Parametrizations for triangular Gk spline surfaces of low degree

In this article, we present regularly parametrized <i>G</i><sup><i>k</i></sup> free-form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4<i>k</i> + 1, ⌈3k/2 + 1⌉r}, where <i>r</i> ∈ &U2115; can be chosen arbitrarily and determines the flexibility at extraordinary points. The <i>G</i><sup><i>k</i></sup> splines presented in this article depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this work and we present two classes of singular and regular parametrizations. Also, we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only <i>n</i> patches, in principle.

[1]  M. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1978 .

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

[3]  Richard F. Riesenfeld,et al.  A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[5]  Jörg M. Hahn.  Geometric continuous patch complexes , 1989, Comput. Aided Geom. Des..

[6]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[7]  Ruibin Qu.  Recursive subdivision algorithms for curve and surface design (subdivision algorithms) , 1990 .

[8]  C. D. Boor,et al.  Box splines , 1993 .

[9]  U. Reif.  A degree estimate for subdivision surfaces of higher regularity , 1996 .

[10]  Hartmut Prautzsch,et al.  Freeform splines , 1997, Computer Aided Geometric Design.

[11]  Malcolm A. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1998 .

[12]  Hartmut Prautzsch,et al.  Smoothness of subdivision surfaces at extraordinary points , 1998, Adv. Comput. Math..

[13]  J. Clark,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[14]  U. Reif.  TURBS—Topologically Unrestricted Rational B-Splines , 1998 .

[15]  Ulrich Reif,et al.  Degree estimates for Ck‐piecewise polynomial subdivision surfaces , 1999, Adv. Comput. Math..

[16]  H. Prautzsch,et al.  Triangular G2-Splines , 2000 .

[17]  Leif Kobbelt,et al.  √3-subdivision , 2000, SIGGRAPH.

[18]  Peter Schröder,et al.  A unified framework for primal/dual quadrilateral subdivision schemes , 2001, Comput. Aided Geom. Des..

[19]  Hartmut Prautzsch,et al.  Box Splines , 2002, Handbook of Computer Aided Geometric Design.

[20]  W. Boehm,et al.  Bezier and B-Spline Techniques , 2002 .

[21]  Jörg Peters,et al.  C2 free-form surfaces of degree (3, 5) , 2002, Comput. Aided Geom. Des..

[22]  John Hart,et al.  ACM Transactions on Graphics , 2004, SIGGRAPH 2004.

[23]  Jörg Peters,et al.  Shape characterization of subdivision surfaces--basic principles , 2004, Comput. Aided Geom. Des..

[24]  Jörg Peters,et al.  Shape characterization of subdivision surfaces--case studies , 2004, Comput. Aided Geom. Des..

[25]  G. Umlauf.  A Technique for Verifying the Smoothness of Subdivision Schemes , 2022 .