A representation of one line segment

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,^[1] infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves")^[2] containing an infinite number of points. 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 In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Euclidean geometry, exactly one line can be found that passes through any two points. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume The line provides the shortest connection between the points. ^[3]

In two dimensions, two different lines can either be parallel, meaning they never meet, or may intersect at one and only one point. In Euclidean geometry, the Intersection of a line and a line can be the Empty set, a point, or a line In three or more dimensions, lines may also be skew, meaning they don't meet, but also don't define a plane. In Geometry, skew lines are two lines that do not intersect but are not Parallel. 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 Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, the term linear function can refer to either of two different but related concepts In two dimensions, the characteristic equation is often given by the slope-intercept form:

where:

m is the slope of the line. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable Slope is used to describe the steepness incline gradient or grade of a straight line.

b is the y-intercept of the line. In Coordinate geometry, the y -intercept is the y-value of the point where the Graph of a function or relation intercepts the y -axis

x is the independent variable of the function y. Dependent variables and independent variables refer to values that change in relationship to each other

In three dimensions, a line is described by parametric equations:

where:

x, y, and z are all functions of the independent variable t. In Mathematics, parametric equations are a method of defining a curve

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.

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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 Geometry, skew lines are two lines that 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). In Mathematics, a coefficient is a Constant multiplicative factor of a certain object A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable Important properties of these lines are their slope, x-intercept and y-intercept. Slope is used to describe the steepness incline gradient or grade of a straight line. This article is about the zeros of a function which should not be confused with the value at zero. In Coordinate geometry, the y -intercept is the y-value of the point where the Graph of a function or relation intercepts the y -axis The eccentricity of a straight line is infinity. In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

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. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, the real numbers may be described informally in several different ways However, one could also use the hyperreal numbers for this purpose, or even the long line of topology. In Topology, the long line (or Alexandroff line) is a Topological space analogous to the Real line, but much longer Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

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 /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In geometry, a ray starts at one point, then goes on forever in one direction. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume

Notes

1. ^ The no-width length is the second definition in Book I of Euclid's Elements. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry It is described as ideal (epistemological sense) because it is a concept only. The dimensions of small physical objects are limited by the Planck length; that is, real lines are impossible. The Planck length, denoted by \scriptstyle\ell_P \, is the unit of Length approximately 1
2. ^ "Geometry". The Encyclopaedia Britannica: Eleventh Edition 11. (1910). Cambridge University Press. 717.   In Analytic geometry a straight line is a first-order curve; that is, no term in its equation has an exponent greater than 1. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Second- and greater order curves better fit the English-language sense.
3. ^ This principle is stated explicitly by Archimedes in the second assumption of Book I, On the Sphere and Cylinder. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer It is implicitly proved by Proposition 20, Book I, Euclid's Elements. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid forms a triangle with any other point off the line at a vertex and shows that the one side between the points is shorter than the sum of the two other sides.

See also

• Line segment
• Affine function
• Diffraction
• Glossary of Riemannian and metric geometry#R for its meaning in Riemannian geometry. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology. Elliptic geometry is also sometimes called Riemannian geometry.
• Incidence (geometry)
• Minimal line representation
• Ridge detection and Hough transform for algorithms for detecting lines in digital images

External links

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