Topology during Subdivision of Bezier Curves I: Angular Convergence & Homeomorphism

TL;DR

This paper proves that the exterior angles of subdivided Bezier control polygons decrease at a specific rate, enabling the determination of topological features and establishing homeomorphism between the curve and its control polygon.

Contribution

It introduces a novel angular convergence rate for control polygons during subdivision and provides formulas to determine subdivision steps for topological equivalence.

Findings

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

Control polygons become homeomorphic to Bezier curves after sufficient subdivision

Closed-form formulas for subdivision iterations to ensure topological accuracy

Abstract

For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision 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 topological features. We use it to show homeomorphism between a Bezier curve and its control polygon under subdivision. The constructive geometric proofs yield closed-form formulas to compute sufficient numbers of subdivision iterations to obtain small exterior angles and achieve homeomorphism.