Line (geometry)

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the 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 intersect in at most one point; two different planes can intersect in at most one line. 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.

One can show that 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. Important properties of these lines are their slope, x-intercept and y-intercept.

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.

See also: incidence (geometry).