Shape factors and shoulder points for shape control of rational Bézier curves. (April 2023) - Sánchez-Reyes, Javier
Abstract: The weights of rational Bézier curves cannot be regarded as true independent shape factors since they do not enjoy invariance with respect to Moebius (i.e., rational linear) reparametrizations, which do not change the curve shape. However, the existence of such shape factors, also called shape invariants, is well-known. They are associated with each inner control point and are computed as the ratio of weight ratios for three consecutive control points. We show that these shape factors, in addition to their invariance to Moebius reparameterization, provide a more convenient shape control than the customary weights since they exert a more localized push/pull. Each shape factor amounts to that of the conic defined by a triplet of consecutive control points and weights. Thus, shape factors can be controlled in a geometric way using existing techniques for conics by setting the conic rho-factor via moving the associated shoulder point. Each shoulder point moves along a radial direction through its corresponding control point, furnishing a more practical shape handle than sliding the traditional weight points (aka Farin points) on the polygon legs. Highlights: Shape factors of rational Bézier curves must be invariant to Moebius reparameterization. They are defined as the ratio of weight ratios for three consecutive control points. They provide a more convenient shape control (localized push/pull) than the weights. Each factor is that of the conic defined by three consecutive control points and weights. They can be controlled via the conic rho-factor, moving the associated shoulder point.