Wonderful triangle: a simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design

Abstract Many important algorithms in computer aided geometric design can be viewed as change of basis procedures. These include evaluation, subdivision, knot insertion, knot deletion, interpolation, degree elevation, and differentiation. We provide a simple, unified, algorithmic approach to such change of basis procedures for a wide class of polynomial and spline bases. Along the way we shall observe that many well-known algorithms and formulas for polynomials and splines, including synthetic division, forward differencing, Horner's method, the de Casteljau algorithm, the de Boor algorithm, Boehm's derivative algorithm, Boehm's knot insertion algorithm, the Oslo algorithm, Marsden's identity, and the binomial theorem are simply variations on a common theme.

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