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Line (mathematics)

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	Topics Line (mathematics) Topic Home Examples Lines in a Cartesian plane can be described algebraically by linear equationLinear equation A linear equation is an equation involving only the sum of constants or products of constants and the first power of a vari... s and linear functionLinear function A linear function can refer to two slightly different concepts.... s. In two dimensions, the characteristic equation is often given by the slope-intercept form: where: m is the slopeSlope The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straigh... of the line. b is the y-interceptY-intercept Summary The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis ... of the line. x is the independent variableFacts About Independent variable In an experimental design, the independent variable is the variable which is manipulated or selected by the experimenter to ... of the function y. In three dimensions, a line is 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 vectorVector (spatial) In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a... (a, b, c) is a parallel to the line. Formal definitions This intuitive concept of a line can be formalized in various ways. If geometryGeometry Geometry arose as the field of knowledge dealing with spatial relationships.... is developed axiomatically (as in EuclidEuclid Euclid , a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I... 's ElementsEuclid's Elements Euclid's Elements is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathemat... and later in David HilbertFacts About David Hilbert David Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19t... '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 spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... Rⁿ (and analogously in all other vector spaceVector space In mathematics, a vector space is a collection of objects that, informally speaking, may be scaled and added.... s), we define a line L as a subset of the form where a and b are given vectorVector space In mathematics, a vector space is a collection of objects that, informally speaking, may be scaled and added.... s 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 coefficientCoefficient In mathematics, a coefficient is a constant multiplicative factor of a certain object.... s a, b and c such that a and b are not both zero (see Linear equationLinear equation A linear equation is an equation involving only the sum of constants or products of constants and the first power of a vari... for other forms). Important properties of these lines are their slopeSlope The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straigh... , x-intercept and y-interceptY-intercept The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis ... . The eccentricityEccentricity (mathematics) Overview In mathematics, eccentricity is a parameter associated with every conic section.... of a straight line is infinityInfinity he word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts which arise i... . More abstractly, one usually thinks of the real lineReal line In mathematics, the real line is simply the set R of real numbers.... as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numberFacts About Real number In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... s. However, one could also use the hyperreal numberHyperreal number The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers ... s for this purpose, or even the long lineLong line (topology) In topology, the long line is a topological space analogous to the real line, but much longer.... of topologyTopology Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation ; these are ... . 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 geodesicGeodesic In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".... s on differentiable manifoldManifold A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in... s. Ray In Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... , a ray, or half-line, given two distinct pointPoint (geometry) A spatial point is an entity with a location in space but no extent.... s 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 geometryGeometry Geometry arose as the field of knowledge dealing with spatial relationships.... , a ray starts at one pointPoint (geometry) A spatial point is an entity with a location in space but no extent.... , then goes on forever in one direction. In topologyTopology Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation ; these are ... , a ray in a space X is a continuos embedding . It is used to define the important concept of end (topology) of the space. See also Line segmentLine 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 betw... Affine function DiffractionFacts About Diffraction Diffraction refers to the various phenomena associated with wave propagation, such as the bending, spreading and interferenc... Glossary of Riemannian and metric geometry#RGlossary of Riemannian and metric geometry This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of diffe... for its meaning in Riemannian geometryRiemannian geometry In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i.e.... . Incidence (geometry)Incidence (geometry) In geometry, the relations of incidence are those such as 'lies on' between points and lines, and 'intersects'.... Minimal line representationRobotics conventions Overview There are a lot of conventions used in the Robotics research field. This article summarises these conventions.... Ridge detectionRidge detection In a 2-D function, a ridge is a connected set of points that are maximal in at least one dimension.... and Hough transformHough transform The Hough transform is a feature extraction technique used in digital image processing.... for algorithms for detecting lines in digital images External links at cut-the-knotCut-the-knot cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variet...