Line (geometry)

In geometry, the notion of line or straight

line was introduced by ancient
mathematicians to represent straight
objects (i.e., having no curvature) with
negligible width and depth. Lines are an
idealization of such objects, which are
often described in terms of two points

(e.g., ) or referred to using a single

letter (e.g., ).[1][2]
The red and blue lines on this graph have the same
slope (gradient); the red and green lines have the
same y-intercept (cross the y-axis at the same place).

A representation of one line segment.

Until the 17th century, lines were defined

as the "[…] first species of quantity, which
has only one dimension, namely length,
without any width nor depth, and is
nothing else than the flow or run of the
point which […] will leave from its
imaginary moving some vestige in length,
exempt of any width. […] The straight line
is that which is equally extended between
its points."[3]

Euclid described a line as "breadthless

length" which "lies equally with respect to
the points on itself"; he introduced several
postulates as basic unprovable properties
from which he constructed all of geometry,
which is now called Euclidean geometry to
avoid confusion with other geometries
which have been introduced since the end
of the 19th century (such as non-
Euclidean, projective and affine geometry).

In modern mathematics, given the

multitude of geometries, the concept of a
line is closely tied to the way the geometry
is described. For instance, in analytic
geometry, a line in the plane is often
defined as the set of points whose
coordinates satisfy a given linear equation,
but in a more abstract setting, such as
incidence geometry, a line may be an
independent object, distinct from the set
of points which lie on it.
When a geometry is described by a set of
axioms, the notion of a line is usually left
undefined (a so-called primitive object).
The properties of lines are then
determined by the axioms which refer to
them. One advantage to this approach is
the flexibility it gives to users of the
geometry. Thus in differential geometry, a
line may be interpreted as a geodesic
(shortest path between points), while in
some projective geometries, a line is a 2-
dimensional vector space (all linear
combinations of two independent
vectors). This flexibility also extends
beyond mathematics and, for example,
permits physicists to think of the path of a
light ray as being a line.

Definitions versus
descriptions
All definitions are ultimately circular in
nature, since they depend on concepts
which must themselves have definitions, a
dependence which cannot be continued
indefinitely without returning to the
starting point. To avoid this vicious circle,
certain concepts must be taken as
primitive concepts; terms which are given
no definition.[4] In geometry, it is frequently
the case that the concept of line is taken
as a primitive.[5] In those situations where
a line is a defined concept, as in
coordinate geometry, some other
fundamental ideas are taken as primitives.
When the line concept is a primitive, the
behaviour and properties of lines are
dictated by the axioms which they must
satisfy.

In a non-axiomatic or simplified axiomatic

treatment of geometry, the concept of a
primitive notion may be too abstract to be
dealt with. In this circumstance, it is
possible to provide a description or mental
image of a primitive notion, to give a
foundation to build the notion on which
would formally be based on the (unstated)
axioms. Descriptions of this type may be
referred to, by some authors, as definitions
in this informal style of presentation.
These are not true definitions, and could
not be used in formal proofs of
statements. The "definition" of line in
Euclid's Elements falls into this category.[6]
Even in the case where a specific
geometry is being considered (for
example, Euclidean geometry), there is no
generally accepted agreement among
authors as to what an informal description
of a line should be when the subject is not
being treated formally.
In Euclidean geometry
When geometry was first formalised by
Euclid in the Elements, he defined a
general line (straight or curved) to be
"breadthless length" with a straight line
being a line "which lies evenly with the
points on itself".[7] These definitions serve
little purpose, since they use terms which
are not by themselves defined. In fact,
Euclid himself did not use these
definitions in this work, and probably
included them just to make it clear to the
reader what was being discussed. In
modern geometry, a line is simply taken as
an undefined object with properties given
by axioms,[8] but is sometimes defined as
a set of points obeying a linear
relationship when some other
fundamental concept is left undefined.

In an axiomatic formulation of Euclidean

geometry, such as that of Hilbert (Euclid's
original axioms contained various flaws
which have been corrected by modern
mathematicians),[9] a line is stated to have
certain properties which relate it to other
lines and points. For example, for any two
distinct points, there is a unique line
containing them, and any two distinct lines
intersect in at most one point.[10] In two
dimensions (i.e., the Euclidean plane), two
lines which do not intersect are called
parallel. In higher dimensions, two lines
that do not intersect are parallel if they are
contained in a plane, or skew if they are
not.

Any collection of finitely many lines

partitions the plane into convex polygons
(possibly unbounded); this partition is
known as an arrangement of lines.

On the Cartesian plane …

Lines in a Cartesian plane or, more

generally, in affine coordinates, can be
described algebraically by linear
equations.

In two dimensions, the equation for non-

vertical lines is often given in the slope-
intercept form:

where:

m is the slope or gradient of the line.

b is the y-intercept of the line.
x is the independent variable of the
function y = f(x).

The slope of the line through points

and , when
, is given by
and the equation of this line can be written
.

In , every line (including vertical

lines) is described by a linear equation of
the form

with fixed real coefficients a, b and c such

that a and b are not both zero. Using this
form, vertical lines correspond to the
equations with b = 0.

There are many variant ways to write the

equation of a line which can all be
converted from one to another by
algebraic manipulation. These forms (see
Linear equation for other forms) are
generally named by the type of information
(data) about the line that is needed to
write down the form. Some of the
important data of a line is its slope, x-
intercept, known points on the line and y-
intercept.

The equation of the line passing through

two different points and
may be written as

If x0 ≠ x1, this equation may be rewritten as

or

In three dimensions, lines can not be

described by a single linear equation, so
they are frequently described by
parametric equations:

where:
 x, y, and z are all functions of the
independent variable t which ranges
over the real numbers.
(x0, y0, z0) is any point on the line.
a, b, and c are related to the slope of the
line, such that the vector (a, b, c) is
parallel to the line.

They may also be described as the

simultaneous solutions of two linear
equations

such that and

are not proportional (the relations
 imply
). This follows since in three
dimensions a single linear equation
typically describes a plane and a line is
what is common to two distinct
intersecting planes.

In normal form …

The normal form (also called the Hesse

normal form,[11] after the German
mathematician Ludwig Otto Hesse), is
based on the normal segment for a given
line, which is defined to be the line
segment drawn from the origin
perpendicular to the line. This segment
joins the origin with the closest point on
the line to the origin. The normal form of
the equation of a straight line on the plane
is given by:

where θ is the angle of inclination of the

normal segment (the oriented angle from
the unit vector of the x axis to this
segment), and p is the (positive) length of
the normal segment. The normal form can
be derived from the general form
by dividing all of the
coefficients by
Unlike the slope-intercept and intercept
forms, this form can represent any line but
also requires only two finite parameters, θ
and p, to be specified. If p > 0, then θ is
uniquely defined modulo 2π. On the other
hand, if the line is through the origin (c = 0,
p = 0), one drops the c/|c| term to
compute sinθ and cosθ, and θ is only
defined modulo π.

In polar coordinates …

In polar coordinates on the Euclidean

plane the slope-intercept form of the
equation of a line is expressed as:

where m is the slope of the line and b is

the y-intercept. When θ = 0 the graph will
be undefined. The equation can be
rewritten to eliminate discontinuities in
this manner:

In polar coordinates on the Euclidean

plane, the intercept form of the equation of
a line that is non-horizontal, non-vertical,
and does not pass through pole may be
expressed as,
where and represent the x and y
intercepts respectively. The above
equation is not applicable for vertical and
horizontal lines because in these cases
one of the intercepts does not exist.
Moreover, it is not applicable on lines
passing through the pole since in this
case, both x and y intercepts are zero
(which is not allowed here since and
are denominators). A vertical line that
doesn't pass through the pole is given by
the equation
Similarly, a horizontal line that doesn't
pass through the pole is given by the
equation

The equation of a line which passes

through the pole is simply given as:

where m is the slope of the line.

As a vector equation …

The vector equation of the line through

points A and B is given by
(where λ is a scalar).
If a is vector OA and b is vector OB, then
the equation of the line can be written:
.

A ray starting at point A is described by

limiting λ. One ray is obtained if λ ≥ 0, and
the opposite ray comes from λ ≤ 0.

In Euclidean space …

In three-dimensional space, a first degree

equation in the variables x, y, and z defines
a plane, so two such equations, provided
the planes they give rise to are not parallel,
define a line which is the intersection of
the planes. More generally, in n-
dimensional space n-1 first-degree
equations in the n coordinate variables
define a line under suitable conditions.

In more general Euclidean space, Rn (and

analogously in every other affine space),
the line L passing through two different
points a and b (considered as vectors) is
the subset

The direction of the line is from a (t = 0) to

b (t = 1), or in other words, in the direction
of the vector b − a. Different choices of a
and b can yield the same line.
Collinear points …

Three points are said to be collinear if they

lie on the same line. Three points usually
determine a plane, but in the case of three
collinear points this does not happen.

In affine coordinates, in n-dimensional

space the points X=(x1, x2, ..., xn), Y=(y1, y2,
..., yn), and Z=(z1, z2, ..., zn) are collinear if
the matrix

has a rank less than 3. In particular, for

three points in the plane (n = 2), the above
matrix is square and the points are
collinear if and only if its determinant is
zero.

Equivalently for three points in a plane, the

points are collinear if and only if the slope
between one pair of points equals the
slope between any other pair of points (in
which case the slope between the
remaining pair of points will equal the
other slopes). By extension, k points in a
plane are collinear if and only if any (k–1)
pairs of points have the same pairwise
slopes.
In Euclidean geometry, the Euclidean
distance d(a,b) between two points a and b
may be used to express the collinearity
between three points by:[12][13]

The points a, b and c are collinear if and

only if d(x,a) = d(c,a) and d(x,b) = d(c,b)
implies x=c.

However, there are other notions of

distance (such as the Manhattan distance)
for which this property is not true.

In the geometries where the concept of a

line is a primitive notion, as may be the
case in some synthetic geometries, other
methods of determining collinearity are
needed.

Types of lines …

In a sense,[14] all lines in Euclidean

geometry are equal, in that, without
coordinates, one can not tell them apart
from one another. However, lines may play
special roles with respect to other objects
in the geometry and be divided into types
according to that relationship. For
instance, with respect to a conic (a circle,
ellipse, parabola, or hyperbola), lines can
be:
 tangent lines, which touch the conic at a
single point;
secant lines, which intersect the conic at
two points and pass through its interior;
exterior lines, which do not meet the
conic at any point of the Euclidean
plane; or
a directrix, whose distance from a point
helps to establish whether the point is
on the conic.

In the context of determining parallelism in

Euclidean geometry, a transversal is a line
that intersects two other lines that may or
not be parallel to each other.
For more general algebraic curves, lines
could also be:

i-secant lines, meeting the curve in i

points counted without multiplicity, or
asymptotes, which a curve approaches
arbitrarily closely without touching it.

With respect to triangles we have:

the Euler line,

the Simson lines, and
central lines.

For a convex quadrilateral with at most

two parallel sides, the Newton line is the
line that connects the midpoints of the two
diagonals.

For a hexagon with vertices lying on a

conic we have the Pascal line and, in the
special case where the conic is a pair of
lines, we have the Pappus line.

Parallel lines are lines in the same plane

that never cross. Intersecting lines share a
single point in common. Coincidental lines
coincide with each other—every point that
is on either one of them is also on the
other.

Perpendicular lines are lines that intersect

at right angles.
In three-dimensional space, skew lines are
lines that are not in the same plane and
thus do not intersect each other.

In projective geometry
In many models of projective geometry,
the representation of a line rarely
conforms to the notion of the "straight
curve" as it is visualised in Euclidean
geometry. In elliptic geometry we see a
typical example of this.[15] In the spherical
representation of elliptic geometry, lines
are represented by great circles of a
sphere with diametrically opposite points
identified. In a different model of elliptic
geometry, lines are represented by
Euclidean planes passing through the
origin. Even though these representations
are visually distinct, they satisfy all the
properties (such as, two points
determining a unique line) that make them
suitable representations for lines in this
geometry.

Extensions

Ray …

Given a line and any point A on it, we may

consider A as decomposing this line into
two parts. Each such part is called a ray
and the point A is called its initial point. It
is also known as half-line, a one-
dimensional half-space. The point A is
considered to be a member of the ray.[16]
Intuitively, a ray consists of those points
on a line passing through A and
proceeding indefinitely, starting at A, in one
direction only along the line. However, in
order to use this concept of a ray in proofs
a more precise definition is required.

Given distinct points A and B, they

determine a unique ray with initial point A.
As two points define a unique line, this ray
consists of all the points between A and B
(including A and B) and all the points C on
the line through A and B such that B is
between A and C.[17] This is, at times, also
expressed as the set of all points C such
that A is not between B and C.[18] A point D,
on the line determined by A and B but not
in the ray with initial point A determined by
B, will determine another ray with initial
point A. With respect to the AB ray, the AD
ray is called the opposite ray.

Thus, we would say that two different

points, A and B, define a line and a
decomposition of this line into the disjoint
union of an open segment (A, B) and two
rays, BC and AD (the point D is not drawn
in the diagram, but is to the left of A on the
line AB). These are not opposite rays since
they have different initial points.

In Euclidean geometry two rays with a

common endpoint form an angle.

The definition of a ray depends upon the

notion of betweenness for points on a line.
It follows that rays exist only for
geometries for which this notion exists,
typically Euclidean geometry or affine
geometry over an ordered field. On the
other hand, rays do not exist in projective
geometry nor in a geometry over a non-
ordered field, like the complex numbers or
any finite field.

Line segment …

A line segment is a part of a line that is

bounded by two distinct end points and
contains every point on the line between
its end points. Depending on how the line
segment is defined, either of the two end
points may or may not be part of the line
segment. Two or more line segments may
have some of the same relationships as
lines, such as being parallel, intersecting,
or skew, but unlike lines they may be none
of these, if they are coplanar and either do
not intersect or are collinear.

Geodesics …

The "shortness" and "straightness" of a

line, interpreted as the property that the
distance along the line between any two of
its points is minimized (see triangle
inequality), can be generalized and leads
to the concept of geodesics in metric
spaces.

See also
Affine function
 Curve
Distance between two lines
Distance from a point to a line
Imaginary line (mathematics)
Incidence (geometry)
Line coordinates
Line (graphics)
Line segment
Locus
Plane (geometry)
Polyline
Rectilinear (disambiguation)

Notes
1. "Compendium of Mathematical
Symbols" . Math Vault. 2020-03-01.
Retrieved 2020-08-16.
2. Weisstein, Eric W. "Line" .
mathworld.wolfram.com. Retrieved
2020-08-16.
3. In (rather old) French: "La ligne est la
première espece de quantité, laquelle
a tant seulement une dimension à
sçavoir longitude, sans aucune latitude
ni profondité, & n'est autre chose que
le flux ou coulement du poinct, lequel
[…] laissera de son mouvement
imaginaire quelque vestige en long,
exempt de toute latitude. […] La ligne
droicte est celle qui est également
estenduë entre ses poincts." Pages 7
and 8 of Les quinze livres des
éléments géométriques d'Euclide
Megarien, traduits de Grec en
François, & augmentez de plusieurs
figures & demonstrations, avec la
 corrections des erreurs commises és
autres traductions, by Pierre Mardele,
Lyon, MDCXLV (1645).
4. Coxeter 1969, p. 4
5. Faber 1983, p. 95
. Faber 1983, p. 95
7. Faber, Appendix A, p. 291.
. Faber, Part III, p. 95.
9. Faber, Part III, p. 108.
10. Faber, Appendix B, p. 300.
11. Bôcher, Maxime (1915), Plane Analytic
Geometry: With Introductory Chapters
on the Differential Calculus , H. Holt,
p. 44, archived from the original on
2016-05-13.
12. Alessandro Padoa, Un nouveau
système de définitions pour la
géométrie euclidienne, International
Congress of Mathematicians, 1900
13. Bertrand Russell, The Principles of
Mathematics, p. 410
14. Technically, the collineation group acts
transitively on the set of lines.
15. Faber, Part III, p. 108.
1 . On occasion we may consider a ray
without its initial point. Such rays are
called open rays, in contrast to the
typical ray which would be said to be
closed.
17. Wylie, Jr. 1964, p. 59, Definition 3
1 . Pedoe 1988, p. 2

References

Wikisource has the text of the 1911

Encyclopædia Britannica article Line.

Coxeter, H.S.M (1969), Introduction to

Geometry (2nd ed.), New York: John
Wiley & Sons, ISBN 0-471-18283-4
 Faber, Richard L. (1983), Foundations of
Euclidean and Non-Euclidean Geometry,
New York: Marcel Dekker, ISBN 0-8247-
1748-1
Pedoe, Dan (1988), Geometry: A
Comprehensive Course, Mineola, NY:
Dover, ISBN 0-486-65812-0
Wylie, Jr., C.R. (1964), Foundations of
Geometry, New York: McGraw-Hill,
ISBN 0-07-072191-2

External links

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related to Lines.
 "Line (curve)" , Encyclopedia of
Mathematics, EMS Press, 2001 [1994]
Equations of the Straight Line at Cut-
the-Knot

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