![Gerade](https://upload.wikimedia.org/wikipedia/commons/7/76/Gerade.svg)
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 c ...
, 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 deviates from being a plane.
For curves, the canonic ...
. Thus, lines are
one-dimensional objects, though they may exist in
two
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
,
three, 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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
in everyday life, which has two
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
to denote its ends. Lines can be referred by two points that lay on it (e.g.,
) or by a single letter (e.g.,
).
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
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
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
,
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
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 variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
, 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. An ''incidenc ...
, 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 that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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 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 and multili ...
, a line may be interpreted as a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
(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 that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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 that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
. 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, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
), 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, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
, or
skew
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not ...
if they are not.
On an
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into
convex polygons (possibly unbounded); this partition is known as an
arrangement of lines.
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 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, ...
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
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), 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
A convention is a set of agreed, stipulated, or generally accepted standards, norms, social norms, or criteria, often taking the form of a custom.
In a social context, a convention may retain the character of an "unwritten law" of custom (for e ...
'' determine a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
, but in the case of three collinear points this does ''not'' happen.
In
affine coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
, 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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
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
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
''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) 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
![Tangent_to_a_curve](https://upload.wikimedia.org/wikipedia/commons/0/0f/Tangent_to_a_curve.svg)
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 (a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
,
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 mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
, or
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
), lines can be:
*
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s, 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
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...
is the line that connects the midpoints of the two
diagonals.
[ ()]
For a
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
with vertices lying on a conic we have the
Pascal line
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ...
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
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct line (geometry), li ...
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
In elementary geometry, two geometric objects are perpendicular if they intersection, intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ...
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 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
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 that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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 mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
falls into this category.
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.
Definition
Linear equation
![Linear_Function_Graph](https://upload.wikimedia.org/wikipedia/commons/0/0e/Linear_Function_Graph.svg)
Lines in a Cartesian plane or, more generally, in
affine coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
, are characterized by linear equations. More precisely, every line
(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 of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
(''x'', ''y'') satisfy a linear equation; that is,
where ''a'', ''b'' and ''c'' are fixed
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s (called
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
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
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
and
may be written as
If , this equation may be rewritten as
or
In
two dimensions, the equation for non-vertical lines is often given in the ''
slope-intercept form'':
where:
* ''m'' is the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
or
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of the line.
* ''b'' is the
y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable ''x'' and the vertical axis represents a variable ''y'', a ''y''-intercept or vertical intercept is a point where the graph of a function or relatio ...
of the line.
* ''x'' is the
independent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
of the function .
The slope of the line through points
and
, when
, is given by
and the equation of this line can be written
.
Parametric equation
Parametric equations are also used to specify lines, particularly in those in
three dimensions
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 of an element (i.e., point). This is the informal ...
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:
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 (''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 variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s
such that
and
are not proportional (the relations
imply
). This follows since in three dimensions a single linear equation typically describes a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
and a line is what is common to two distinct intersecting planes.
Hesse normal form
![Hesse_normalenform](https://upload.wikimedia.org/wikipedia/commons/e/e5/Hesse_normalenform.svg)
The ''normal form'' (also called the ''Hesse normal form'', after the German mathematician
Ludwig Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse norma ...
), 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:
where
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
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,
and , to be specified. If , then
is uniquely defined modulo . On the other hand, if the line is through the origin (), one drops the term to compute
and
, and it follows that
is only defined modulo .
Other representations
Vectors
The vector equation of the line through points A and B is given by
(where λ is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
).
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.
Polar coordinates
![Parametres polaires droite](https://upload.wikimedia.org/wikipedia/commons/5/5c/Parametres_polaires_droite.svg)
In a
Cartesian plane,
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
are related to
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
by the parametric equations:
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
with and
Here, is the (positive) length of the
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
perpendicular to the line and delimited by the origin and the line, and
is the (oriented) angle from the -axis to this segment.
It may be useful to express the equation in terms of the angle
between the -axis and the line. In this case, the equation becomes
with and
These equations can be derived from the
normal form of the line equation by setting
and
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 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
of the points of a line passing through the origin and making an angle of
with the -axis, are the pairs
such that
Projective geometry
![Great_circle_hemispheres](https://upload.wikimedia.org/wikipedia/commons/0/04/Great_circle_hemispheres.png)
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.
In the spherical representation of elliptic geometry, lines are represented by
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
s of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean
planes
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''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), 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) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s in
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s.
Extensions
Ray
![Ray (A, B, C)](https://upload.wikimedia.org/wikipedia/commons/8/8e/Ray_%28A%2C_B%2C_C%29.svg)
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 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 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 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 ...
s or any
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.
Line segment
![Adcinak](https://upload.wikimedia.org/wikipedia/commons/6/65/Adcinak.svg)
A
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
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.
Number line
![Number line with x smaller than y](https://upload.wikimedia.org/wikipedia/commons/7/77/Number_line_with_x_smaller_than_y.svg)
A point on number line corresponds to a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and vice versa. Usually,
integers 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
In general, an imaginary line is usually any sort of geometric line that has only an abstract definition and does not physically exist. In fact, they are used to properly identify places on a map.
Some outside geography do exist, such as th ...
representing
imaginary numbers 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 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, is formed by the ...
, a geometrical representation of the set of
complex numbers.
In graphics design
See also
*
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 angles.
More generally, ...
*
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
*
Distance from a point to a line
*
Imaginary line (mathematics)
In complex geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.
It is a special case of an imaginary curve.
An imaginary line is ...
*
Incidence (geometry)
*
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
*
Generalised circle
*
Locus
*
Plane (geometry)
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
*
Polyline
References
External links
*
Equations of the Straight Lineat
Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
{{DEFAULTSORT:Line (Geometry)
Elementary geometry
Analytic geometry
Mathematical concepts