INTERNATIONAL ISO

STANDARD 14405-3

First edition
2016-12-15

Geometrical product specifications

(GPS) — Dimensional tolerancing —
Part 3:
Angular sizes
Spécification géométrique des produits (GPS) — Tolérancement
dimensionnel —
Partie 3: Tailles angulaires
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Reference number
ISO 14405-3:2016(E)

© ISO 2016
 ISO 14405-3:2016(E)


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Contents Page

Foreword......................................................................................................................................................................................................................................... iv
Introduction...................................................................................................................................................................................................................................v
1 Scope.................................................................................................................................................................................................................................. 1
2 Normative references....................................................................................................................................................................................... 2
3 Terms and definitions...................................................................................................................................................................................... 2
4 Specification modifiers and symbols............................................................................................................................................11
5 Default specification operator for angular size.................................................................................................................13
5.1 General......................................................................................................................................................................................................... 13
5.2 ISO default specification operator for angular size............................................................................................... 14
5.3 Drawing-specific default specification operator for angular size............................................................. 15
6 Drawing indication...........................................................................................................................................................................................16
6.1 Drawing indication for special specification operators for angular size............................................ 16
6.2 Indication of the toleranced feature on which the angular size characteristic is defined... 17
Annex A (normative) Association criteria for two-line angular size (for a revolute or
prismatic feature of size)...........................................................................................................................................................................18
Annex B (informative) Differences between two planes considered as an angular feature of
size and two planes considered as two single features..............................................................................................19
Annex C (informative) Overview diagram for angular size.........................................................................................................20
Annex D (informative) Relation to ISO GPS matrix..............................................................................................................................21
Bibliography.............................................................................................................................................................................................................................. 22
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Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment,
as well as information about ISO’s adherence to the World Trade Organization (WTO) principles in the
Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html.
The committee responsible for this document is Technical Committee ISO/TC 213, Dimensional and
geometrical product specifications and verification.
ISO 14405 consists of the following parts, under the general title Geometrical product specification
(GPS) — Dimensional tolerancing:
— Part 1: Linear sizes
— Part 2: Dimensions other than linear sizes
— Part 3: Angular sizes
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Introduction
This part of ISO 14405 is a geometrical product specification (GPS) standard and is to be regarded as a
general GPS standard (see ISO 14638). In the general GPS matrix, it influences chain links “Symbols and
indications”, “Feature requirements” and “Feature properties” of the size chain of standards.
The ISO/GPS Masterplan given in ISO  14638 gives an overview of the ISO/GPS system of which this
part of ISO 14405 is a part. The fundamental rules of ISO/GPS given in ISO 8015 apply to this part of
ISO 14405 and the default decision rules given in ISO 14253-1 apply to specifications made in accordance
with this part of ISO 14405, unless otherwise indicated.
For more detailed information on the relation of this part of ISO 14405 to other standards and to the
GPS matrix model, see Annex D.
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 INTERNATIONAL STANDARD ISO 14405-3:2016(E)

Geometrical product specifications (GPS) — Dimensional

tolerancing —
Part 3:
Angular sizes

1 Scope
This part of ISO  14405 establishes the default specification operator for angular size and defines a
number of special specification operators for features of angular size: cone (truncated, i.e. frustum, or
not), wedge (truncated or not), two opposite straight lines (intersection of a wedge/truncated wedge and
a plane perpendicular to the intersection straight line of the two planes of the wedge/truncated wedge,
intersection of a cone/frustum and a plane containing the axis of revolution of the cone/frustum). See
Figure 1 and Figure 2.
This part of ISO 14405 also defines the specification modifiers and the drawing indications for these
angular sizes.
This part of ISO 14405 covers the following angular sizes:
— local angular size:
— angular size between two lines;
— portion angular size;
— global angular size:
— direct global angular size:
— least squares angular size;
— minimax angular size;
— rank order angular size/indirect global angular size:
— maximum angular size;
— minimum angular size;
— average angular size;
— range of angular sizes;
— mid-range angular size;
— median angular size;
— standard deviation of angular size.
This part of ISO 14405 defines the meaning of tolerances of angular sizes indicated as
— + and/or - limit deviations, e.g. 0°/-0,5°, or
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— indicated with upper limit of size (ULS) and/or lower limit of size (LLS), e.g. 35° max. or 15° min.,
34°/36°,

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— with or without modifiers.

This part of ISO 14405 provides a set of tools to express several types of angular size characteristics.
It does not give any information on the relationship between a function or a use and an angular size
characteristic.

2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 8015, Geometrical product specifications (GPS) — Fundamentals — Concepts, principles and rules
ISO 17450-1, Geometrical product specifications (GPS) — General concepts — Part 1: Model for geometrical
specification and verification
ISO 17450-2, Geometrical product specifications (GPS)  — General concepts  — Part  2: Basic tenets,
specifications, operators, uncertainties and ambiguities
ISO 17450-3, Geometrical product specification (GPS) — General concepts — Part 3: Toleranced Features
ISO 14405-1:2016, Geometrical product specifications (GPS)  — Dimensional tolerancing  — Part  1:
Linear sizes
ISO 14405-2, Geometrical product specifications (GPS) — Dimensional tolerancing — Part 2: Dimensions
other than linear sizes

3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 8015, ISO 17450-1, ISO 17450-2,
ISO 17450-3, ISO 14405-1, ISO 14405-2 and the following apply.
For association criteria, it is assumed that the terms “least squares” and “Gaussian” are equivalent, as
well as “minimax” and “Chebyshev”. In this document, the terms “least squares” and “minimax” are
retained. The least squares criterion is understood without material constraint throughout this part of
ISO 14405.
3.1
angular size
angular dimension of a cone or between two coplanar opposite straight lines or between two opposite
non parallel planes
Note 1 to entry: The angular size is defined from nominal features or from associated features which are angular
features of size.

Note 2 to entry: See example of angular size on Figure 1 and Figure 2.

Note 3 to entry: Definition of “angular feature of size (feature of angular size)” is given in ISO 17450-1, the angle
dimension cannot be 0° or 180°.

Note 4 to entry: Angular features of size are of two types.

— Revolute angular feature of size: a cone or a frustum. Two opposite straight lines are established from
a longitudinal section of a cone/frustum with a plane containing the associated axis of revolution of the
cone/frustum.

— Prismatic angular feature of size: a wedge (truncated or not). Two opposite straight lines are established
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from a cross section of a wedge/truncated wedge with a plane perpendicular to the intersection straight line
of the two associated planes of the wedge/truncated wedge.

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Note 5 to entry: Figure 1 and Figure 2 illustrate angular features of size type wedge, cone, frustum and two lines.

Note 6 to entry: Figure 3a and Figure 3b illustrate the case of an angular feature of size and an angular distance
between two planes which is not an angular feature of size and shows that an angular feature of size exists when
the material directions are opposed (when one of the features is rotated around their intersection line to coincide
with the other feature, then the material is on the opposite sides for the two features). See also Annex B.

Note 7 to entry: Envelope requirement cannot be applied for angular features of size.

a)   Wedge b)   Cone

c)   Truncated wedge d)   Frustum

Figure 1 — Examples of areal angular features of size

a)   Two opposite lines of a wedge b)   Two opposite generatrix lines of a cone

Figure 2 — Examples of angular features of size

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a)   Example # 1 b)   Example # 2

Key
1 Feature of angular size
2 Non-feature of angular size

Figure 3 — Examples of possible angular features of size

3.2
local angular size
local angular size characteristic
angular size characteristic having a unique value for a specific location and a non-unique value along
and/or around the angular feature of size
Note 1 to entry: For a given feature, an infinite number of local angular sizes exists.

Note 2 to entry: In Figure 4, an example of local angular size is shown.

Note 3 to entry: Two local angular size associations can be specified: least squares local angular size and minimax
local angular size. See Annex A.
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Key
1 real angular feature of size
2 associated planes with (1)
3 intersection straight line of (2)
4 perpendicular cross section to (3)
5 two extracted lines
6 two associated straight lines
7 two-line angular size

Figure 4 — Two-line angular size

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Key
1 real angular feature of size
2 portion of (1)
3 associated planes with (2)
4 intersection straight line of (3)
5 portion angular size

Figure 5 — Portion angular size

3.2.1
two-line angular size
local angular size between two lines
angle between two associated straight lines established from two extracted lines obtained from the
intersection with an intersection plane defined from the associated angular feature of size
Note 1 to entry: See Figure 4.

Note 2 to entry: The process for establishing the two-line angular size depends on the invariance class of the
feature: revolute surface or prismatic surface.

Note 3 to entry: The default specification operator defining the two-line angular size is described in Annex A.

3.2.1.1
two-line revolute angular size
two-line angular size (3.2.1) where the straight lines are associated with two extracted lines resulting
from the intersection of an extracted revolute feature with a plane containing its associated axis
Note 1 to entry: The axis of the associated revolute feature is the “directly associated median line” described in
ISO 22432:2011, 3.5.1.2.4.
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3.2.1.2
two-line prismatic angular size
two-line angular size (3.2.1) where the straight lines are associated with two extracted lines resulting
from the intersection of two extracted surfaces with a plane perpendicular to their associated straight
line of intersection
3.2.2
portion angular size
global angular size (3.3) for a given portion of the extracted angular feature of size
Note  1  to entry:  A portion angular size is a local angular size when taking into account the complete angular
feature of size and a global angular size when taking into account only the specific portion.

Note 2 to entry: See Figure 5.

Note 3 to entry: A portion angular size can be a direct global angular size (3.3.1) or an indirect global angular size
(3.3.2).

Note 4 to entry: The operator for defining a portion is not covered by this part of ISO 14405.

3.3
global angular size
global angular size characteristic
angular size characteristic having one unique value for the entire toleranced angular feature of size
3.3.1
direct global angular size
direct global angular size characteristic
global angular size defined from one association on the angular feature of size
Note 1 to entry: If the angular feature of size is a cone, the associated feature is a cone. If the angular feature of
size is a wedge, the associated features are a pair of planes. One plane is associated independently to each side of
the wedge.

Note 2 to entry: See Figure 6.

Note 3 to entry: Different association criteria lead to different results. The association criteria described in this
part of ISO 14405 are the least squares criterion and minimax criterion.
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Key
1 real angular features of size
2 associated planes with (1)
3 intersection straight line of (2)
4 direct global angular size defined in a plane perpendicular to the intersection straight line

Figure 6 — Direct global angular sizes

3.3.1.1
least squares global angular size
direct global angular size using the least squares association criterion
Note 1 to entry: See Figure 7 and Figure 9.

3.3.1.2
minimax global angular size
direct global angular size using the minimax association criterion
Note 1 to entry: See Figure 8 and Figure 10.

Note 2 to entry: Figure 7 to Figure 10 are the illustrations of two global angular sizes defined on the same real
angular feature of size.
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Key
1 real angular features of size
2 associated planes with (1)
3 intersection straight line of (2)
4 least squares global angular size

Figure 7 — Least squares global angular size 3D view

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Key
1 real angular features of size
2 associated planes with (1)
3 intersection straight line of (2)
4 minimax global angular size

Figure 8 — Minimax global angular size 3D view

Key
1 least squares global angular size

Figure 9 — Least squares global angular size 2D view

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Key
1 minimax global angular size

Figure 10 — Minimax global angular size 2D view

3.3.2
indirect global angular size
indirect global angular size characteristic
rank-order angular size
rank order angular size characteristic
global angular size defined mathematically from a homogeneous set of local size values obtained along
and/or around the angular feature of size
Note 1 to entry: A rank-order angular size can define a unique value along and/or around the angular feature of
size from a local angular size for a portion or for the entire angular feature of size.

Note 2 to entry: A rank-order angular size can define a local angular size from another local angular size (for
example a rank order angular size from a two-line angular size taken in any portion).

Note 3 to entry: For further details on the subtypes of rank order size, see ISO 14405-1.

Note 4 to entry: A rank-order angular size can be, for example, the average of all two-line angular size values on
an extracted conical surface.

4 Specification modifiers and symbols

For the purposes of this part of ISO  14405, the specification modifiers and symbols given in Table 1
and Table 2 apply. To define in a dimensional specification a specific type of angular size characteristic
available for upper and/or lower limit specification, modifiers or symbols shall be used in the sequence
given in Table 3 (see also Annex C).
The combination of these modifiers and symbols is described in Clause 5 and Clause 6.
Rules for the presentation of graphical symbols are given in ISO 14405-1:2016, Annex A and Annex E.
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Table 1 — Specification modifiers for angular size

Modifier Description
Two-line angular size with minimax
association criterion
Two-line angular size with least squares
association criterion
Global angular size with least squares asso-
ciation criterion
Global angular size with minimax associa-
tion criterion
Maximum angular sizea
Minimum angular sizea
Average angular sizea
Median angular sizea
Mid-range angular sizea
Range of angular sizesa
Standard deviation of angular sizea b
a Rank-order angular size can be used as a supplement to portion angular
size or global portion angular size or local angular size.
b SQ from root mean square.

Table 2 — General specification modifiers for angular size

Examples of indication
Description Symbol
Prismatic angular feature of size Revolute angular feature of size
Any restrict-
ed portion of
/linear distance 35° ±1°/15a 35° ±1°/15a
angular fea-
ture of size
Any restrict-
ed portion of
/angular distance Not applicable 35° ±1°/15°a
angular fea-
ture of size
Specific fixed
SCS 45° ±2° SCS Not applicable
cross section
More than
one angular
Number × 2× 45° ±2° 2× 45° ±2°
feature of
size
Common
toleranced
CT 2× 45° ±2° CT 2× 45° ±2° CT
feature of
angular size
Free-state
b 35° ± 1° 35° ± 1°
condition
Between 35° ± 1° A B 35° ± 1° A B
a /linear distance applies to prismatic features of size and to revolute features of size along the axis of the revolute
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feature. /angular distance applies to revolute feature of size.

b See ISO 10579.

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Table 3 — Sequence of modifiers to refer to a specific angular size characteristic

Type of angular
Additional defini-
size Subtype Associated modifiers
tion
characteristic
Two-line
angular size or
With least squares
association crite- Example:
riona
Local angular size With minimax
Portion an-
association crite- Example:
gular size of
riona
length L
Rank order angular
size of local angle ; Example:
angular size estab-
lished on a portion
With least squares
Example
Direct global association criterion
angular size With minimax asso-
Global angular ciation criterion Example
size Rank order angular Examples
Indirect global size based upon
 
angular size local angular size

a For the second association operation, the first association operation is always defined by the least squares criterion.

5 Default specification operator for angular size

5.1 General
When the basic GPS indication is used for angular size, the default specification operator for angular
size applies. The default specification operator for angular size can be as follows:
— the ISO default GPS specification operator (see 5.2 and ISO 8015);
— the drawing-specific default GPS specification operator (see 5.3);
— the altered default GPS specification operator (see ISO 8015).
The basic GPS specification for angular size has no specification modifier attached and can be one of
four types; see Table 4.
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Table 4 — Different basic GPS specifications for angular size

GPS basic specification for angular size Examples
35° ±1°

Nominal angular size ± deviation limitsa

       +1°
35°−2°
36°
Values of upper and lower limits of angular sizea
34°
Values of upper or lower limits of angular sizea 45° max 32° min
A nominal angular size which is neither indicated in brackets 45° and
nor as a theoretically exact dimension (TED) and a reference
to general tolerancing in or near the drawing title blockb. ISO 2768-f (in or near the drawing title block)
a Nominal angular size and deviation limit should be indicated with the unit indication (decimal degrees or
degrees/minutes/seconds).
b See ISO 2768-1 for information on general tolerancing.

5.2 ISO default specification operator for angular size

The ISO default specification operator for angular size is the “two-line angular size” with minimax
association criterion, see also Annex A.
The ISO default specification operator for angular size applies when there is no indication on the
drawing referring to another default specification for angular size as defined in 5.3.
If the two-line angular size is applied for both specified limits, the modifier can be omitted.
See Figure 11 a) and Figure 11 b).
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a) Indication b) Interpretation
Key
1 real feature
2 associated feature with minimax criterion without material constraint
3 associated feature with minimax criterion with outside material constraint
4 associated feature with minimax criterion with inside material constraint
5 two-line angular size
6 angular dimension

Figure 11 — ISO default specification operator for angular size

In case of minimax criterion, there is no impact on the result of evaluation of angular size when material
constraint is used or not. See also Figure 11.
The tolerance applies to all cross sections where an angle exists along the two real integral surfaces
and all such angles shall be contained within the tolerance interval.

5.3 Drawing-specific default specification operator for angular size

When a drawing-specific default specification operator for angular size applies, it shall be indicated on
the drawing in or near the title block in the following order:
— reference to ISO 14405;
— the specification modifier(s) for the chosen default specification definition of angular size.
See also Figure 12.
In order to specify different default specification operators for linear and/or angular size, it shall be
indicated before the reference to this International Standard by adding the words “linear size” and/or
“angular size”.
EXAMPLE “Angular size ISO 14405”.

See also Figure 12 and Figure 13.

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NOTE 1 The default specification operator for angular size for this drawing is changed to the least squares
angular size.

NOTE 2 The changed default specification operator is the default angular size, unless otherwise stated.

NOTE 3 The default specification for this drawing applies to angular sizes only.

Figure 12 — Example of change of default specification operator for angular size for the
entire drawing

To facilitate the reading of the drawing, it is possible to show all other types of modifiers used on the
drawing by listing them in brackets after the specific default specification indication. See Figure 13.

Figure 13 — Example of change of default specification operator for both linear and angular
size for the entire drawing

6 Drawing indication

6.1 Drawing indication for special specification operators for angular size
A tolerance indication for angular size applies to one single complete feature of angular size. It is
possible to indicate
— the tolerance applies to any restricted part or a fixed restricted part of the feature of angular size, or
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— the tolerance applies to more than one feature of angular size; see Figure 14 a) and Figure 14 b).

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a) Same requirement for two separate b) Common requirement for the two
features of angular size features of angular size
evaluated individually considered as one (CT modifier)

Figure 14 — Example of tolerance applied to more than one angular feature of size

When the ISO default specification operator for angular size does not apply, the tolerance indication
shall include specification modifiers (see Table 1 and Table 2) indicating which special specification
operator(s) apply.
The rules for sequence and indication of modifiers (see Table 1 and Table 2) for angular size shall
respect those defined in ISO 14405-1:2016, Clause 6.

6.2 Indication of the toleranced feature on which the angular size characteristic is
defined
When the toleranced feature is the complete feature, no additional indication is necessary (see
Figure 15).

Figure 15 — Example of requirement for the complete angular feature of size

When the tolerance feature is a part of the complete feature, indications to define a cross-section or a
portion according to ISO 14405-1:2016, 7.2 to 7.3 shall be applied.
When a requirement applies individually to more than one angular feature of size, then rules about the
use of the modifier for the number of repeated features “n×” from ISO 14405-1:2016, 7.6 shall be applied.
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When a requirement applies simultaneously to multiple angular features of size, considered as one,
then the CT modifier shall be applied in accordance with ISO 14405-1:2016, 7.7.

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Annex A
(normative)

Association criteria for two-line angular size (for a revolute or

prismatic feature of size)

Two-line angular size

Two sequential association operations are used to define a two-line angular size. See Table A.1.
a) To associate the real angular features of size with a feature of the same nature.
b) To associate the extracted lines (coming from intersection of the real angular features of size
and the intersection plane) with two straight lines. See Figure A.1 for an example with minimax
criterion.
For the first association (a): the association criterion is the least squares association criterion
(without material constraint, see Clause 3) for the two extracted surfaces of a prismatic feature of size
(independently of each other) and for the extracted surface of a revolute feature of size.
NOTE The first association with a least squares criterion is to establish either a stable intersection line for
the prismatic feature of size, see Figure 4, or a stable axis for the revolute feature of size.

For the second association (b): the second association criterion can be either a minimax association
criterion (default case) or a least squares association criterion (without material constraint, see
Clause 3). The two lines are associated independently of each other.

Key
1 extracted lines
2 associated lines

Figure A.1 — Two-line angular size: Second association with minimax criterion

Table A.1 — Symbols for two-line angular size

Second association
First association specification dependent
mandatory
Minimax = default case Least squares
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Least squares
Two-line angular size
no symbol

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Annex B
(informative)

Differences between two planes considered as an angular feature

of size and two planes considered as two single features

Drawing indication Feature of angular size

Yes
The two planes are considered together to build a feature
of size and the angle is considered as variable.

No
The two planes are considered independently. One plane
is orientated from the other. The TED angle defines the
tolerance zone.

No
The two planes are considered independently. One plane
is orientated from the other. The TED angle defines the
tolerance zone.

No
The two planes are considered simultaneously. The TED
angle defines the combined tolerance zone.
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Annex C
(informative)

Overview diagram for angular size

Figure C.1 presents the relations between angular size characteristics.

Key


→ rank order operator
dd dimension defining the portion

Figure C.1 — Overview diagram for angular size

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Annex D
(informative)

Relation to ISO GPS matrix

D.1 General
For full details about the GPS matrix model, see ISO 14638.
The ISO/GPS Masterplan given in ISO  14638 gives an overview of the ISO/GPS system of which this
part of ISO 14405 is a part. The fundamental rules of ISO/GPS given in ISO 8015 apply to this part of
ISO 14405 and the default decision rules given in ISO 14253-1 apply to specifications made in accordance
with this part of ISO 14405, unless otherwise indicated.

D.2 Information about the standard and its use

This part of ISO 14405 establishes the default definition of angular size, special definitions and drawing
indications for dimensional angular specification applied to a feature of angular size (e.g. wedge, cone
or type two opposite integral lines defined by intersection of a plane with the surface of the feature of
angular size).

D.3 Position in the GPS matrix model

This part of ISO 14405 is a general GPS standard which influences chain links “Symbols and indications”,
“Feature requirements” and “Feature properties” of the size chain of standards in the general GPS
matrix, as illustrated in Table D.1.

Table D.1 — Fundamental and general ISO GPS standards

Chain links
A B C D E F G
Conform-
Symbols Feature Measure-
Feature ance and Measure- Calibra-
and require- ment
properties non-con- ment tion
indications ments equipment
formance
Size • • •
Distance
Form
Orientation
Location
Run-out
Profile sur-
face texture
Areal surface
texture
Surface im-
perfections
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 ISO 14405-3:2016(E)


Bibliography

[1] ISO 2768-1, General tolerances  — Part  1: Tolerances for linear and angular dimensions without
individual tolerance indications
[2] ISO 10579, Geometrical product specifications (GPS)  — Dimensioning and tolerancing  — Non-
rigid parts
[3] ISO 14253-1, Geometrical product specifications (GPS) — Inspection by measurement of workpieces
and measuring equipment — Part 1: Decision rules for proving conformance or non-conformance
with specifications
[4] ISO 14638, Geometrical product specifications (GPS) — Matrix model
[5] ISO 22432:2011, Geometrical product specifications (GPS) — Features utilized in specification and
verification
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 ISO 14405-3:2016(E)

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ICS 17.040.10
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