Weighted radial displacement: A geometric look at Bézier conics and quadrics Original Research Article

We describe a simple geometric construction for Bézier conics and quadrics, based on a tool called Weighted Radial Displacement (WRD). The shape of a rational Bézier curve or surface is modified via WRD by choosing an arbitrary point O and displacing the control points along radial directions through O, changing simultaneously the weights. To construct a conic through O, take 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 Bézier patch on a nondegenerate quadric, which defines a stereographic projection, can be obtained through this method. We present a practical algorithm to detect such quadratic Bézier patches lying on quadrics. Bézier patches on degenerate quadrics are also derived via a WRD.