specification.
feature references are not present.
A-10.4 Application of Uniform Tolerance Zones
A-10.13 Tolerance Zone Transformation
Examples were added illustrating profile tolerance zone
The term “tolerance zone transformation” was intro-
derivation for equally disposed, unequally disposed,
duced in the context of composite profile, to allow descrip-
unilateral inside, and unilateral outside dispositions.
tion of zones that can translate relative to the basic
A-10.5 Actual Zone Definition reference-body.
A definition of an actual profile zone was added in which A-10.14 Candidate Configurations
the generating line segment is adjusted to just envelop the
The term “candidate configuration” was introduced to
actual surface.
describe a particular state of a profile tolerance system.
66
ASME Y14.5.1-2019
A-10.15 Default Optimization Criteria A-11.2 All Figures Redrawn

A default criteria was defined for optimization of profile All figures were redrawn to show improvements for
tolerance systems for the calculation of actual values, clarity and readability.
which minimizes the maximum value of the growth pa-
rameter g in the system. This is a generalization of the A-11.3 Terminology
minimum-zone criteria defined for single-feature charac- Revised in accordance with ASME Y14.5-2009.
teristics that can be applied to multiple features and mixed
tolerance values.
A-12 NONMANDATORY APPENDICES
A-10.16 Application Examples With No Datum
A-12.1 Nonmandatory Appendix A
Reference Frame
Nonmandatory Appendix A was added to reflect the
Application of the actual zone, growth parameter, opti-
revisions in this revision of the Standard.
mization, and actual value concepts were illustrated in a
series of multi-figure examples. A single-feature system A-12.2 Nonmandatory Appendix B
with no datum reference frame was explained, along
with multi-feature systems with and without grouping. Nonmandatory Appendix B, “Mathematical Datum
Simulators Referenced at RMB,” was added to this revi-
A-10.17 Optimization and Actual Value sion.
Calculation Examples With Datum
Reference Frames A-12.3 Nonmandatory Appendix C
Further multi-figure examples were added to illustrate Nonmandatory Appendix C, “Former Practices,” was
the optimization concepts in the context of datum refer- added to this revision.
ence frames. Examples included a fully constrained DRF,
A-12.4 Nonmandatory Appendix D
partially constrained with simultaneous requirements,
and composite profile lower segment. Nonmandatory Appendix D, “Concepts Related to Size,”
was added to this revision.
A-10.18 Sharp Corners
A-12.5 Nonmandatory Appendix E
A mathematical description was added for the extension
that occurs where the true profile has a sharp corner, with Nonmandatory Appendix E, “A Selection of Mathema-
figures illustrating the tolerance zone derivation and tical Concepts,” was added to this revision.
actual zone definition.
A-12.6 Nonmandatory Appendix F
A-10.19 Nonuniform Tolerance Zone
Nonmandatory Appendix F, “Potential Misuse of the
A definition was added for conformance to a nonuni- Swept-Sphere Definition of Size,” was added to this revi-
form tolerance zone. sion.
A-11 SECTION 9, TOLERANCES OF RUNOUT
A-11.1 Renumbered and Made Into Separate

Section
Section 5 was previously Section 6 and included Orien-
tation, Profile and Runout
67
ASME Y14.5.1-2019
NONMANDATORY APPENDIX B
MATHEMATICAL DATUM SIMULATORS REFERENCED AT RMB:
DEFINITIONS AND PROPERTIES
B-1 INTRODUCTION B-2.1 Concepts of Simulator Definitions

It is very common for tolerances to have datum features In practical datum simulator definitions, it is helpful to
referenced at RMB. Per this Standard (see Section 4), in define the concept of signed distance. The absolute value
these scenarios a possibly-infinite set of candidate datum of the signed distance is the same as normally understood
feature simulators is associated with the datum feature. distance. The sign conveys information about inside
Any of the candidate simulators may be used to evaluate versus outside. In this document, the following convention
the tolerances, and so the task is to identify which simu- is used: negative distances are inside the surface of the
lator, if any, enables the tolerances to pass. datum simulator while positive distances are outside
However, in many scenarios it is desirable to specify a the surface of the datum simulator. See Figure B-1,
single datum feature simulator associated with a datum which illustrates positive and negative distances.
feature referenced at RMB. Having a single datum feature Using the concept of signed distance, a signed residual
simulator corresponds more closely to typical mechanical function rF(P|S) can be defined. That is, given a datum
techniques of evaluating tolerances. For example, a feature’s surface F, and a candidate datum simulator S
metrology engineer might place a datum plane feature and a point P on the feature’s surface F, the residual rF
on a physical datum plane simulator (surface plate), (P|S) is defined as the signed distance from the point P
and if the part “rocks” shims would be used to stabilize to the simulator’s surface S. The first of the desired simu-
the part. lator attributes, the simulator being external to the mate-
Due to the desirability of a single datum feature simu- rial, is then equivalent to requiring the signed residuals to
lator, the purpose of this Appendix is to allow design engi- be positive.
neers to specify a well-defined single simulator associated
with datum features. This Appendix provides several B-2.2 Nonexternal Simulator Definitions
mathematical definitions for single simulators corre- The simplest definitions for single simulators do not
sponding to datums referenced at RMB. The definitions’ guarantee a simulator that is external to the datum
properties are analyzed, and a recommendation is made feature. In fact, they guarantee the simulator intersects
for which definition is likely to be the best in most cases. the datum feature unless the feature has perfect form.
While these nonexternal definitions do not satisfy the re-
B-2 REQUIREMENTS FOR A SINGLE SIMULATOR quirements for a good simulator, they may be useful in
There are several desirable attributes of a definition for some cases, explained below. More importantly, these
a single datum feature simulator. First, the simulator must nonexternal definitions are relatively simple, so they
be external to the material (externality). Second, the simu- form a good base for understanding the external defini-
lator must contact the part (contact). Third, in cases where tions provided later on.
the datum feature “rocks,” the solution chosen must be a B-2.2.1 Least Squares. The simplest definition for a
single solution that minimizes the separation between the single simulator is that of least squares (also called Gaus-
datum feature and the datum simulator (plausibility). sian). The chosen simulator is the surface S that minimizes
Fourth, minute changes in the manufactured part the sum (integral) of squared distances between S and the
should not correspond to large changes in the datum simu- datum feature. That is, S is the surface (e.g., plane, cylin-
lator (repeatability or stability); in other words, two simi- der) that minimizes
larly-manufactured parts should have similar datum 2
simulators.
P F
( )
rF P S dF
See Figure B-2, which illustrates what a least squares

simulator looks like. Two-dimensional simplifications of
common feature types are shown. The least squares
68
ASME Y14.5.1-2019
simulator’s greatest strength is its repeatability across B-2.3 External Simulator Definitions
parts: similarly manufactured parts will almost always
have highly similar datum simulators. Like the simple nonexternal simulator definitions, the
external simulator definitions minimize a norm. However,
B-2.2.2 Min-Max. Another definition for a single simu- they utilize a progression strategy or a constraint to
lator is known as min-max; it is also known as Chebyshev, ensure externality.
minimax, and minimum zone. The min-max simulator
surface S is the surface that minimizes the maximum abso- B-2.3.1 Progression Strategies. Perhaps the simplest
lute distance between S and the datum feature. That is, S way to create an external simulator is to compute a simple
minimizes nonexternal simulator, and allow it to progress in some
way such that it becomes external. The most familiar
max rF(P S) example is sometimes known as translated least
P F
squares: a least squares plane can be translated until it
Per this definition, the worst deviation (also known as is external to the material. Other kinds of progression
residual) is as small as possible. This property is why min- besides translation may be desirable. For example, a
max fitting is used in many tolerance calculations (cylin- least squares cylinder may grow or shrink until it is
dricity, flatness, perpendicularity, etc.). See Figure B-3 for external to the material. See Figure B-5 for illustration.
an illustration of min-max datum feature simulators. It is not always desirable to progress the simulator until
it is wholly external to the material. It may be beneficial to
B-2.2.3 L1 Minimization. The last nonexternal simu- approximate how mating parts fit together. For example, a
lator defined in this Appendix is the L1 (or L1 or L1) mini- datum plane feature typically mates with a corresponding
mization; it is also known as least absolute deviations or as datum plane feature on another part. Typically, the peaks
minimal sum of absolute deviations. The L1 simulator is and valleys of one part will not perfectly line up with those
the surface S that minimizes the sum (integral) of the abso- of the other, allowing some nesting in the texture. This is
lute distances between S and the datum feature. That is, S depicted in Figure B-6. In these situations, it may be
minimizes preferable to not require the datum simulator to be strictly
rF(P S) dF external to the material and instead take a least squares
P F simulator, and translate/progress it until it is mostly, but
not fully, external to the material, as shown in Figure B-7.
The greatest strength of the L1 association is that it can Datum simulators chosen by progression strategies
ignore relatively strong aberrations relatively easily. may be useful, but they are typically not plausible, in
Nicks and ridges in the surface have relatively little the sense that they only rarely correspond to a simulator
effect on the simulator. In a sense, the least squares simu- a metrology engineer might have chosen. For example, a
lator is a “mean” or average of the datum feature surface, least squares plane that is translated to be external to the
while the L1 simulator is a “median” of the datum feature material will typically have only one point of contact with
surface. See Figure B-4 for an illustration of L1 simulators. the simulator. By contrast, a metrology engineer would
B-2.2.4 Norms and Names. The name “L1” comes from usually choose a simulator with two or three points of
the concept of p-norms within vector spaces, where p is a contact.
number between 1 and ∞. The p-norm dp(F, S) of the B-2.3.2 Constrained Norm-Minimizing Strategies.
distances between a datum feature surface F and a simu- Another way to define an external simulator is to add
lator surface S is given by constraints to the optimization problem. Any norm can
be minimized, but a constraint is added such that the simu-
dp(F , S) = p r F ( P S ) p dF lator is required to minimize the norm while being
P F
external to the material.
Any Lp simulator minimizes the corresponding p-norm For example, a constrained least squares (i.e., a
of the distance between the feature surface and the simu- constrained L2) simulator can be defined as the simulator
lator surface. Thus, the above definitions — least squares, S that minimizes
min-max, and L1 minimization — can be rewritten in
rF(P S) 2 dF
consistent terms, where least squares corresponds to P F
the L2 minimization, and min-max corresponds to the
L∞ minimization. subject to the constraint that the simulator lies outside the
material (i.e., subject to the constraint that rF(P|S) ≥ 0 for
every point P on the datum feature).
In other words, minimize the integrated squared
distances, while requiring all the signed distances to
be non-negative. The three main constrained simulator
definitions — constrained L1, constrained L2, and
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ASME Y14.5.1-2019
constrained L∞ — all have interesting properties. In each have very different constrained L1 simulators, depending
case, the objective function comes from the corresponding on which face has a slightly smaller objective function. By
L1, L2, or L∞ nonexternal definition, but the externality contrast, both the constrained least squares and
constraint is added to the optimization problem, that of constrained min-max simulators equalize the gaps on
requiring all signed distances to be non-negative. In most the left and right much like a metrology engineer
cases these constrained definitions are not the same as might do with shims. In general, in two-point rocking
simply progressing the unconstrained definitions, cases, the constrained L1 simulator can have dramatically
which were discussed previously. poor repeatability from part to part, and poor plausibility
For planar features, the constrained L1 case somewhat due to a failure to equalize the rocking condition.
corresponds to a simulator chosen by gravity. If a rigid The poor repeatability of the constrained L1 simulator
infinitely thin sheet were to have the exactly same applies to all types of datum features. Figure B-12 shows a
shape as the datum feature surface, and if this rigid circular 2D simplification and Figure B-13 shows an
sheet were placed on a perfectly flat and level datum simu- opposed-parallel-lines 2D simplification. In each case,
lator, it would be the same result as the constrained L1. the constrained L1 simulator gives a nonequalized solu-
This does not yield a plausible result when the datum tion, and small differences in the actual part can make the
feature surface “rocks” on the datum feature simulator. L1 simulator flip to a different nonequalized solution. The
There is a detailed comparison of how the constrained constrained least squares and constrained min-max simu-
simulators behave below. To summarize that comparison, lators always equalize the solution.
the constrained L1 definition has excellent externality and There are also several situations where the constrained
contact, but only middling plausibility, and poor repeat- min-max simulator performs poorly. These are cases
ability/stability across similar parts. See Figure B-8 for an where a large deviation from perfect form dominates.
illustration of constrained L1 simulators. See Figures B-14, B-15, and B-16 for illustrations. The
By contrast, the constrained min-max (constrained L∞) constrained L1 and constrained least squares simulators
simulator is the same as an unconstrained min-max simu- are both plausible, but the constrained min-max simulator
lator that has been progressed to be external to the mate- moves quite a bit to minimize the extreme deviation. The
rial. Since this definition minimizes the worst-case constrained min-max simulator has relatively poor
distance, it is sensitive to outliers. This simulator has repeatability in these cases as well: different parts will
excellent externality, moderate repeatability across have slightly different dominant deviations, and the simu-
similar parts, and poor contact and plausibility. See lator will rotate to match that. The repeatability is not as
Figure B-9 for an illustration. poor as the constrained L1 V-shaped case.
The constrained least squares simulator (constrained In general, these examples illustrate how the
L2) behaves remarkably well. In cases where there are constrained simulators behave. In many situations,
clear points of contact that a metrology engineer where there are sufficiently good points of contact, and
would choose, the constrained least squares simulator the largest deviation is not too dominant, all three
chooses them. In cases where the part “rocks,” the constrained simulators give the same answer.
constrained least squares simulator chooses an equalized However, in rocking situations, the constrained L1 simu-
solution that a metrology engineer might have chosen. In lator tends to be implausible and not repeatable. In cases
summary of the detailed comparison below, the where the largest deviation is dominant, the constrained
constrained least squares simulator has excellent extern- min-max simulator is implausible and not repeatable. The
ality and contact. It has good plausibility, and good repeat- constrained least squares simulator gives results that are
ability across parts. See Figure B-10 for an illustration of generally plausible and repeatable across various parts.
constrained least squares simulators.
B-2.5 External Envelope Definitions for Datum
B-2.4 Comparison of Constrained Definitions Planes
The strength of the constrained L1 simulator is that it The above definitions all operate directly on the surface
makes full contact with the datum feature. But there are of the part. In some cases, this may be undesirable. For
several situations where the constrained L1 simulator example, if the surface of the datum feature has a recession
performs poorly. An example is that of a planar datum (a void) in it, it might not affect the simulator a metrology
that has two obvious points of contact, and the datum engineer would choose, but it would dominate the
simulator can rock back and forth across the line constrained min-max simulator, and somewhat affect
between those two points. This is a very similar situation the constrained least squares simulator. See
as a four-legged chair that can rock. Figure B-11 illustrates Figure B-17 for an illustration. There is a deep void
a 2D simplification: a V-shaped planar feature. The outside of the constrained L1 points of contact. The
constrained L1 simulator chooses one of the two faces. constrained min-max simulator swings wildly to minimize
Which face is the constrained L1 simulator will often that deviation. The constrained least squares also moves
vary from part to part. Even highly similar parts can towards the void a little bit.
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ASME Y14.5.1-2019
The issue of recessions or voids has led to an idea called is dominated by the edges of the feature, which is
“void-filling.” This idea is only well-defined at present for usually not plausible. Therefore, it is not recommended
datum planes. The convex hull of the surface is deter- to use the constrained min-max simulator with the
mined, and the external-facing portion of that hull can external envelope. Meanwhile, the constrained L1 simu-
be named the “external envelope” or the “void-filled lator is nearly the same whether applied to the surface or
surface.” Datum simulators can then be applied to the to its external envelope. However, the constrained least
external envelope surface. The primary goal of using squares simulator works well when applied to the external
the external envelope is to improve the plausibility of envelope. It ignores voids, and nearly ignores edges. It
the simulator. Figure B-18 illustrates how the constrained remains repeatable, while in many cases improving plau-
simulators can improve when applied to the external sibility.
envelope surface. The constrained L1 simulator is not
affected at all by the change, the constrained least B-2.6 Summary and Recommendations
squares simulator is altered slightly, while the constrained When referencing datums at RMB, it is often beneficial
min-max simulator gets dramatically better, although still to define a single datum simulator instead of a set of candi-
not plausible. date datum simulators. In this Appendix, several possible
Any of the above simulator definitions can be applied to definitions are described, utilizing the L1, L2, and L∞
the external envelope surface instead of the feature norms, optionally utilizing a progression strategy or
surface. For example, a simple (unconstrained) least externality constraint.
squares simulator applied to the external envelope For most datums referenced at RMB, it is recommended
would be mostly external, and highly repeatable. to use the constrained least squares (constrained L2)
Void-filling can improve the plausibility of the simulator. In cases of planar datums referenced at
constrained min-max simulator in some cases, as RMB, the constrained least squares simulator
shown in Figure B-18. On the other hand, it can make (constrained L2) applied to the external envelope is rec-
the simulator less plausible in other cases, as shown in ommended.
Figure B-19. The orientation of the constrained min-
max simulator when applied to the external envelope
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ASME Y14.5.1-2019
Figure B-1 Signed Distances
Figure B-2 Least Squares Simulators
Figure B-3 Min–Max Simulators
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ASME Y14.5.1-2019
Figure B-4 L1 Simulators
Figure B-5 Progressed Least Squares Simulators
Figure B-6 Mating of Actual Parts
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ASME Y14.5.1-2019
Figure B-7 Mostly Progressed Least Squares Simulators
Figure B-8 Constrained L1 Simulators
Figure B-9 Constrained Min–Max Simulators
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ASME Y14.5.1-2019
Figure B-10 Constrained Least Squares Simulators
Figure B-11 Constrained Simulators for a V-Shaped Feature
Figure B-12 Constrained Simulators for a Peanut-Shaped Circular Feature
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ASME Y14.5.1-2019
Figure B-13 Constrained Simulators for an Hourglass-Shaped Opposed Lines Feature
Figure B-14 Constrained Simulators for a Skew-Convex Linear Feature
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ASME Y14.5.1-2019
Figure B-15 Constrained Simulators for a Dented Circular Feature
Figure B-16 Constrained Simulators for a Skew-Convex Opposed Lines Feature
Figure B-17 Large Voids Can Harm Constrained Simulators
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ASME Y14.5.1-2019
Figure B-18 Void-Filling Can Improve Constrained Simulators
Figure B-19 Void-Filling Can Degrade the Constrained Min-Max Simulator
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ASME Y14.5.1-2019
NONMANDATORY APPENDIX C
FORMER PRACTICES
C-1 PROFILE CONTROL profile tolerances either t + or t − equals zero. t + and

t− are always non-negative numbers.
Per ASME Y14.5-2009
A profile is the outline of an object in a given plane (two- The contribution of the nominal surface point P N
dimensional figure). Profiles are formed by projecting a towards the total tolerance zone is a line segment
three-dimensional figure onto a plane or taking cross normal to the nominal surface and bounded by points
sections through the figure. The elements of a profile at distances t+ and t− from P N . The profile tolerance
are straight lines, arcs, and other curved lines. With
profile tolerancing, the true profile may be defined by zone is the union of line segments obtained from each
basic radii, basic angular dimensions, basic coordinate of the points on the nominal surface.
dimensions, basic size dimensions, undimensioned draw- (b) Conformance. A surface conforms to a profile toler-
ings, or formulas. ance t0 if all points P S of the surface conform to either of
(a) Definition. A profile tolerance zone is an area the intermediate tolerances t+ or t− disposed about some
(profile of a line) or a volume (profile of a surface) gener- corresponding point PN on the nominal surface. A point PS
ated by offsetting each point on the nominal surface in a
direction normal to the nominal surface at that point. For conforms to the intermediate tolerance t+ if it is between
unilateral profile tolerances the surface is offset totally in PN and PN + N t+. A point PS conforms to the intermediate
one direction or the other by an amount equal to the tolerance t− if it is between PN and PN N t . Mathema-
profile tolerance. For bilateral profile tolerances the
surface is offset in both directions by a combined tically, this is the condition that there exists some P N on
amount equal to the profile tolerance. The offsets in the nominal surface and some u, −t− ≤ u ≤ t+, for which
each direction may, or may not, be disposed equally. P S = PN + N u.
For a given point P N on the nominal surface there is a (c) Actual Value. For both unilateral and bilateral
unit vector N normal to the nominal surface whose posi- profile tolerances two actual values are necessarily calcu-
tive direction is arbitrary; it may point either into or out of lated: one for surface variations in the positive direction
the material. A profile tolerance t consists of the sum of and one for the negative direction. For each direction, the
two intermediate tolerances t+ and t−. The intermediate actual value of profile is the smallest intermediate toler-
tolerances t+ and t− represent the amount of tolerance to ance to which the surface conforms.
be disposed in the positive and negative directions of the NOTE: No single actual value may be calculated for comparison to
the tolerance value in the feature control frame, except in the
surface normal N , respectively, at P N . For unilateral case of unilateral profile tolerances.
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ASME Y14.5.1-2019
NONMANDATORY APPENDIX D
CONCEPTS RELATED TO SIZE
D-1 GENERAL described by any larger size. In the case of regular features
of size, this can be accomplished by using the same spine to
Although the concept of size seems simple, there are a generate the two boundaries.
number of important concepts related to size that interact Because imperfect (actual) features require two param-
in complex ways. This Appendix attempts to describe eters, ordering by size of actual features is not always
some of the underlying ideas that support the definition possible (see Figure D-1) because containment is not
and use of size in the main body of this Standard. always possible.
Table D-1 describes the relationship between the
D-2 ORDERING OF SIZE VALUES boundaries of nominal and actual features of different
A nominal (theoretical) feature of size may have perfect sizes. The header row identifies nominal features of
form, in which case a unique perfect-form boundary is size with both perfect and imperfect geometry (of the
described for any value of the size parameter. Actual spine) and actual features with imperfect surfaces. The
(imperfect) features of size do not have a unique size pa- next row gives examples of surfaces for each category.
rameter, and are instead described by a pair of size values The following row lists uniqueness of the generated
that describe the limits of the feature’s size. boundary for given size values, and the final row describes
The ordering of size values corresponds directly to set containment relationships between these boundaries.
containment of the perfect-form boundaries described by These concepts are shown graphically in Table D-2. The
these values for the size parameter. A perfect-form header row identifies three different feature instances:
boundary described by a smaller size value can be perfect form, imperfect form with constant actual local
contained in the perfect-form boundary described by size, and actual imperfect features. The next row has illus-
any larger size value. trations showing how size is realized, and the following
For nominal features of size that have a single size value row indicates whether these features are unique for a
for the entire feature, but where perfect form is not certain size value. The final row states conditions
required, the containment of a smaller size value under which containment can be attained for features
boundary will be assured for at least one boundary with different sizes.
Figure D-1 Size Ordering for Perfect and Imperfect Features of Size
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ASME Y14.5.1-2019
Table D-1 Uniqueness and Containment Relationships for Features of Size — Part 1
Features With a
Single Size Value
Feature Boundaries F(s1) Nominal Features (Other Parameters, Such as Actual Features
and F(s2) With Sizes s1 > s2 With Perfect Form the Spine, May Vary) (Imperfect Form)
Example features Spheres, 2-D line Cylindrical FOS tolerance zones, Actual part features will be imperfect,
profiles and 3-D parallel-plane FOS tolerance regardless of the classification of their
surface profiles, MMC zones, swept (extruded) nominal surface.
extent of features profiles (e.g., a 2-D line profile
where Rule #1 applies on a 3-D surface)
Uniqueness si and F(si) have 1:1 si and F(si) have 1: many Actual features F each have one
correspondence correspondence actual mating size and one
actual least material size.
Containment of feature surfaces F (s1) ⊃ F (s2) For any F(s1), there exists at Perfect form sizes (actual
least one F(s2) such that mating size, actual minimum
F (s1) ⊃ F (s2). material size) satisfy
For any F(s2), there exists at least relationship of F (s1) ⊃ F (s2).
one F(s1) such that F (s1) ⊃ F (s2).
Table D-2 Uniqueness and Containment Relationships for Features of Size — Part 2
Features With the
Nominal Features Same Size at
Property With Perfect Form All Cross Sections Actual (Imperfect) Features
Graphical
representation
Size = S2
Size = S1 Smax Smin
Size = S2
Unique? Yes: one size = one geometry No: one size = many shapes One feature = one maximum size and one minimum size.
(e.g., different spines) Multiple possible shapes, especially for minimum size.
Containment? S1 > S2 implies feature with S1 > S2 implies feature with size S1 The actual feature can be contained in some ideal feature
size S1 can contain feature can contain at least one feature with maximum size, and will contain at least one
with size S2 with size S2 feature with the minimum size at all cross sections
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ASME Y14.5.1-2019
NONMANDATORY APPENDIX E
A SELECTION OF MATHEMATICAL CONCEPTS
E-1 WHY THIS APPENDIX? the slope can’t be defined. The idea of continuity can also
be applied to surfaces. See Figures E-2 and E-3.
This Standard presents some fairly complex mathema-
tical tools to describe tolerance zones, actual values, and
other tolerancing-related concepts. The terms used in this
E-4 SUPPORT
Standard are intended to convey precise concepts that are A surface of support is defined in Section 1 of this Stan-
mathematically rigorous, but the fundamental ideas are dard. This means that the surface is entirely on one side of
often quite simple. Some of these terms are described a feature. This idea is introduced because — unlike real
in this Appendix as an aid to the reader. The descriptions part surfaces — theoretical surfaces (such as the xy-plane)
are not rigorous, but simply intended to give a high-level don’t have material on either side. The concept of support
view of the terms. is used to allow us to create a theoretical datum surface
that doesn’t interfere with the real part.
E-2 MANIFOLDS
E-5 REGULARIZED SETS
A manifold is a geometric object that, no matter where
we look at it, locally looks like a line, plane, or solid, To make a lot of geometric modeling work, we need to
depending on its dimension. A one-dimensional manifold keep careful track of the solids we are modeling and the
locally looks like a line; a two-dimensional manifold locally boundaries of these solids. Boundaries need to be mani-
looks like a plane, and so on. folds so there is a continuous boundary everywhere
Manifolds are used to describe well-behaved lines and between the solid and the surrounding air. When we
surfaces in this Standard. For example, a line shaped like a think of CAD packages constructing solids from primitives,
figure 8 is not a one-dimensional manifold, because if we we would like the result of joining these primitives to have
look closely at the intersection point, it doesn’t look like a the same nice boundary properties.
line. Similarly, a planar surface with a “hanging edge” is not Regularized set operations, such as union, treat the
a 2D manifold because the edge doesn’t look like a plane. interior and the boundary of objects differently.
On the other hand, a sphere is a two-dimensional manifold, Figure E-4 shows a two-dimensional object, its interior,
because anywhere we look at it — if we are close enough and its boundary. The interior is two-dimensional, and
— the surface is like a plane (it has two sides). Figure E-1 the boundary is a one-dimensional manifold.
shows examples of what is and is not a one‐dimensional In Figure E-5 we see the result of performing the union
manifold. operation between two objects that are touching along
Models of workpiece surfaces are usually two-dimen- one edge. The edges between the objects are joined,
sional manifolds, so that useful properties can be defined, and the resulting boundary is no longer a one-dimensional
such as inside vs. outside. The interior of bounded objects manifold. This is a problem when we want to perform
such as line segments can be manifolds, but the endpoints mathematical operations on either the interior or the
(boundaries) must be excluded as they no longer look like boundary of the result.
a line at the boundary. More discussion of interiors and The solution to the problem shown in Figure E-5 is to
boundaries is in subsection E-5. perform a union of the object interiors, and shown in
Figure E-6, and then find the boundary of the resulting
E-3 CONTINUITY two-dimensional object. The regularized set operation
results in maintaining the well-behaved properties of
The idea of continuity reflects that a property exists the object and its boundary.
continuously along a line or over a surface. A line is contin- Let’s refer to the two objects above as A and B, with
uous if it has no breaks in it. The tangent to a line is the interiors and boundaries iA, iB, bA, and bB, respectively.
“slope” of the line, or the direction that we’d be going if we So A = iA ∪ bA and B = iB ∪ bB. The regularized union
were traveling along the line. A line is tangent-continuous of objects A and B is not A ∪ B as shown in
if the slope can be uniquely defined everywhere along the Figure E-5, but the union of the interiors, surrounded
line — this means that there are no “sharp corners” where by the boundary of this union as shown in Figure E-6.
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ASME Y14.5.1-2019
Regularization of sets usually involves finding the interior wrapped in its boundary, is exactly what is shown in
of the object, and wrapping it with a boundary. For Figure E-6.
example, the interior of the result in Figure E-5,
Figure E-1 Examples of One-Dimensional Manifold (Left) and Not (Right)
Figure E-2 A Line Which Is Continuous (Left) and Not (Right)
Figure E-3 A Line Which Is Tangent Continuous (Left) and Not (Right)
Figure E-4 An Object, Its Interior, and Its Boundary
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ASME Y14.5.1-2019
Figure E-5 The Result of Union Between Two Objects, and the Non-Manifold Boundary
Figure E-6 The Interior of Two Objects, and Their Union With a Manifold Boundary
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ASME Y14.5.1-2019
NONMANDATORY APPENDIX F
POTENTIAL MISUSE OF THE SWEPT-SPHERE DEFINITION OF
SIZE
F-1 GENERAL cation, Rule #1 requires that the MMC spine be straight.
Two possible spines are shown in Figure F-2; recall that
Spines are a fundamental building block in the defini- any spines that result in conformance of the feature of size
tion of size, in determining conformance to size specifica- to its specification are acceptable.
tions, and finding actual size values. This Appendix The regions (or volumes) swept by the spheres moving
provides some cautionary guidance on how spines are along their respective spines are shown in Figure F-3. This
used in constructing zones for size evaluation. particular choice of spines resulted in the feature surface
Size based on the swept-sphere concept provides a way being completely outside the LMC region, and completely
to relate a pair of size values to the imperfect geometry of inside the MMC region.
an actual part. The spines often are indirectly specified; for As a consequence of the spines chosen, the set difference
example, the standard states “if there exist two spines …” between the swept regions results in a size tolerance zone
but does not describe how to find these spines. This that contains the surface of the feature of size (shown in
indirect (or existential) definition of the size zone is Figure F-4), indicating conformance to the size specifica-
similar to the zone created for flatness: the zone only tion.
has to exist, the standard does not state how to find it.
This Appendix describes three cases (there may be
F-2 THE SHORT SPINE
more) where the spine definition allows conformance
or actual value definitions that are inconsistent with If the LMC spine does not extend the entire length of the
the intent of ASME Y14.5 and with common sense. feature of size, a tolerance zone may be formed where the
These are to be avoided, although mathematical rules feature is contained in the tolerance zone, although it does
for doing this are not provided. not meet the size requirements. The MMC and LMC
The first two of these cases are based on research spheres specified by a size tolerance, and an actual
performed by H. Voelcker and K. Suresh reported in part to be evaluated for conformance are shown in
the journal Manufacturing Review in 1994. Figure F-5.
The following 2D example shows conformance to a size In this case shown in Figure F-6, a spine is chosen for the
tolerance zone. In this case perfect form at MMC is shown. LMC evaluation that does not extend the entire length of
The 2D part can be thought of as a slice of a 3D external the feature.
feature of size (i.e., a pin or a plate). In Figures F-1 through Figure F-7 shows the regions swept by the LMC and
F-4, the use of the LMC ball, MMC ball, and their respective MMC spheres along the length of their respective spines.
spines is shown to create a zone in which conformance to a The size tolerance zone obtained by the difference
size specification can be verified. The figures show the between the two regions completely contains the
manner in which the swept-sphere definition was actual part surface, but it is not the intent of swept-
intended to be used. sphere size to allow this determination. See Figure F-8.
The paragraphs that follow describe how the spine-
based definition of size limits might be misused to incor- F-3 THE SPACE-FILLING SPINE
rectly determine conformance to the size specification.
These cases are classified as For external features of size where perfect form at MMC
(a) the short spine (subsection F-2), is not required, it is possible for the MMC spine to meander
(b) the space-filling spine (subsection F-3), and and create a zone larger than was intended by the size
(c) the nonsurround case (subsection F-4). specification. The analogous case for internal features
The MMC and LMC spheres specified by a size tolerance, of size occurs regardless of whether perfect form at
and an actual part to be evaluated for conformance are MMC is required. The MMC and LMC spheres specified
shown in Figure F-1. The first step in constructing the by a size tolerance, and an actual part to be evaluated
size tolerance zone is to choose spines along which the for conformance are shown in Figure F-9. In this case,
spheres are to be swept. In the absence of other specifi- the part clearly exceeds the MMC size limit.
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ASME Y14.5.1-2019
In Figure F-10, the LMC and MMC spines are shown, but Figure F-1 MMC and LMC Spheres, and a Part to Be Tested
the MMC spine does not follow the general shape of the for Conformance
feature axis, rather it meanders around, with the swept
region filling a large area as shown in Figure F-11.
Because of the large area swept by the MMC sphere, the
part surface is completely contained in the difference
between the MMC and LMC regions, resulting in an incor-
rect determination of conformance, shown in Figure F-12.
F.1
F-4 THE NONSURROUND CASE
The final case examined in this Appendix is where a Figure F-2 LMC and MMC Spines
feature of size has a very different size than was intended
to be controlled by the size specification. This is shown in
Figure F-13, where the part is very small compared to the
specified size limits.
The incorrect conformance determination comes from
the entire part surface being contained in the difference F.1
between the MMC and LMC regions as shown in
Figure F-14. This is referred to as violating the surrounds Figure F-3 LMC and MMC Swept Regions
condition, as the part itself does not surround the LMC
region (for an external feature of size).
F-5 POSSIBLE REMEDIES

F.1
While the cases described in this Appendix are given as
cautionary examples, the development of rules for appro-
Figure F-4 Conformance to the Size Tolerance
priates spines and containment can be envisioned for
future use. For example, the spines may be required to
cover the full extent of a feature of size, and the swept
LMC region might be required to be a subset of the
feature of size.
F.1
Figure F-5 MMC and LMC Spheres, and a Part to Be Tested

for Conformance
F.2
Figure F-6 LMC and MMC Spines
F.2
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ASME Y14.5.1-2019
Figure F-7 LMC and MMC Swept Regions Figure F-11 LMC and MMC Swept Regions
F.2
Figure F-8 Incorrect Determination of Conformance to

the Size Tolerance F.3

the Size Tolerance
F.2
Figure F-9 MMC and LMC Spheres, and a Part to Be Tested

for Conformance
F.3
Figure F-13 MMC and LMC Spheres, and a Part to Be
Tested for Conformance
Figure F-10 LMC and MMC Spines
F.4
F.3
the Size Tolerance
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ASME Y14.5.1-2019
INTENTIONALLY LEFT BLANK
88
Y14 ENGINEERING PRODUCT DEFINITION AND RELATED
DOCUMENTATION PRACTICES
Y14.1-2012 Decimal Inch Drawing Sheet Size and Format

Y14.1M-2012 Metric Drawing Sheet Size and Format
Y14.2-2014 (R2020) Line Conventions and Lettering
Y14.3-2012 (R2018) Orthographic and Pictorial Views
Y14.5-2018 Dimensioning and Tolerancing
Y14.5.1-2019 Mathematical Definition of Dimensioning and Tolerancing Principles
Y14.5.2-2017 Certification of Geometric Dimensioning and Tolerancing Professionals
Y14.6-2001 (R2018) Screw Thread Representation
Y14.8-2009 (R2014) Castings, Forgings, and Molded Parts
Y14.24-2012 Types and Applications of Engineering Drawings
Y14.31-2014 (R2019) Undimensioned Drawings
Y14.34-2013 (R2018) Associated Lists
Y14.35-2014 (R2019) Revision of Engineering Drawings and Associated Documents
Y14.36-2018 Surface Texture Symbols
Y14.37-2019 Product Definition for Composite Parts
Y14.38-2019 Abbreviations and Acronyms for Use in Product Definition and Related Documents
Y14.41-2019 Digital Product Definition Data Practices
Y14.43-2011 Dimensioning and Tolerancing Principles for Gages and Fixtures
Y14.44-2008 (R2014) Reference Designations for Electrical and Electronics Parts and Equipment
Y14.46-2017 Product Definition Practices for Additive Manufacturing
Y14.47-2019 Model Organization Practices
Y14.100-2017 Engineering Drawing Practices
Y32.7-1972 (R2014) Graphic Symbols for Railroad Maps and Profiles
Y32.18-1972 (R2013) Symbols for Mechanical and Acoustical Elements as Used in Schematic Diagrams
The ASME Publications Catalog shows a complete list of all the Standards published by the Society. For a complimentary catalog, or the latest
information about our publications, call 1-800-THE-ASME (1-800-843-2763).
ASME Y14.5.1-2019

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ASME Y14.5.

Engineering Product Definition and

Two Park Avenue • New York, NY • 10016 USA

The next edition of this Standard is scheduled for publication in 2022.

ASME is the registered trademark of The American Society of Mechanical Engineers.

No part of this document may be reproduced in any form,

The American Society of Mechanical Engineers

Section 1 Scope and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

STANDARDS COMMITTEE OFFICERS

STANDARDS COMMITTEE PERSONNEL

SUBCOMMITTEE 5.1 — MATHEMATICAL DEFINITION OF DIMENSIONING AND TOLERANCING

Secretary, Y14 Standards Committee

1.1 SCOPE (g) The phrase “unless otherwise specified” or “UOS”

1.4.2 Algebraic Notation or

A vector can be expanded into scalar components. Let î, r( P ) = (P )

1.5.5 Candidate Datum Set 1.5.13 Element, Line

1.5.8 Derived Median Line 1.5.17 Feature

1.5.21 Perfect Form 1.5.29 True Position

1.7 FORMAT (b) Conformance — mathematical definition of the

(a) Definition — narrative and mathematical descrip-

Figure 1-2 Example: Cutting Surfaces to Evaluate Circular Runout

Figure 1-3 Examples of a Planar Surface of Support

2.1 SURFACE POINTS Per ASME Y14.5-2009, an IFOSb is “a directly toleranced

Figure 2-1 Irregular Features of Size (Type a), Collection of Features

Figure 2-4 Examples of Surface Attributes Leading to

Figure 2-6 Profile of a Surface Tolerance Describing an External IFOSb

There are no concepts in Section 3 of ASME Y14.5-2009

4.1 GENERAL 4.4.1 Datum Feature Simulator Requirements

Table 4-2 Point as Primary Datum (Spherical Datum Feature)

Table 4-3 Axis as Primary Datum (Cylindical Datum Feature)

Table 4-4 Plane as Primary Datum (Planar or Width Datum Feature)

Table 4-6 Axis and Plane as Primary Datum (Linear

Table 4-7 Axis Point and Plane as Primary Datum

Figure 4-1 Example From Table 4-3 — Axis as Primary Datum

Figure 4-2 Example of Testing Whether a Plane Is a Valid Datum Plane

Figure 4-4 Example of Translation Modifier

5.1 GENERAL Form tolerances are applicable to single (individual)

5.2 FORM CONTROL 5.4.1 Straightness

(a) Definition. A cylindricity tolerance specifies that all

5.4.4 Cylindricity (b) Conformance. A feature conforms to a cylindricity

Figure 5-1 Evaluation of Straightness of a Cylindrical Surface

Figure 5-2 Evaluation of Straightness of a Planar Surface

tion zone. and

Figure 6-3 Cylindrical Orientation Zone With Respect to a

Figure 6-4 Linear Orientation Zone Bounded by Parallel Lines

( ) is the distance to the true position

sponding feature, including any effects of feature size.

(b) Conformance. A cylindrical feature conforms to a elongated-hole cylinders (unconstrained in location or

Figure 7-1 First Illustration of the Difference Between

Table 7-2 Size of Position Tolerance Zone — Surface Interpretation

Figure 7-5 Tolerance Zone and Conformance: Holes at

Table 7-3 Size of Position Tolerance Zone — Resolved Geometry Interpretation

Table 7-4 Definition of Verifying Volume for Projected

External r( P ) > w r( P ) < w

Table 7-6 Size of Conical Tolerance Zone — Surface

Table 7-8 Definition of Bidirectional Tolerance Zone —

External r( P ) > b r( P ) < b

Table 7-9 Size of Bidirectional Tolerance Zone — Surface Interpretation

Figure 7-11 Definition of the Tolerance Zone for Polar