Angular Convergence during Bezier Curve Approximation

TL;DR

This paper analyzes how the exterior angles of piecewise linear approximations of Bezier curves decrease at a specific rate during subdivision, aiding in understanding curve properties like self-intersections.

Contribution

It establishes the rate of angular convergence of PL approximations of Bezier curves during subdivision, which was previously not well characterized.

Findings

Exterior angles converge to 0 at rate O(√(1/2^i))

Angular convergence helps determine self-intersections and knot types

Provides theoretical foundation for curve analysis methods

Abstract

Properties of a parametric curve in R^3 are often determined by analysis of its piecewise linear (PL) approximation. For Bezier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to the curve in distance . The exterior angles of PL curves under subdivision are shown to converge to 0 at the rate of $O(\sqrt{\frac{1}{2^i}})$, where i is the number of subdivisions. This angular convergence is useful for determining self-intersections and knot type.