Halfline

A half-line, also called a ray, is a part of a straight line that contains a fixed starting point, called the endpoint, and extends infinitely in one direction along the line.

Formally, in Euclidean space R^n, choose a point P and a nonzero direction vector v. The half-line determined by P and v is the set {P + t v : t ≥ 0}. The point P is included, and as t increases, the points move away from P along the line determined by v. In the real line, half-lines take the form [a, ∞) or (-∞, b], depending on the chosen endpoint and direction.

Key properties include that a half-line is unbounded and connected, and it is a closed subset of the Euclidean topology. It is homeomorphic to [0, ∞). A half-line is fully determined by its endpoint and direction; reversing the direction yields the opposite half-line on the same line.

Relation to other geometric objects: the entire line is the union of two opposite half-lines sharing the same endpoint, and a half-line together with a finite line segment can form a line segment ending at the endpoint. A half-line differs from a line segment, which has finite length, and from a full line, which extends in both directions.

Applications and contexts include geometry, topology, and computer graphics (for example, ray tracing uses half-lines to model light paths). In calculus and analysis, half-lines often serve as domains for functions defined on a ray emanating from a point. See also line, ray, line segment, and line topology.