A representation of one line

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. Look up line in Wiktionary, the free dictionary. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links Linear_functions2. ... Image File history File links Linear_functions2. ... In general English usage, length (symbol: l) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or... Wiktionary has related dictionary definitions, such as: long Long may mean: long integer, a type variable in computer science long (finance), a position in finance Another name for the Chinese dragon, a mythical creature in Chinese mythology Long is the name of: Long Island, an island in New York, United... 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 Straight, a poker hand containing five cards in sequential order a heterosexual person a type of punch used in boxing, also commonly... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Look up connection, connected, connectivity in Wiktionary, the free dictionary. ...

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. 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. ... Intersecting airplane trails. ... In geometry, skew lines are two lines in Euclidean space that do not intersect but are not parallel. ... This article is about the mathematical construct. ...

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: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... Look up vorarephilia in Wiktionary, the free dictionary. ...

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: This article is about the mathematical term. ... The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis of the coordinate system. ... In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. ...

where:

x, y, and z are all functions of the independent variable t.

x₀, y₀, and z₀ 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.

This article is about vectors that have a particular relation to the spatial coordinates. ...

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. For other uses, see Geometry (disambiguation). ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...

In Euclidean space Rⁿ (and analogously in all other vector spaces), we define a line L as a subset of the form Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

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. 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. ... In geometry, two lines are said to be skew lines if they do not intersect but are not parallel. ...

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. For other senses of this word, see coefficient (disambiguation). ... Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... This article is about the mathematical term. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis of the coordinate system. ... (This page refers to eccentricity in mathematics. ... For other uses, see Infinity (disambiguation). ...

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. In mathematics, the real line is simply the set of real numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In topology, the long line is a topological space analogous to the real line, but much longer. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in 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. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

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. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... For other uses, see Geometry (disambiguation). ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...

Image File history File links Ray_(A,_B,_C). ...

See also

• Line segment
• Affine function
• Diffraction
• Glossary of Riemannian and metric geometry#R for its meaning in Riemannian geometry.
• Incidence (geometry)
• Minimal line representation

The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... The word linear comes from the Latin word linearis, which means created by lines. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In geometry, the relations of incidence are those such as lies on between points and lines (as in point P lies on line L), and intersects (as in line L1 intersects line L2, in three-dimensional space). ... There are a lot of conventions used in the Robotics research field. ...

External links

• Detailed explanation of the line at MathWorld Encyclopedia
• Equations of the Straight Line at cut-the-knot

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...