epsilonquasirectilinear

Epsilonquasirectilinear is a proposed property of a curve that measures how closely it behaves like a straight line within a fixed tolerance ε. Formally, let γ: [a,b] → R^n be a rectifiable curve and let L(γ|[s,t]) denote the length of the subcurve from γ(s) to γ(t). The curve γ is said to be ε-quasirectilinear if, for all s and t with a ≤ s ≤ t ≤ b, the excess length satisfies L(γ|[s,t]) − ||γ(t) − γ(s)|| ≤ ε. Equality ε = 0 occurs precisely when every subcurve is a straight line segment, meaning γ is itself an affine path.

Properties and relationships: The definition is invariant under Euclidean isometries, since lengths and distances are preserved by rotations and translations. Under uniform scaling by a factor c, the excess becomes multiplied by c, so ε must be scaled accordingly. Any subcurve of an ε-quasirectilinear curve remains ε-quasirectilinear. The concept provides a quantitative measure of how far a curve is from being straight: smaller ε indicates a path that more closely approximates a straight line across all subintervals.

Examples and interpretation: A straight line γ(t) = p + t v is 0-quasirectilinear. Curves that wiggle with very small total deviation from the chord between endpoints can be ε-quasirectilinear for a correspondingly small ε. In practice, ε-quasirectilinear curves are interpreted as nearly straight, enabling simple polyline approximations with guaranteed maximum deviation.

Applications: The notion is used in path planning, robotics, and computer graphics to bound error when replacing curved trajectories with polylines and in analyses requiring a quantitative straightness criterion. See also rectifiable curves, curve approximation, and total curvature.