Quasirectilinearity

Quasirectilinearity is a term used in geometry to describe a property of curves or paths that stay close to being straight. In many formulations, a curve gamma: [0,1] -> R^n is called C-quasirectilinear if for every pair of parameters s ≤ t the image gamma([s,t]) lies within a Euclidean distance at most C|t−s| from the straight line segment joining gamma(s) and gamma(t). This provides a quantitative bound on how far subpaths may deviate from the direct chord, with C controlling the allowed bending per unit length. Some authors prefer a formulation in terms of the total turning angle or curvature, requiring the variation of the tangent direction along any subpath to be bounded by a constant.

The concept is closely related to quasi-geodesics in metric geometry, though quasirectilinearity is often used in Euclidean settings as a descriptive property rather than a formal axiom. It is stable under reparameterizations that preserve subpath lengths, and straight lines satisfy the condition with C = 0.

Examples range from the ideal straight line, which is trivially quasirectilinear, to gently undulating curves whose amplitude and frequency produce a small, bounded deviation relative to subpath length. Curves with large curvature or rapid direction changes over short intervals typically fail to be quasirectilinear for finite C.

Applications of quasirectilinearity appear in path planning, computer graphics, and the analysis of trajectories where near-straight behavior is desirable or assumed. The term is not universally standardized, and different fields may adopt related but distinct formulations, such as bounds on distance to chords or on curvature. See also quasi-geodesic, curvature, and path planning.