Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon Original Research Article

The maximal distance between a Bézier segment and its control polygon is bounded in terms of the differences of the control point sequence and a constant that depends only on the degree of the polynomial. The constants derived here for various norms and orders of differences are the smallest possible. In particular, the bound in terms of the maximal absolute second difference of the control points is a sharp upper bound for the Hausdorff distance between the control polygon and the curve segment. It provides a straightforward proof of quadratic convergence of the sequence of control polygons to the Bézier segment under subdivision or degree-fold degree-raising, and establishes the explicit convergence constants, and allows analyzing the optimal choice of the subdivision parameter for adaptive refinement of quadratic and cubic segments and yields efficient bounding regions.