LINE

Line is a fundamental concept in geometry and mathematics, referring to a straight one-dimensional figure that extends without end in both directions. In Euclidean space, a line can be defined as the set of points satisfying a linear equation, or equivalently by a point and a direction vector, or by two distinct points. In a plane it is determined uniquely by any two points; in three-dimensional space lines can be described by a point and a direction, or by a pair of points.

Common representations include the coordinate form y = mx + b for non-vertical lines and the general form ax + by + c = 0; parametric form r = p + t d; and vector form. Key properties include that any two distinct lines in a plane either intersect at a single point or are parallel; in three-dimensional space lines can be parallel, intersecting, or skew (neither parallel nor intersecting).

Line is used in various contexts beyond pure geometry. A line segment is a finite portion with endpoints, and a ray starts at a point and extends in one direction. A line of text is a sequence of characters ending with a newline. In music, a line can refer to a melodic or vocal line. A production line or assembly line denotes a sequence of workstations in manufacturing, while a telephone line denotes a communication channel.

In non-Euclidean geometries, lines may be geodesics, such as great circles on a sphere. In projective geometry, lines have a dual relationship with points, reflecting different foundational viewpoints about incidence and structure.