Line (mathematics)

From Example Problems
Jump to: navigation, search

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. Two different lines can either be parallel and never meet, or may intersect at one and only one point. 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. "Everything that satisfies the axioms for a line is a line." 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=\{{\mathbf  {a}}+t{\mathbf  {b}}\mid t\in {\mathbb  {R}}\}

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 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=\{(x,y)\mid ax+by=c\}

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.

Line segment

In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics).

When the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal.

The midpoint of a line segment is its 'middle' point: the unique point at an equal distance from the two end points.

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.

  O----O-----*--->
  A    B     C

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

External links

ca:semirecta da:Linje de:Gerade eo:Linio es:Recta fi:Suora fr:Droite (mathématiques) he:קו io:Lineo it:Retta ja:半直線 nl:Lijn (meetkunde) pl:Prosta simple:Line sl:Premica sv:Rät linje ta:கோடு tr:Doğru vi:Quang tuyến zh:线段