Line (geometry)

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OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, a line is an infinitely long object with no width, depth, or
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a Surface (mathematics), surface deviates from being a plane (g ...
. Thus, lines are one-dimensional objects, though they may exist in two,
three 3 is a number, numeral, and glyph. 3, three, or III may also refer to: * AD 3, the third year of the AD era * 3 BC, the third year before the AD era * March, the third month Books * ''Three of Them'' (Russian: ', literally, "three"), a 1901 no ...
, or higher
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
spaces. The word ''line'' may also refer to a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., $\overleftrightarrow$) or by a single letter (e.g., $\ell$).
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several
postulate An axiom, postulate, or assumption is a statement that is taken to be true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated c ...
s as basic unprovable properties from which he constructed all of geometry, which is now called
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
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 mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric space, metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main propertie ...
). 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 In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, a line in the plane is often defined as the set of points whose coordinates satisfy a given
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b,a_1,\ldots,a_n are the coefficients, ...
, but in a more abstract setting, such as
incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence (geometry), inc ...
, 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
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s, 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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...
, a line may be interpreted as a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
(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.

# Properties

When geometry was first formalised by
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in the '' Elements'', he defined a general line (now called a ''
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
'') to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". 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
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s, but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined. In an
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
atic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), 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. In two
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s (i.e., the Euclidean
plane Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
), two lines which do not intersect are called
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...
. In higher dimensions, two lines that do not intersect are parallel if they are contained in a
plane Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
, or skew if they are not. On an
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometry, geometric setting in which two real number, real quantities are required to determine the position (geometry), position of each point (mathematics), ...
, a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into
convex polygon In geometry, a convex polygon is a polygon that is the Boundary (topology), boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In ...
s (possibly unbounded); this partition is known as an
arrangement of lines In music Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, d ...
.

## In higher dimensions

In
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, 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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...
variables define a line under suitable conditions. In more general
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, R''n'' (and analogously in every other
affine space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
), the line ''L'' passing through two different points ''a'' and ''b'' (considered as vectors) is the subset $L = \left\$ 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 Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
, but in the case of three collinear points this does ''not'' happen. In affine coordinates, in ''n''-dimensional space the points ''X'' = (''x''1, ''x''2, ..., ''x''''n''), ''Y'' = (''y''1, ''y''2, ..., ''y''''n''), and ''Z'' = (''z''1, ''z''2, ..., ''z''''n'') are collinear if the matrix $\begin 1 & x_1 & x_2 & \cdots & x_n \\ 1 & y_1 & y_2 & \cdots & y_n \\ 1 & z_1 & z_2 & \cdots & z_n \end$ has a
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
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 In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In p ...
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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
''d''(''a'',''b'') between two points ''a'' and ''b'' may be used to express the collinearity between three points by: :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 A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences ...
) for which this property is not true. In the geometries where the concept of a line is a
primitive notion In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
, as may be the case in some synthetic geometries, other methods of determining collinearity are needed.

## Types

In a sense, 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
(a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
,
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
,
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, or
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A h ...
), lines can be: *
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
s, which touch the conic at a single point; *
secant line Secant is a term in mathematics derived from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (th ...
s, 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 curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s, lines could also be: * ''i''-secant lines, meeting the curve in ''i'' points counted without multiplicity, or *
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
s, which a curve approaches arbitrarily closely without touching it. With respect to
triangles A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
we have: * the
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line (mathematics), line determined from any triangle that is not equilateral triangle, equilateral. It is a Central line (geometry), central line of the triangle, and it passes thr ...
, * the Simson lines, and * central lines. For a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
with at most two parallel sides, the
Newton line In Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach co ...
is the line that connects the midpoints of the two
diagonal In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field o ...
s. () For a
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") or sexagon (from Latin , meaning "six") is a six-sided polygon or 6-gon creating the outline of a cube. The total of the internal angles of any ...
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 In geometry, parallel lines are coplanar straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curve ...
are lines in the same plane that never cross.
Intersecting lines In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field o ...
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 angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s. In
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
,
skew lines In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges ...
are lines that are not in the same plane and thus do not intersect each other.

## In axiomatic systems

The concept of line is often considered in geometry as a
primitive notion In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
in axiomatic systems, meaning it is not being defined by other concepts. In those situations where a line is a defined concept, as in
coordinate geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, 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
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s 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 The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
falls into this category. Even in the case where a specific geometry is being considered (for example,
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
), 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.

# Definition

## Linear equation

Lines in a Cartesian plane or, more generally, in affine coordinates, are characterized by linear equations. More precisely, every line $L$ (including vertical lines) is the set of all points whose
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...
(''x'', ''y'') satisfy a linear equation; that is, $L = \,$ where ''a'', ''b'' and ''c'' are fixed
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s (called
coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
s) such that ''a'' and ''b'' are not both zero. Using this form, vertical lines correspond to equations with ''b'' = 0. One can further suppose either or , by dividing everything by if it is not zero. There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the ''standard form''. If the constant term is put on the left, the equation becomes $ax + by - c = 0,$ and this is sometimes called the ''general form'' of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms. These 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 $P_0\left( x_0, y_0 \right)$ and $P_1\left(x_1, y_1\right)$ may be written as $(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0).$ If , this equation may be rewritten as $y=(x-x_0)\,\frac+y_0$ or $y=x\,\frac+\frac\,.$In
two dimensions In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, the equation for non-vertical lines is often given in the ''
slope-intercept form In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
'': $y = mx + b$ where: * ''m'' is the
slope In mathematics, the slope or gradient of a line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...
or
gradient In vector calculus, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "d ...
of the line. * ''b'' is the
y-intercept In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used i ...
of the line. * ''x'' is the
independent variable Dependent and independent variables are variables in mathematical modeling, statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample d ...
of the function . The slope of the line through points $A\left(x_a, y_a\right)$ and $B\left(x_b, y_b\right)$, when $x_a \neq x_b$, is given by $m = \left(y_b - y_a\right)/\left(x_b - x_a\right)$ and the equation of this line can be written $y = m \left(x - x_a\right) + y_a$.

## Parametric equation

Parametric equations are also used to specify lines, particularly in those in three dimensions or more because in more than two dimensions lines ''cannot'' be described by a single linear equation. In three dimensions lines are frequently described by parametric equations: $\begin x &= x_0 + at \\ y &= y_0 + bt \\ z &= z_0 + ct \end$ where: * ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers. * (''x''0, ''y''0, ''z''0) is any point on the line. * ''a'', ''b'', and ''c'' are related to the slope of the line, such that the direction
vector Vector most often refers to: *Euclidean vector In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
(''a'', ''b'', ''c'') is parallel to the line. Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector. As a note, lines in three dimensions may also be described as the simultaneous solutions of two
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b,a_1,\ldots,a_n are the coefficients, ...
s $a_1 x + b_1 y + c_1 z - d_1 = 0$ $a_2 x + b_2 y + c_2 z - d_2 = 0$ such that $\left(a_1,b_1,c_1\right)$ and $\left(a_2,b_2,c_2\right)$ are not proportional (the relations $a_1 = t a_2, b_1 = t b_2, c_1 = t c_2$ imply $t = 0$). This follows since in three dimensions a single linear equation typically describes a
plane Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
and a line is what is common to two distinct intersecting planes.

## Hesse normal form

The ''normal form'' (also called the ''Hesse normal form'', 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 Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
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: $x \cos \varphi + y \sin \varphi - p = 0 ,$ where $\varphi$ is the angle of inclination of the normal segment (the oriented angle from the unit vector of the -axis to this segment), and is the (positive) length of the normal segment. The normal form can be derived from the standard form $ax + by = c$ by dividing all of the coefficients by $\frac\sqrt.$ Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, $\varphi$ and , to be specified. If , then $\varphi$ is uniquely defined modulo . On the other hand, if the line is through the origin (), one drops the term to compute $\sin\varphi$ and $\cos\varphi$, and it follows that $\varphi$ is only defined modulo .

# Other representations

## Vectors

The vector equation of the line through points A and B is given by $\mathbf = \mathbf + \lambda\, \mathbf$ (where λ is a scalar). If a is vector OA and b is vector OB, then the equation of the line can be written: $\mathbf = \mathbf + \lambda \left(\mathbf - \mathbf\right)$. 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.

## Polar coordinates

In a
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
,
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point (mathematics), point on a plane (mathematics), plane is determined by a distance from a reference point and an angle from a reference direction ...
are related to
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
by the parametric equations:$x=r\cos\theta, \quad y=r\sin\theta.$ In polar coordinates, the equation of a line not passing through the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
—the point with coordinates —can be written $r = \frac p ,$ with and $\varphi-\pi/2 < \theta < \varphi + \pi/2.$ Here, is the (positive) length of the
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
perpendicular to the line and delimited by the origin and the line, and $\varphi$ is the (oriented) angle from the -axis to this segment. It may be useful to express the equation in terms of the angle $\alpha=\varphi+\pi/2$ between the -axis and the line. In this case, the equation becomes $r=\frac p ,$ with and $0 < \theta < \alpha + \pi.$ These equations can be derived from the normal form of the line equation by setting $x = r \cos\theta,$ and $y = r \sin\theta,$ and then applying the angle difference identity for sine or cosine. These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to the
right triangle A right triangle (American English) or right-angled triangle (British English, British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one an ...
that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides. The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates $\left(r, \theta\right)$ of the points of a line passing through the origin and making an angle of $\alpha$ with the -axis, are the pairs $\left(r, \theta\right)$ such that $r\ge 0,\qquad \text \quad \theta=\alpha \quad\text\quad \theta=\alpha +\pi.$

## Projective geometry

In many models of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In
elliptic geometry Elliptic geometry is an example of a geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathem ...
we see a typical example of this. In the spherical representation of elliptic geometry, lines are represented by
great circle In mathematics, a great circle or orthodrome is the circle, circular Intersection (geometry), intersection of a sphere and a Plane (geometry), plane incidence (geometry), passing through the sphere's centre (geometry), center point. Any Circula ...
s 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. The "shortness" and "straightness" of a line, interpreted as the property that the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
along the line between any two of its points is minimized (see
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
), can be generalized and leads to the concept of
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s in
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s.

# 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. 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''. This is, at times, also expressed as the set of all points ''C'' on the line determined by ''A'' and ''B'' such that ''A'' is not between ''B'' and ''C''. 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 In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of th ...
of an open segment 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 In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
. 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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
or
affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric space, metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main propertie ...
over an
ordered field In mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-co ...
. On the other hand, rays do not exist in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
nor in a geometry over a non-ordered field, like the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s or any
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
.

## Line segment

A
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
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 In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Ho ...
and either do not intersect or are
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
.

## Number line

A point on number line corresponds to a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
and vice versa. Usually,
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, an imaginary line representing
imaginary numbers An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an imagina ...
can be drawn perpendicular to the number line at zero.. The two lines forms the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, ...
, a geometrical representation of the set of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
.

# In graphics design

*
Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angle In Eucli ...
*
Curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
*
Distance between two parallel lines The distance between two Parallel (geometry), parallel Line (geometry), lines in the plane (geometry), plane is the minimum distance between any two points. Formula and proof Because the lines are parallel, the perpendicular distance between th ...
* Distance from a point to a line * Imaginary line (mathematics) *
Incidence (geometry) In geometry, an incidence Relation (mathematics), relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidenc ...
*
Line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
* Generalised circle * Locus *
Plane (geometry) In mathematics, a plane is a Euclidean space, Euclidean (flatness (mathematics), flat), two-dimensional surface (mathematics), surface that extends indefinitely. A plane is the two-dimensional analogue of a point (geometry), point (zero dimensi ...
*
Polyline In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its Vertex (geometry), vertices. The curve itself consists of th ...