Matrix weighted rational curves and surfaces

Abstract

Rational curves and surfaces are powerful tools for shape representation and geometric modeling. However, the real weights are generally difficult to choose except for a few special cases such as representing conics. This paper presents an extension of rational curves and surfaces by replacing the real weights with matrices. The matrix weighted rational curves and surfaces have the same structures as the traditional rational curves and surfaces but the matrix weights can be defined in geometric ways. In particular, the weight matrices for the extended rational Bézier, NURBS or the generalized subdivision curves and surfaces are computed using the normal vectors specified at the control points. Similar to the effects of control points, the specified normals can be used to control the curve or the surface's shape efficiently. It is also shown that matrix weighted NURBS curves and surfaces can pass through their control points, thus curve or surface reconstruction by the extended NURBS model no longer needs solving any large system but just choosing control points and control normals from the input data.