On rationally supported surfaces

Abstract

We analyze the class of surfaces which are equipped with rational support functions. Any rational support function can be decomposed into a symmetric (even) and an antisymmetric (odd) part. We analyze certain geometric properties of surfaces with odd and even rational support functions. In particular it is shown that odd rational support functions correspond to those rational surfaces which can be equipped with a linear field of normal vectors, which were discussed by Sampoli et al. (Sampoli, M.L., Peternell, M., Jüttler, B., 2006. Rational surfaces with linear normals and their convolutions with rational surfaces. Comput. Aided Geom. Design 23, 179–192). As shown recently, this class of surfaces includes non-developable quadratic triangular Bézier surface patches (Lávička, M., Bastl, B., 2007. Rational hypersurfaces with rational convolutions. Comput. Aided Geom. Design 24, 410–426; Peternell, M., Odehnal, B., 2008. Convolution surfaces of quadratic triangular Bézier surfaces. Comput. Aided Geom. Design 25, 116–129).