Shape preserving alternatives to the rational Bézier model

Abstract

We discus several alternatives to the rational Bézier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding B-bases in general spaces. We also revisit the C-Bézier curves presented by Zhang (1996), which coincide with the helix spline segments developed by Pottmann and Wagner (1994), and are nothing else than curves expressed in the normalized B-basis of the space P 1= span{1,t, cost, sint} . Such curves provide a valuable alternative to the rational Bézier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2= span{1,t, cost, sint, cos2t, sin2t} , Q= span{1,t,t 2, cost, sint} and I= span{1,t, cost, sint,t cost,t sint} are also suitable for shape preserving design, and we find their normalized B-basis.