Cutting corners always works

Abstract

The convergence of corner-cutting algorithms, such as de Rham's trisection algorith, Chaikin's algorithm, degree-raising for the B-form, and knot-insertion for splines, is customarily established by using specific features of the algorithm in question. This note gives a general convergence proof that uses only the fact that, in the typical step, a corner-cutting algorithm replaces a convex (-cave) curve segment by a convex (-cave) curve segment with smaller curvature. Somewhat surprisingly, the proof is harder for closed curves than for open curves.