Blending, smoothing and interpolation of irregular meshes using N-sided Varady patches

EHending, Smoothing and Interpolation of Irregular Meshes Using N-sided Varady Patches Xuefu Wang* Fuhua (Frank) Cheng** Brian A. Barsky*** *Softimage Co. **University of Kentucky ***University of California, Berkeley *xwang@softimage.com *:*cheng@cs.uky.edu ***barsky@cs.berkeley.edu Blending, smoothing and interpolation of irregular meshes are important modeling problems. Traditionally they are solved by using parametric (NURBS ) or subdivision surfaces, despite NlJRBS represenation imposes somfe topological restrictions, and wbdivision surfaces are mostly used for smoothing and interpolation. N-sided krady surface representation is not new, but its use in solving modeling problems has been limited. In this paper a unified method for blending, smoothing and interpolation of irregular meshes of arbitrary topological lype using Varudy patches is presented. The new approach treats all these problems as a surface fitting problem by first restructuring the given mesh to obtain a new mesh suitable for network curves construction, and then filling all the holes framed by the network curves with n-sided Varady patches. The smoothness of the resulting surface is achieved through a global energy minimization process. The resulting surface is C’ continuous. Several new techniques are introduced; these include restructuring techniques for a given mesh regarding blending and interpolation problems, ,a network curves construction technique, an energy formulation technique for n-sided Varady patches, and an energy computation technique for Varady surfaces. Advantages of the new method include: (1) a unified solution to three important problems in geometric modeling; (2) carrying good features of Varady patches (such 21s no need in domain decomposition in the hole filling process and having an additional degree of freedom in adjusting the shape of an n-sided patch); (3) overall smoothness and more even distribution of curvature because of the global energy minimization process; (4) presenting new, effective way for solving modeling problems using Varady patches. CR

[1]  Christophe Schlick,et al.  X-splines: a spline model designed for the end-user , 1995, SIGGRAPH.

[2]  Carlo H. Séquin,et al.  Functional optimization for fair surface design , 1992, SIGGRAPH.

[3]  Jörg Peters,et al.  Smooth free-form surfaces over irregular meshes generalizing quadratic splines , 1993, Comput. Aided Geom. Des..

[4]  Ahmad H. Nasri,et al.  Polyhedral subdivision methods for free-form surfaces , 1987, TOGS.

[5]  John A. Gregory,et al.  A C2 polygonal surface patch , 1989, Comput. Aided Geom. Des..

[6]  Der Min Tsay,et al.  Corner Blending of Free-Form N-Sided Holes , 1998, IEEE Computer Graphics and Applications.

[7]  Tony DeRose,et al.  Subdivision surfaces in character animation , 1998, SIGGRAPH.

[8]  John A. Gregory,et al.  A pentagonal surface patch for computer aided geometric design , 1984, Comput. Aided Geom. Des..

[9]  Jiaguang Sun,et al.  Blending Polyhedra with NURBS , 1993 .

[10]  Josef Hoschek,et al.  Gn∗-functional splines for interpolation and approximation of curves, surfaces and solids , 1990, Comput. Aided Geom. Des..

[11]  Tamás Várady.  Survey and new results in n-sided patch generation , 1987 .

[12]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[13]  J. Hopcroft,et al.  Quadratic blending surfaces , 1985 .

[14]  A. Requicha,et al.  CONSTANT-RADIUS BLENDING IN SOLID MODELLING , 1984 .

[15]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

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

[17]  A. Robin Forrest,et al.  Curves and surfaces for computer-aided design , 1968 .

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

[19]  Malcolm A. Sabin,et al.  Non-uniform recursive subdivision surfaces , 1998, SIGGRAPH.

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

[21]  Tony DeRose,et al.  Parametric surface interpolation , 1992, IEEE Computer Graphics and Applications.

[22]  Ahmad H. Nasri,et al.  Surface interpolation on irregular networks with normal conditions , 1991, Comput. Aided Geom. Des..

[23]  Bongsik Choi,et al.  Constant-radius blending in surface modelling , 1989 .

[24]  Tony DeRose,et al.  Necessary and sufficient conditions for tangent plane continuity of Bézier surfaces , 1990, Comput. Aided Geom. Des..

[25]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.

[26]  Jindong Chen,et al.  Modeling with cubic A-patches , 1995, TOGS.

[27]  Daniel J. Filip,et al.  Blending parametric surfaces , 1989, TOGS.

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

[29]  Jörg Peters,et al.  Smoothing polyhedra made easy , 1995, TOGS.

[30]  Tamás Várady,et al.  Overlap patches: a new scheme for interpolating curve networks with n-sided regions , 1991, Comput. Aided Geom. Des..

[31]  Charles T. Loop.  Generalized B-spline surfaces of arbitrary topological type , 1992 .

[32]  Matthias Eck,et al.  Automatic reconstruction of B-spline surfaces of arbitrary topological type , 1996, SIGGRAPH.

[33]  G. Greiner.  Surface construction based on variational principles , 1994 .

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

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

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

[37]  John Woodwark.  Blends in geometric modelling , 1987 .

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

[39]  Carlo H. Séquin,et al.  Local surface interpolation with Bézier patches: errata and improvements , 1991, Comput. Aided Geom. Des..

[40]  Jörg Peters,et al.  The simplest subdivision scheme for smoothing polyhedra , 1997, TOGS.

[41]  Fumihiko Kimura,et al.  Non-four-sided patch expressions with control points , 1984, Comput. Aided Geom. Des..

[42]  Larry L. Schumaker,et al.  Minimal energy surfaces using parametric splines , 1996, Comput. Aided Geom. Des..

[43]  Tony DeRose,et al.  A multisided generalization of Bézier surfaces , 1989, TOGS.

[44]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[45]  Markus Schichtel.  G/sup 2/ blend surfaces and filling of N-sided holes , 1993, IEEE Computer Graphics and Applications.

[46]  Nira Dyn,et al.  Using parameters to increase smoothness of curves and surfaces generated by subdivision , 1990, Comput. Aided Geom. Des..

[47]  Charles T. Loop,et al.  Smooth spline surfaces over irregular meshes , 1994, SIGGRAPH.

[48]  Hans Hagen,et al.  Automatic smoothing with geometric surface patches , 1987, Comput. Aided Geom. Des..

[49]  Ahmad H. Nasri,et al.  Boundary-corner control in recursive-subdivision surfaces , 1991, Comput. Aided Des..

[50]  Hiroaki Chiyokura,et al.  Design of solids with free-form surfaces , 1983, SIGGRAPH.

[51]  Insung Ihm,et al.  Smoothing polyhedra using implicit algebraic splines , 1992, SIGGRAPH.

[52]  Jos Stam,et al.  Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values , 1998, SIGGRAPH.

[53]  Tony DeRose,et al.  Efficient, fair interpolation using Catmull-Clark surfaces , 1993, SIGGRAPH.

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

[55]  John F. Hughes,et al.  Modeling surfaces of arbitrary topology using manifolds , 1995, SIGGRAPH.

[56]  J. Peters,et al.  C 1 -surface splines , 1995 .

[57]  Fuhua Cheng,et al.  A blending model for parametrically defined geometric objects , 1991, SMA '91.

[58]  Bahram Ravani,et al.  Optimal twist vectors as a tool for interpolating a network of curves with a minimum energy surface , 1990, Comput. Aided Geom. Des..

[59]  Lizhuang Ma,et al.  Smoothing of free-form surfaces with Bézier patches , 1995, Comput. Aided Geom. Des..

[60]  Nicholas M. Patrikalakis,et al.  Blending Rational B-Spline Surfaces , 1989, Eurographics.