Line (geometry)

Key characteristics

Several properties identify a straight line in classical geometry:

• It consists of an endless collection of positions or locations, mathematically modeled as points.
• Any two distinct points on a plane determine a unique line; conversely, a line contains infinitely many points (points).
• Lines are straight — the shortest path between two points on the same line lies entirely on that line.
• Pairs of lines may be parallel, perpendicular, intersecting, or in three dimensions, skew.

Types and related segments

Common variants derived from the ideal line are useful in constructions and proofs:

• Line: extends infinitely in two directions.
• Ray: starts at a point and continues infinitely in one direction.
• Line segment: has two endpoints and a finite length.

Algebraic description and uses

In coordinate geometry a line can be described by linear equations (for example, slope-intercept or parametric forms) and by vector expressions in higher dimensions. Lines are central in analytic geometry, linear algebra (as affine subspaces), computer graphics, engineering drawings and surveying. They also appear in physics to represent idealized trajectories, axes, or symmetry directions.

History and importance

The notion of a straight line appears in ancient geometry and was formalized in Euclid’s axioms as a primary undefined term with characteristic properties. Over centuries the concept expanded: Descartes linked lines to algebra, and modern geometry treats lines as fundamental objects in many systems of axioms. Their simplicity makes them indispensable for proofs, constructions and modeling across mathematics and applied disciplines.

For further reading, see introductions to Euclidean geometry and analytic geometry: length, width, geometric figure, endless, points.