On rationally supported surfaces

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., Juttler, 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 Bezier surface patches (Lavicka, 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 Bezier surfaces. Comput. Aided Geom. Design 25, 116-129).

[1]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[2]  Leo F. Boron,et al.  Theory of Convex Bodies , 1988 .

[3]  Jörg Peters,et al.  The 42 equivalence classes of quadratic surfaces in affine n-space , 1998, Comput. Aided Geom. Des..

[4]  Robert E. Barnhill,et al.  Surfaces in Computer Aided Geometric Design , 1983 .

[5]  On C$^2$-smooth Surfaces of Constant Width , 2007, 0704.3248.

[6]  Glen Mullineux,et al.  The Mathematics of Surfaces VI , 1996 .

[7]  Bert Jüttler,et al.  Surfaces with Piecewise Linear Support Functions over Spherical Triangulations , 2007, IMA Conference on the Mathematics of Surfaces.

[8]  Rida T. Farouki,et al.  Pythagorean-Hodograph Curves , 2002, Handbook of Computer Aided Geometric Design.

[9]  Wendelin L. F. Degen.  The Types of Triangular Bézier Surfaces , 1994, IMA Conference on the Mathematics of Surfaces.

[10]  Bert Jüttler,et al.  Curves and surfaces represented by polynomial support functions , 2008, Theor. Comput. Sci..

[11]  J. Gravesen.  Surfaces Parametrised by the Normals , 2006 .

[12]  Bert Jüttler,et al.  Rational surfaces with linear normals and their convolutions with rational surfaces , 2006, Comput. Aided Geom. Des..

[13]  Josef Hoschek,et al.  Handbook of Computer Aided Geometric Design , 2002 .

[14]  H. Groemer.  Geometric Applications of Fourier Series and Spherical Harmonics , 1996 .

[15]  Bohumír Bastl,et al.  Rational hypersurfaces with rational convolutions , 2007, Comput. Aided Geom. Des..

[16]  Ragni Piene,et al.  Monoid hypersurfaces , 2006 .

[17]  Helmut Pottmann,et al.  Rational curves and surfaces with rational offsets , 1995, Comput. Aided Geom. Des..

[18]  Josef Schicho.  Elementary Theory of Del Pezzo Surfaces , 2005 .

[19]  T. Dokken,et al.  Computational Methods for Algebraic Spline Surfaces , 2008 .

[20]  Larry L. Schumaker,et al.  Fitting scattered data on sphere-like surfaces using spherical splines , 1996 .

[21]  Jens Gravesen,et al.  Surfaces parametrized by the normals , 2007, Computing.

[22]  Helmut Pottmann,et al.  A Laguerre geometric approach to rational offsets , 1998, Comput. Aided Geom. Des..

[23]  T. Sakkalis,et al.  Pythagorean hodographs , 1990 .

[24]  Martin Peternell,et al.  Convolution surfaces of quadratic triangular Bézier surfaces , 2008, Comput. Aided Geom. Des..

[25]  R. Connelly.  In Handbook of Convex Geometry , 1993 .

[26]  Ragni Piene,et al.  Geometric Modeling and Algebraic Geometry , 2007 .

[27]  Hans Hagen,et al.  Curve and Surface Design , 1992 .