Topology during Subdivision of Bezier Curves I: Angular Convergence & Homeomorphism · arXivDesk
1211.0353Nov 2, 2012This result has been combined with another result in a single paper "Isotopic Equivalence from Bezier Curve Subdivision.", arXiv:1211.0354
Topology during Subdivision of Bezier Curves I: Angular Convergence & Homeomorphism
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(2i1)
Nearby in the stack
, 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.