maxcurvature

Max curvature, often written as max curvature or kappa_max, is a quantity used in differential geometry to describe the greatest bending of a curve over a specified portion of its domain. For a smooth plane curve r(s) parameterized by arc length s, the curvature is κ(s) = ||dT/ds||, where T is the unit tangent. The maximum curvature on an interval I is κ_max = max_{s ∈ I} κ(s). For a space curve r(t), curvature is κ(t) = ||r'(t) × r''(t)|| / ||r'(t)||^3.

Examples and intuition: A circle of radius R has constant curvature κ = 1/R, so κ_max = 1/R. A straight line has κ ≡ 0, so κ_max = 0. The concept provides a single scalar measure of how sharply a curve bends somewhere along its length.

Computation and applications: In discrete or computational settings, curvature is often approximated from samples or polygonal data; the maximum of these approximations serves as an estimate of κ_max. Applications include robotics path planning, shape analysis, computer-aided design, and computer graphics, where controlling or detecting high-curvature regions informs smoothing, feature detection, and motion planning.

Surfaces and generalizations: On surfaces, the maximum normal curvature at a point is the largest absolute value among the directional curvatures, equal to the larger magnitude of the principal curvatures k1 and k2. Notationally, κ_max or max κ is used, with meaning depending on whether one discusses curves, surfaces, or discrete data.