Weighted radial displacement: A geometric look at Bezier conics and quadrics

We describe a simple geometric construction for Bezier conics and quadrics, based on a tool called Weighted Radial Displacement (WRD). The shape of a rational Bezier curve or surface is modified via WRD by choosing an arbitra ry point O and displacing the control points along radial directions throug h O, changing simultaneously the weights. To construct a conic through O, t ake an arbitrary segment representing a curve of degree n = 1, degree raise it to n = 2 and apply a WRD. Analogously, if a degree-elevated triangle is modified using a WRD, we get a quadric through O. Any quadratic Bezier pat ch on a nondegenerate quadric, which defines a stereographic projection, ca n be obtained through this method. We present a practical algorithm to dete ct such quadratic Bezier patches lying on quadrics. Bezier patches on degen erate quadrics are also derived via a WRD. (C) 2000 Elsevier Science B.V. A ll rights reserved.