Sánchez-Reyes, J.; Paluszny, M. Weighted radial displacement: A geometric look at Bézier conics and quadrics. (English) Zbl 0939.68132 Comput. Aided Geom. Des. 17, No. 3, 267-289 (2000). Summary: 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. Cited in 4 Documents MSC: 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry Keywords:rational triangular quadratic Bézier patch; conic; quadric; weighted radial displacement; stereographic projection; degree elevation; Möbius reparameterization PDFBibTeX XMLCite \textit{J. Sánchez-Reyes} and \textit{M. Paluszny}, Comput. Aided Geom. Des. 17, No. 3, 267--289 (2000; Zbl 0939.68132) Full Text: DOI