BEYOND BÉZIER CURVES

Publisher Summary This chapter elaborates the different aspects of Bezier curves. These curves are deservedly popular, because the control points have a geometrical relationship to the curve, unlike polynomial coefficients. By associating a homogeneous weight with each control point, it is possible to describe rational curves and hence, conies as well. Surfaces can be described with tensor products of curves, giving four-sided Bezier patches, and solids as well as higher dimensional volumes can be described by an extension of the same technique. This is not the only way to proceed, however. Instead, one can construct Bezier triangles, tetrahedra, and so on. The Bezier simplices have been exploited by Loop, and De Rose as a way to construct patches with any number of sides, and from these, surfaces of arbitrary topology. Evaluation to find points on Bezier simplices can be accomplished by a generalization of the recursive geometric de Casteljau algorithm for curves. One tricky part in doing this is simply managing the bookkeeping, because the algorithms, and data structures are most naturally expressed in terms of what are called multi-indices.