Construction and smoothing of triangular Coons patches with geodesic boundary curves

Given three regular space curves r"1(t), r"2(t), r"3(t) for t@?[0,1] that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch R(u"1,u"2,u"3) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study (Farouki et al., 2009a) concerned with tensor-product patches are adapted to identify constraints on the given curves for the existence of such geodesic-bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically-blended triangular Coons interpolation scheme. A formulation of thin-plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic-bounded triangular surface patches.

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

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

[3]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

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

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

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

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

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

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

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

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

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

[13]  J. A. Gregory,et al.  A C1 triangular interpolation patch for computer-aided geometric design , 1980 .

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

[15]  Rida T. Farouki,et al.  Construction of Bézier surface patches with Bézier curves as geodesic boundaries , 2009, Comput. Aided Des..

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

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