Line (mathematics)
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A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes "straight curves"). In Euclidean geometry, exactly one line can be found that passes through any two points. The line provides the shortest connection between the points.
Three or more points that lie on the same line are called collinear. 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 planes intersect in at most one line.
Lines in a Cartesian plane can be described algebraically by linear equations and linear functions.
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
- [L = \+t\mathbf\mid t\in\mathbb\}]
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
- [L=\ \,]
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.
In geometric optics a ray or a (light) beam is a line or curve that describes the direction in which light or other electromagnetic radiation is propagated. The ray is perpendicular to the wavefront in wave optics.
In most media, light rays are straight lines. Light passing from one medium to another undergoes refraction or total internal reflection following Snell's law.
See also
- Line segment
- Affine function
- Linear equation
- Linear function
- Diffraction
- Glossary of Riemannian and metric geometry#R for its meaning in Riemannian geometry.
- incidence (geometry).
External links
- [Detailed explanation of the line at MathWorld Encyclopedia]
- [Equations of the Straight Line] at cut-the-knot
- [Definition of Ray. Math Open Reference] with interactive applet.
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