Straightline

A straightline, or straight line, is a one-dimensional geometric object that extends without end in both directions and has no curvature. It is the shortest path between any two points on the line and, in Euclidean space, is uniquely determined by any two distinct points or by a linear equation.

In the plane, a straight line can be described by several equivalent forms. If it passes through two points (x1, y1) and (x2, y2) with x1 ≠ x2, its slope is m = (y2 − y1)/(x2 − x1) and the line consists of all points (x, y) satisfying y − y1 = m(x − x1). Common representations include y = mx + b (slope-intercept form) and the standard form Ax + By = C.

In three-dimensional space, a line is described by a point r0 and a direction vector v, using the parametric form r = r0 + t v, where t is a real parameter. A line can also be defined as the intersection of two planes or by a system of linear equations.

Key properties include behavior under parallelism and perpendicularity: lines are parallel if they share the same direction and never meet, and perpendicular if their direction vectors are orthogonal. The distance from a point (x0, y0) to the line Ax + By = C is |Ax0 + By0 − C| divided by sqrt(A^2 + B^2).

Beyond Euclidean geometry, straight lines appear as geodesics in non-Euclidean spaces and as fundamental objects in projective geometry. They are central to analytic geometry, vector calculus, and many applications in science and engineering.