Subdivision Methods for Geometric Design: A Constructive Approach

From the Publisher: The world's leading animation houses rely increasingly on subdivision methods for creating realistic-looking complex shapes. However, until now there was no one book devoted to this powerful geometric modeling technique. Subdivision Methods for Geometric Design does the job with authority and precision, providing all that is needed to understand how subdivision works its magic, and how to make that magic work. Throughout the book, icons cue readers to visit a companion Web site loaded with interactive exercises, implementations of the book's images, and supplementary material. Rich in theory, analysis, and practical information, this book is the complete resource for subdivision methods. Features The result of a collaboration between a leading university researcher and an industry practitioner. The only book devoted exclusively and comprehensively to this important new technology. Provides solid background and theoretical analysis of subdivision as well as a wide variety of specific applications. Addresses algorithms for Bezier and uniform B-Spline curves, Catmull-Clark subdivision for quad meshes, and regularity tests for polyhedral meshes. Via the companion Web site, (www.subdivision.com), provides opportunities for readers to experiment hands-on with implementations in a richly interactive environment. Includes a foreword by Tony DeRose, recipient of the 1999 ACM Computer Graphics Achievement Award for his seminal work in subdivision methods. Author Biography: Joe Warren, Professor of Computer Science at Rice University since 1986, is one of the world's leading experts on subdivision. Of his nearly 50 computer science papers-published in prestigious forums such as SIGGRAPH, Transactions on Graphics, Computer-Aided Geometric Design, and The Visual Computer-a dozen specifically address subdivision and its applications to computer graphics. Prof. Warren received both his M.S. and Ph.D. in Computer Science at Cornell University. His research interests focus on mathematical methods for representing geometric shape. Henrik Weimer is a research scientist at the DaimlerChrysler Corporate Research Center in Berlin, where he works on knowledge-based support for the design and creation of engineering products. Dr. Weimer obtained his Ph.D. in Computer Science from Rice University.

[1]  Dimitris N. Metaxas,et al.  Realistic Animation of Liquids , 1996, Graphics Interface.

[2]  Tom Lyche,et al.  Cones and recurrence relations for simplex splines , 1987 .

[3]  Dimitris N. Metaxas,et al.  Modeling the motion of a hot, turbulent gas , 1997, SIGGRAPH.

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

[5]  Ahmad H. Nasri,et al.  Interpolating meshes of boundary intersecting curves by subdivision surfaces , 2000, The Visual Computer.

[6]  Klaus Höllig,et al.  B-splines from parallelepipeds , 1982 .

[7]  Nira Dyn,et al.  Interpolation of scattered Data by radial Functions , 1987, Topics in Multivariate Approximation.

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

[9]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

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

[11]  Peter Schröder,et al.  Wavelets in computer graphics , 1996, Proc. IEEE.

[12]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[13]  Lyle Ramshaw,et al.  Blossoms are polar forms , 1989, Comput. Aided Geom. Des..

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

[15]  C. Micchelli,et al.  Recent Progress in multivariate splines , 1983 .

[16]  Helmut Pottmann,et al.  Symmetric Tchebycheffian B-spline schemes , 1994 .

[17]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[18]  Leif Kobbelt,et al.  Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology , 1996, Comput. Graph. Forum.

[19]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[20]  Wolfgang Böhm.  On the efficiency of knot insertion algorithms , 1985, Comput. Aided Geom. Des..

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

[22]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[23]  Wolfgang Dahmen,et al.  Computation of inner products of multivariate B-splines , 1981 .

[24]  Joe Warren,et al.  Sparse Filter Banks for Binary Subdivision Schemes , 1996 .

[25]  Peter Schröder,et al.  Multiresolution signal processing for meshes , 1999, SIGGRAPH.

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

[27]  Tom Lyche,et al.  Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics , 1980 .

[28]  Joe D. Warren,et al.  Subdivision Schemes for Thin Plate Splines , 1998, Comput. Graph. Forum.

[29]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[30]  Murray R. Spiegel,et al.  Schaum's outline of theory and problems of calculus of finite differences and difference equations , 1971 .

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

[32]  I. N. Sneddon,et al.  Boundary value problems , 2007 .

[33]  Richard F. Riesenfeld,et al.  Discrete box splines and refinement algorithms , 1984, Comput. Aided Geom. Des..

[34]  Gavin S. P. Miller,et al.  Rapid, stable fluid dynamics for computer graphics , 1990, SIGGRAPH.

[35]  Tom Lyche,et al.  Control curves and knot insertion for trigonometric splines , 1995, Adv. Comput. Math..

[36]  Gerald E. Farin,et al.  NURBS: From Projective Geometry to Practical Use , 1999 .

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

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

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

[40]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[41]  Henning Biermann,et al.  Piecewise smooth subdivision surfaces with normal control , 2000, SIGGRAPH.

[42]  C. Micchelli.  On a numerically efficient method for computing multivariate B-splines , 1979 .

[43]  Tom Lyche,et al.  Construction of Exponential Tension B-splines of Arbitrary Order , 1991, Curves and Surfaces.

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

[45]  Luiz Velho,et al.  4-8 Subdivision , 2001, Comput. Aided Geom. Des..

[46]  C. Micchelli,et al.  Uniform refinement of curves , 1989 .

[47]  J. Warren,et al.  Subdivision methods for geometric design , 1995 .

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

[49]  Zhang Jiwen,et al.  C-curves: An extension of cubic curves , 1996, Comput. Aided Geom. Des..

[50]  S. Rippa,et al.  Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions , 1986 .

[51]  Joe D. Warren,et al.  Subdivision schemes for fluid flow , 1999, SIGGRAPH.

[52]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[53]  G. Wahba,et al.  Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation , 1980 .

[54]  Norishige Chiba,et al.  Visual simulation of water currents using a particle-based behavioural model , 1995, Comput. Animat. Virtual Worlds.

[55]  J. Thorpe,et al.  Lecture Notes on Elementary Topology and Geometry. , 1967 .

[56]  D. Schweikert.  An Interpolation Curve Using a Spline in Tension , 1966 .

[57]  Charles A. Micchelli,et al.  Computing curves invariant under halving , 1987, Comput. Aided Geom. Des..

[58]  Chandrajit L. Bajaj,et al.  Automatic parameterization of rational curves and surfaces III: Algebraic plane curves , 1988, Comput. Aided Geom. Des..

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

[60]  R. N. Desmarais,et al.  Interpolation using surface splines. , 1972 .

[61]  Ulrich Reif,et al.  Curvature integrability of subdivision surfaces , 2001, Adv. Comput. Math..

[62]  Denis Z orin.  Smoothness of Stationary Subdivision on Irregular Meshes , 1998 .

[63]  I. J. Schoenberg,et al.  On Pólya frequency functions IV: The fundamental spline functions and their limits , 1966 .

[64]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[65]  W. Fleming.  Functions of Several Variables , 1965 .

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

[67]  Joe D. Warren,et al.  A subdivision scheme for surfaces of revolution , 2001, Comput. Aided Geom. Des..

[68]  Eric W. Weisstein,et al.  The CRC concise encyclopedia of mathematics , 1999 .

[69]  C. D. Boor,et al.  On splines and their minimum properties , 1966 .

[70]  M. A. Sabin,et al.  Cubic Recursive Division With Bounded Curvature , 1991, Curves and Surfaces.

[71]  Helmut Pottmann,et al.  Helix splines as an example of affine Tchebycheffian splines , 1994, Adv. Comput. Math..

[72]  Henrik WeimerAbstract,et al.  Variational Subdivision for Natural Cubic Splines , 1998 .

[73]  Wolfgang Dahmen,et al.  Subdivision algorithms for the generation of box spline surfaces , 1984, Comput. Aided Geom. Des..

[74]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[75]  E. Kunz.  Introduction to commutative algebra and algebraic geometry , 1984 .

[76]  Y. Meyer,et al.  Wavelets and Filter Banks , 1991 .

[77]  C. Micchelli,et al.  Stationary Subdivision , 1991 .

[78]  George Merrill Chaikin,et al.  An algorithm for high-speed curve generation , 1974, Comput. Graph. Image Process..

[79]  John Charles Fields.  Theory of the algebraic functions of a complex variabel , .

[80]  Gilles Deslauriers,et al.  Symmetric iterative interpolation processes , 1989 .

[81]  Ayman Habib,et al.  Edge and vertex insertion for a class of C1 subdivision surfaces , 1999, Comput. Aided Geom. Des..

[82]  Joe Warren,et al.  Binary Subdivision Schemes for Functions over Irregular Knot Sequences , 1995 .

[83]  M LaneJeffrey,et al.  A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces , 1980 .

[84]  Jim X. Chen,et al.  Toward Interactive-Rate Simulation of Fluids with Moving Obstacles Using Navier-Stokes Equations , 1995, CVGIP Graph. Model. Image Process..

[85]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[86]  Wolfgang Dahmen,et al.  Multivariate B-Splines — Recurrence Relations and Linear Combinations of Truncated Powers , 1979 .

[87]  Nira Dyn,et al.  Analysis of uniform binary subdivision schemes for curve design , 1991 .

[88]  D. T. Kaplan,et al.  Finite-Difference Equations , 1995 .

[89]  Ahmad H. Nasri,et al.  Curve interpolation in recursively generated B-spline surfaces over arbitrary topology , 1997, Comput. Aided Geom. Des..

[90]  Gavin S. P. Miller,et al.  Globular dynamics: A connected particle system for animating viscous fluids , 1989, Comput. Graph..

[91]  Peter Schröder,et al.  Trimming for subdivision surfaces , 2001, Comput. Aided Geom. Des..

[92]  Jörg Peters,et al.  Computing curvature bounds for bounded curvature subdivision , 2001, Comput. Aided Geom. Des..

[93]  Alyn P. Rockwood,et al.  Interactive curves and surfaces - a multimedia tutorial and CAGD , 1996 .

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

[95]  N. Dyn,et al.  Multiresolution Analysis by Infinitely Differentiable Compactly Supported Functions , 1995 .

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

[97]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[98]  Hartmut Prautzsch,et al.  A G2-Subdivision Algorithm , 1996, Geometric Modelling.

[99]  S. S. Abi-Ezzi,et al.  The graphical processing of B-splines in a highly dynamic environment , 1990 .

[100]  Peter Schröder,et al.  A multiresolution framework for variational subdivision , 1998, TOGS.

[101]  Hans-Peter Seidel,et al.  An introduction to polar forms , 1993, IEEE Computer Graphics and Applications.

[102]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[103]  Michael E. Mortenson.  Computer graphics handbook: geometry and mathematics , 1990 .

[104]  A. K. Cline.  Scalar- and planar-valued curve fitting using splines under tension , 1974, Commun. ACM.

[105]  J. Warren.  On algebraic surfaces meeting with geometric continuity , 1986 .

[106]  Leif Kobbelt,et al.  A variational approach to subdivision , 1996, Comput. Aided Geom. Des..

[107]  C. Micchelli,et al.  Banded matrices with banded inverses, II: Locally finite decomposition of spline spaces , 1993 .

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

[109]  Eugene Fiume,et al.  Turbulent wind fields for gaseous phenomena , 1993, SIGGRAPH.

[110]  Hans-Peter Seidel,et al.  Interactive multi-resolution modeling on arbitrary meshes , 1998, SIGGRAPH.

[111]  D. Levin,et al.  Analysis of asymptotically equivalent binary subdivision schemes , 1995 .

[112]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[113]  F. Holt.  Toward a curvature-continuous stationary subdivision algorithm , 1996 .

[114]  Jörg Peters,et al.  Gaussian and Mean Curvature of Subdivision Surfaces , 2000, IMA Conference on the Mathematics of Surfaces.

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

[116]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[117]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[118]  Nira Dyn,et al.  A 4-point interpolatory subdivision scheme for curve design , 1987, Comput. Aided Geom. Des..

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

[120]  Jörg Peters,et al.  Computing Volumes of Solids Enclosed by Recursive Subdivision Surfaces , 1997, Comput. Graph. Forum.

[121]  Adi Levin,et al.  Interpolating nets of curves by smooth subdivision surfaces , 1999, SIGGRAPH.

[122]  K. B. Oldham,et al.  An Atlas of Functions. , 1988 .

[123]  A. A. Ball,et al.  An investigation of curvature variations over recursively generated B-spline surfaces , 1990, TOGS.

[124]  Nira Dyn,et al.  The subdivision experience , 1994 .

[125]  Tony DeRose,et al.  Multiresolution analysis for surfaces of arbitrary topological type , 1997, TOGS.

[126]  Joe Warren,et al.  Subdivision Schemes for Variational Splines , 2000 .

[127]  Adi Levin.  Combined subdivision schemes for the design of surfaces satisfying boundary conditions , 1999, Comput. Aided Geom. Des..

[128]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[129]  P. Alfeld.  Scattered data interpolation in three or more variables , 1989 .

[130]  Peter Schröder,et al.  Spherical wavelets: efficiently representing functions on the sphere , 1995, SIGGRAPH.

[131]  A. A. Ball,et al.  Conditions for tangent plane continuity over recursively generated B-spline surfaces , 1988, TOGS.

[132]  Tunc Geveci,et al.  Advanced Calculus , 2014, Nature.

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

[134]  Requicha,et al.  Solid Modeling: A Historical Summary and Contemporary Assessment , 1982, IEEE Computer Graphics and Applications.

[135]  H. Anton,et al.  Functions of several variables , 2021, Thermal Physics of the Atmosphere.

[136]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[137]  Jakub Wejchert,et al.  Animation aerodynamics , 1991, SIGGRAPH.

[138]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[139]  Manfred R. Trummer,et al.  Multivariate B-Splines , 1990 .

[140]  J. Warren,et al.  A Smooth Subdivision Scheme for Hexahedral Meshes , 2001 .

[141]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[142]  David Salesin,et al.  Wavelets for computer graphics: a primer. 2 , 1995, IEEE Computer Graphics and Applications.

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

[144]  L. Kobbelt.  Fairing by finite difference methods , 1998 .