Smooth free-form surfaces over irregular meshes generalizing quadratic splines

Abstract An algorithm for refining a combinatorily unrestricted mesh of points into a bivariate C 1 surface is given. The algorithm generalizes the construction of quadratic splines from a mesh of control points. An explicit parametrization of the surface with quadratic and cubic pieces is given. When the mesh is regular then a quadratic spline surface is generated. Irregular input meshes with nonquadrilateral mesh cells and more or fewer than four cells meeting at a point are allowed and generate spline spaces that generalize the space of quadratic splines. Consequently, the algorithm can model bivariate open or closed surfaces of general topological structure.

[1]  Tony DeRose,et al.  Generalized B-spline surfaces of arbitrary topology , 1990, SIGGRAPH.

[2]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[3]  M. A. Sabin.  Non-Rectangular Surface Patches Suitable for Inclusion in a B-Spline Surface , 1983, Eurographics.

[4]  J. A. Gregory.  Smooth interpolation without twist constraints , 1974 .

[5]  Tim N. T. Goodman,et al.  Closed surfaces defined from biquadratic splines , 1991 .

[6]  Nira Dyn,et al.  Interpolatory convexity-preserving subdivision schemes for curves and surfaces , 1992, Comput. Aided Des..

[7]  Charles T. Loop,et al.  Smooth Subdivision Surfaces Based on Triangles , 1987 .

[8]  P. Zwart.  Multivariate Splines with Nondegenerate Partitions , 1973 .

[9]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[10]  J. Hahn,et al.  Filling polygonal holes with rectangular patches , 1989 .

[11]  C. Micchelli,et al.  Blossoming begets B -spline bases built better by B -patches , 1992 .

[12]  Klaus Höllig,et al.  G-splines , 1990, Comput. Aided Geom. Des..

[13]  M. Sabin.  The use of piecewise forms for the numerical representation of shape , 1976 .

[14]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

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

[16]  Jarke J. van Wijk,et al.  Bicubic patches for approximating non-rectangular control-point meshes , 1986, Comput. Aided Geom. Des..

[17]  J. Peters.  Smooth interpolation of a mesh of curves , 1991 .

[18]  H. Seidel.  Symmetric recursive algorithms for surfaces: B-patches and the de boor algorithm for polynomials over triangles , 1991 .

[19]  John A. Gregory,et al.  Smooth Parametric Surfaces and n-Sided Patches , 1990 .