Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bezier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bezier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bezier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.

[1]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[2]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[3]  Jörg Peters,et al.  Local smooth surface interpolation: a classification , 1990, Comput. Aided Geom. Des..

[4]  D. Struik.  Lectures on classical differential geometry , 1951 .

[5]  Rubén Dorado,et al.  Constrained design of polynomial surfaces from geodesic curves , 2008, Comput. Aided Des..

[6]  Nathalie Sprynski.  Reconstruction de courbes et surfaces à partir de données tangentielles , 2007 .

[7]  Rida T. Farouki,et al.  Existence conditions for Coons patches interpolating geodesic boundary curves , 2009, Comput. Aided Geom. Des..

[8]  Gerald Farin,et al.  Curves and surfaces for cagd , 1992 .

[9]  Ramon F. Sarraga,et al.  Errata: G1 interpolation of generally unrestricted cubic Bézier curves , 1989, Comput. Aided Geom. Des..

[10]  Ramon F. Sarraga,et al.  G1 interpolation of generally unrestricted cubic Bézier curves , 1987, Comput. Aided Geom. Des..

[11]  S. A. Coons.  SURFACE PATCHES AND B-SPLINE CURVES , 1974 .

[12]  T. Banchoff,et al.  Differential Geometry of Curves and Surfaces , 2010 .

[13]  Marco Paluszny,et al.  Cubic polynomial patches through geodesics , 2008, Comput. Aided Des..

[14]  Carlo H. Séquin,et al.  Local surface interpolation with Bézier patches , 1987, Comput. Aided Geom. Des..

[15]  Nathalie Sprynski,et al.  Surface reconstruction via geodesic interpolation , 2008, Comput. Aided Des..