| Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon (1998) | |||||||||||||||
Abstract | |||||||||||||||
| The maximal distance between a Bezier segment and its control polygon is bounded in terms of the dierences 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 dierences are the smallest possible. In particular, the bound in terms of the maximal absolute second dierence of the control points is a sharp upper bound for the Hausdor 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 Bezier segment under subdivision or degree-fold degreeraising and establishes the explicit convergence constants, and it allows analyzing the optimal choice of the subdivision parameter for adaptive renement of quadratic and cubic segments and yields eÆcient bounding regions. 1 Curved geometry and control polygons A widely used, eÆcient and intuitive way to specify, represent and reason abou... | |||||||||||||||
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