Curvature monotonicity evaluation functions on rational Bézier curves

Abstract

We have derived functions of the lowest possible degree that enable us to evaluate curvature monotonicity for any 2D and 3D rational Bézier curves. We proved that the degree of the function is at most 8n−12 for planar rational Bézier curves of degree n, and is at most 11n−18 for space rational Bézier curves of degree n. These functions are derived in the Bernstein basis, allowing for efficient checking of curvature monotonicity using subdivision or Bézier clipping. As an application, we present real-time visualization of the region of a particular control point that guarantees monotonic variation of curvature over the entire segment of the rational Bézier curve. This allows users to identify where to move the control point to ensure that the curvature changes monotonically.