# Line (mathematics)

Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept.

A line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points. The line provides the shortest connection between the points.

In two dimensions, two different lines can either be parallel, meaning they never meet, or may intersect at one and only one point. In three or more dimensions, lines may also be skew, meaning they don't meet, but also don't define a plane. Two distinct planes intersect in at most one line. Three or more points that lie on the same line are called collinear.

## Examples

Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:
where:
m is the slope of the line.
b is the y-intercept of the line.
x is the independent variable of the function y.

In three dimensions, a line is often described by parametric equations:
where:
x, y, and z are all functions of the independent variable t.
, , and are the initial values of each respective variable.
a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.

## Formal definitions

This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.

In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form

where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.

## Properties

In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines.

In R2, every line L 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 (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity.

More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.

The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.

## Ray

In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. In geometry, a ray starts at one point, then goes on forever in one direction.

Ray

Line or lines may refer to:
• Line (mathematics), an infinitely-extending one-dimensional figure that has no curvature
• a length of rope, cable or chain when put to use (such as a clothesline, anchor line)
• a line or queue of people waiting in a queue area

Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth
In general English usage, long is the adjectival form of length. It may also refer to:

In geography:
• Long, People's Republic of China
• Long Island, New York, United States
In other fields:
• LÃ³ng, the Chinese dragon

Straight is a term which may commonly refer to:
• the quality or state of extending in one direction without turns, bends or curves; or being without influence or interruption
• the personal character of displaying honesty or fairness yes

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
A spatial point is a concept used to define an exact location in space. It has no volume, area or length, making it a zero dimensional object. Points are used in the basic language of geometry, physics, vector graphics (both 2D and 3D), and many other fields.
Connection, Connected, Connectivity, or Connexion may refer to:

Music
• Connected (album) is a 1992 album by the Stereo MC's
• Connected (Foreign Exchange album) is an album by The Foreign Exchange

Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate.
In analytic geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases, and determining equations for the point and line in the latter cases have use, for example, in computer graphics, motion planning, and collision detection.
In geometry, skew lines are two lines that do not intersect but are not parallel. Equivalently, they are lines that are not both in the same plane. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron.
plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. Such an equation is equivalent to equating a first-degree polynomial to zero.
In mathematics, the term linear function can refer to either of two different but related concepts.

## Usage in elementary mathematics

In elementary algebra and analytic geometry, the term linear function
Slope is often used to describe the measurement of the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run
In two-dimensional coordinate geometry, the y-intercept is the point where the graph of a function or relation intercepts the y-axis of the coordinate system.

In other words, the y-intercept of a function is the point at which it intersects the line
In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function.
parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables.
spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
Euclid

Born fl. 300 BC

Residence Alexandria, Egypt
Nationality Greek
Field Mathematics
Known for Euclid's Elements Euclid (Greek:
Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC.
David Hilbert

David Hilbert (1912)
Born January 23 1862
Wehlau (Welawa), Province of Prussia