Mathematical Tools in Computer Graphics with C# Implementations Downloaded from

Bibliography [1] T. Akenine-Möller and E. Haines. Real-Time Rendering. A.K. Peters Ltd., Natick, Massachusetts, second edition, 2002. [2] A. W. Bargteil, T. G. Goktekin, J. F. O brien, and J. A. Strain. A semi-lagrangian contouring method for fluid simulation. ACM Trans. Graph., 25(1):19 38, 2006. [3] M. F. Barnsley. Fractals Everywhere. Academic Press, San Diego, 1988. [4] G. Beylkin. On the fast Fourier transform of functions with singularities. Applied and Computational Harmonic Analysis, 2:363 381, 1995. [5] P. Bézier. Numerical Control: Mathematics and Applications. Wiley, Chichester, UK, 1972. [6] J. Bolz and P. Schröder. Rapid evaluation of Catmull-Clark subdivision surfaces. In Web3D 02: Proceeding of the seventh international conference on 3D Web technology, pages 11 17. ACM Press, 2002. [7] O. P. Bruno and M. M. Pohlman. High order surface representation. In Topics in Computational Wave Propagation, Direct and Inverse Problems, volume 31 of Lecture Notes in Computational Science and Engineering, pages 47 53. Springer-Verlag, 2003. [8] S. Buss. 3-D Computer Graphics. Cambridge University Press, Cambridge, 2003. [9] A. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo. Lossless image compression using integer to integer wavelet transforms. In ICIP 97: Proceedings of the 1997 International Conference on Image Processing (ICIP 97) 3-Volume Set-Volume 1, page 596, Washington, DC, USA, 1997. IEEE Computer Society. 459

460 Bibliography [10] S. Campagna, P. Slusallek, and H.-P. Seidel. Ray tracing of spline surfaces: Bézier clipping, Chebyshev boxing, and bounding volume hierarchy - a critical comparison with new results. The Visual Computer, 13(6):265 282, 1997. [11] M. Carlson, P. J. Mucha, and G. Turk. Rigid fluid: animating the interplay between rigid bodies and fluid. In SIGGRAPH 04: ACM SIGGRAPH 2004 Papers, pages 377 384, New York, NY, USA, 2004. ACM Press. [12] E. Catmull. Subdivision Algorithm for the Display of Curved Surfaces. PhD thesis, University of Utah, 1974. [13] E. Catmull and R. Rom. A class of local interpolating splines. In R. Barnhill and R. Riesenfeld, editors, Computer Aided Geometric Design, pages 317 326, San Francisco, 1974. Academic Press. [14] G. Celniker and D. Gossard. Deformable curve and surface finiteelements for free-form shape design. In SIGGRAPH 91: Proceedings of the 18th annual conference on Computer graphics and interactive techniques, pages 257 266, New York, NY, USA, 1991. ACM Press. [15] G. Chaikin. An algorithm for high speed curve generation. Computer Graphics and Image Processing, 4(3), 1974. [16] F. Cheng and A. Goshtasby. A parallel B-spline surface fitting algorithm. ACM Trans. Graph., 8(1):41 50, 1989. [17] B. V. Cherkassky, A. V. Goldberg, and T. Radzik. Shortest paths algorithms: theory and experimental evaluation. In SODA 94: Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms, pages 516 525, Philadelphia, PA, USA, 1994. Society for Industrial and Applied Mathematics. [18] F. C. Crow. The origins of the teapot. IEEE Computer Graphics and Applications, 7(1):8 19, 1987. [19] I. Daubechies. The wavelet-transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory, 36:961 1005, 1990. [20] A. de Cusatis Junior, L. H. de Figuieredo, and M. Gattass. Interval methods for ray casting implicit surfaces with affine arithmetic. Computer graphics forum, 20(3), 2001. [21] T. DeRose, M. Kass, and T. Truong. Subdivision surfaces in character animation. Computer Graphics, 32(Annual Conference Series):85 94, Aug. 1998.

Bibliography 461 [22] M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. [23] D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356 360, 1978. [24] A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data. Siam Journal on Scientific Computing, 14(6):1368 1393, 1993. [25] A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data II. Applied and Computation Harmonic Analysis, 2:85 100, 1995. [26] N. Dyn, D. Levin, and J. A. Gregory. A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design, 4(4):160 169, 1987. [27] D. S. Ebert, F. K. Musgrave, D. Peachey, K. Perlin, and S. Worley. Texturing and Modeling: A Procedural Approach. Academic Press, San Diego, second edition, 1998. [28] M. Eck and H. Hoppe. Automatic reconstruction of B-spline surfaces of arbitrary topological type. Computer Graphics, 30(Annual Conference Series):325 334, 1996. [29] D. Enright, S. Marschner, and R. Fedkiw. Animation and rendering of complex water surfaces. In SIGGRAPH 02: Proceedings of the 29th annual conference on Computer graphics and interactive techniques, pages 736 744, New York, NY, USA, 2002. ACM Press. [30] M. P. Epstein. On the influence of parametrization in parametric interpolation. SIAM Journal of Numerical Analysis, 13(2):261 268, 1976. [31] J. Erickson and S. Har-Peled. Optimally cutting a surface into a disk. In SCG 02: Proceedings of the eighteenth annual symposium on Computational geometry, pages 244 253, New York, NY, USA, 2002. ACM Press. [32] G. Farin. Curves and Surfaces for Computer Aided Geometric Design A practical guide. Academic Press Inc., fifth edition, 2002. [33] R. Fedkiw, J. Stam, and H. W. Jensen. Visual simulation of smoke. In SIGGRAPH 01: Proceedings of the 28th annual conference on Computer graphics and interactive techniques, pages 15 22, New York, NY, USA, 2001. ACM Press.

462 Bibliography [34] A. Finkelstein and D. H. Salesin. Multiresolution curves. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pages 261 268. ACM Press, 1994. [35] M. S. Floater. Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:232 250, 1997. [36] M. S. Floater. Mean value coordinates. Comput. Aided Geom. Des., 20(1):19 27, 2003. [37] M. S. Floater and K. Hormann. Surface parameterization: a tutorial and survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in multiresolution for geometric modelling, pages 157 186. Springer Verlag, 2005. [38] J. D. Foley, A. van Dam, S. K. Feiner, and H. John F. Computer Graphics, Principles and Practice. Addison Wesley, Reading, Massachusetts, second edition, 1995. [39] A. R. Forrest. The twisted cubic curve: A computer-aided geometric design approach. Computer Aided Design, 12(4):165 172, 1980. [40] N. Foster and D. Metaxis. Realistic animation of liquids. Graphical Models and Image Processing, 58(5):471 483, 1996. [41] A. Fournier and J. Buchanan. Chebyshev polynomials for boxing and intersections of parametric curves and surfaces. Computer Graphics Forum, 13(3):127 142, 1994. [42] I. Friedel, P. Schröder, and M. Desbrun. Unconstrained spherical parameterization. In SIGGRAPH 05: ACM SIGGRAPH 2005 Sketches, page 134, New York, NY, USA, 2005. ACM Press. [43] R. Goldman. Graphics gems, chapter Matrices and transformations, pages 472 475. Academic Press Professional, Inc., San Diego, CA, USA, 1990. [44] R. Goldman. The ambient spaces of computer graphics and geometric modeling. IEEE Comput. Graph. Appl., 20(2):76 84, 2000. [45] R. Goldman. Baseball arithmetic and the laws of pseudoperspective. IEEE Comput. Graph. Appl., 21(2):70 78, 2001. [46] R. Goldman. On the algebraic and geometric foundations of computer graphics. ACM Trans. Graph., 21(1):52 86, 2002.

Bibliography 463 [47] R. Goldman. Deriving linear transformations in three dimensions. IEEE Comput. Graph. Appl., 23(3):66 71, 2003. [48] C. Gotsman, X. Gu, and A. Sheffer. Fundamentals of spherical parameterization for 3d meshes. In SIGGRAPH 03: ACM SIGGRAPH 2003 Papers, pages 358 363, New York, NY, USA, 2003. ACM Press. [49] A. Graps. An introduction to wavelets. IEEE Computational Science and Engineering, 2(2), 1995. [50] G. Greiner and K. Hormann. Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In A. L. Méhauté, C. Rabut, and L. L. Schumaker, editors, Surface Fitting and Multiresolution Methods, Innovations in Applied Mathematics, pages 163 172. Vanderbilt University Press, Nashville, TN, 1997. [51] X. Gu. Parametrization for surfaces with arbitrary topologies. PhD thesis, Harvard University, 2002. [52] X. Gu, S. Gortler, and H. Hoppe. Geometry images. Computer Graphics Proceedings (SIGGRAPH 2002), pages 355 361, 2002. [53] M. Halstead, M. Kass, and T. DeRose. Efficient, fair interpolation using Catmull-Clark surfaces. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pages 35 44. ACM Press, 1993. [54] A. Hardy and W.-H. Steeb. Harmonic interpolation and Lie groups. International Journal of Theoretical Physics, 43(5):1261 1266, 2004. [55] A. Hardy and W.-H. Steeb. Harmonic interpolation, Bézier curves and trigonometric interpolation. Z. Naturforsch., 59a:591 596, 2004. [56] F. H. Harlow and J. E. Welch. Numerical calculation of timedependant viscous incompressible flow of fluids with free surface. Physics of Fluids, 8(12):2182 2189, 1965. [57] M. J. Harris. GPUGems, chapter Fast Fluid Dynamics Simulation on the GPU, pages 637 663. Addison-Wesley, 2004. [58] J. C. Hart. Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer, 12(10):527 545, 1996. [59] P. S. Heckbert. Adaptive radiosity textures for bidirectional ray tracing. In SIGGRAPH 90: Proceedings of the 17th annual conference

464 Bibliography on Computer graphics and interactive techniques, pages 145 154, New York, NY, USA, 1990. ACM Press. [60] N. J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15(2):227 242, 1997. [61] J. D. Hobby. Smooth, easy to compute interpolating splines. Discrete and Computational Geometry, 1(2):123 140, 1986. [62] J. D. Hobby. A user s manual for MetaPost. Technical Report 162, AT&T Bell Laboratories, Murray Hill NJ, 1992. [63] H. Hoppe. Surface Reconstruction from Unorganized Points. PhD thesis, University of Washington, 1994. [64] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle. Piecewise smooth surface reconstruction. Computer Graphics, 28(Annual Conference Series):295 302, 1994. [65] J. Hoschek and D. Lasser. Fundamentals of Computer Aided Geometric Design. A.K. Peters Ltd., Natick, Massachusetts, 1993. [66] A. Iones, A. Krupin, M. Sbert, and S. Zhukov. Fast, realistic lighting for video games. IEEE Computer Graphics and Applications, 23(3):54 64, 2003. [67] H. W. Jensen. Realistic Image Synthesis Using Photon Mapping. A. K. Peters Ltd., Natick, Massachusetts, 2001. [68] H. Jiaxing. On a linear combination of S.N. Bernstein trigonometric interpolation polynomial. Applied Mathematics and Computation, 106(2 3):197 203, 1999. [69] L. W. Johnson and R. D. Riess. Numerical analysis. Addison Wesley, second edition, 1982. [70] K. I. Joy and M. N. Bhetanabhotla. Ray tracing parametric surface patches utilizing numerical techniques and ray coherence. In Proceedings of the 13th annual conference on Computer graphics and interactive techniques, pages 279 285. ACM Press, 1986. [71] J. T. Kajiya. Ray tracing parametric patches. In Proceedings of the 9th annual conference on Computer graphics and interactive techniques, pages 245 254. ACM Press, 1982.

Bibliography 465 [72] D. E. Knuth. Computers and Typesetting, volume D. Addison Wesley, Reading, Massachusetts, 1986. [73] D. E. Knuth. The Art of Computer Programming - Fundamental Algorithms, volume 1. Addison Wesley, Reading, Massachusetts, third edition, 1997. [74] L. Kobbelt. 3 subdivision. Computer Graphics Proceedings (SIG- GRAPH 2000), pages 103 112, 2000. [75] L. Kobbelt, K. Daubert, and H.-P. Seidel. Ray tracing of subdivision surfaces. In Rendering Techniques 98 proceedings of the 9th Eurographics Workshop on Rendering, Berlin, 1998. Springer Verlag. [76] W. Koepf. Efficient computation of Chebyshev polynomials. In M. Wester, editor, Computer Algebra Systems: A Practical Guide, pages 79 99, Chichester, 1999. John Wiley. [77] R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. In Computing, volume 4, pages 187 201, 1969. [78] E. Kreysig. Advanced Engineering Mathematics. John Wiley & Sons, Singapore, eighth edition, 1999. [79] U. Labsik and G. Greiner. Interpolatory 3 subdivision. Computer Graphics Forum (Proceedings of Eurographics 2000), 19(3):131 138, 2000. [80] H. Landis. Production-ready global illumination. In Renderman in Production, pages 87 101. ACM SIGGRAPH Course notes, 2002. [81] S. Lee, G. Wolberg, and S. Shin. Scattered data interpolation with multilevel B-splines. IEEE Transactions on Visualization and Computer Graphics, 3(3):228 244, 1997. [82] S.-L. Lien, M. Shantz, and V. Pratt. Adaptive forward differencing for rendering curves and surfaces. In SIGGRAPH 87: Proceedings of the 14th annual conference on Computer graphics and interactive techniques, pages 111 118. ACM Press, 1987. [83] D. Lischinski and J. Gonczarowski. Improved techniques for ray tracing parametric surfaces. The Visual Computer: International Journal of Computer Graphics, 6(3):134 152, 1990. [84] N. Litke, A. Levin, and P. Schröder. Fitting subdivision surfaces. In VIS 01: Proceedings of the conference on Visualization 01, pages 319 324. IEEE Computer Society, 2001.

466 Bibliography [85] C. Loop. Smooth subdivision based on triangles. Master s thesis, University of Utah, 1987. [86] C. Loop. Generalized B-spline Surfaces of Arbitrary Topological Type. PhD thesis, University of Washington, 1992. [87] F. Losasso, F. Gibou, and R. Fedkiw. Simulating water and smoke with an octree data structure. In SIGGRAPH 04: ACM SIGGRAPH 2004 Papers, pages 457 462, New York, NY, USA, 2004. ACM Press. [88] F. Losasso, H. Hoppe, S. Schaefer, and J. Warren. Smooth geometry images. Eurographics Symposium on Geometry Processing, pages 138 145, 2003. [89] W. Ma and J.-P. Kruth. NURBS curve and surface fitting and interpolation. Mathematical Methods for Curves and Surfaces, pages 315 322, 1995. [90] B. B. Mandelbrot. The Fractal Geometry of Nature. Freeman and Company, New York, 1982. [91] D. Manocha and J. Demmel. Algorithms for intersecting parametric and algebraic curves I: simple intersections. ACM Trans. Graph., 13(1):73 100, 1994. [92] G. Miller. Efficient algorithms for local and global accessibility shading. In SIGGRAPH 94: Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pages 319 326, New York, NY, USA, 1994. ACM Press. [93] R. E. Moore. The automatic analysis and control of error in digital computing based on the use of interval numbers, volume 1, chapter 2, pages 61 130. John Wiley and Sons, 1965. [94] H. P. Moreton and C. H. Séquin. Functional optimization for fair surface design. In SIGGRAPH 92: Proceedings of the 19th annual conference on Computer graphics and interactive techniques, pages 167 176. ACM Press, 1992. [95] A. C. R. Newberry. Interpolation by algebraic and trigonometric polynomials (in technical notes and short papers). Mathematics of Computation, 20(96):597 599, 1966. [96] A. C. R. Newberry. Trigonometric interpolation and curve-fitting. Mathematics of Computation, 24(112):869 876, 1970.

Bibliography 467 [97] T. Nishita, T. W. Sederberg, and M. Kakimoto. Ray tracing trimmed rational surface patches. In Proceedings of the 17th annual conference on Computer graphics and interactive techniques, pages 337 345. ACM Press, 1990. [98] J. Peters. Constructing C 1 surfaces of arbitrary topology using biquadratic and bicubic splines. In N. Sapidis, editor, Designing Fair Curves and Surfaces, pages 277 293. SIAM, 1994. [99] L. Piegl and W. Tiller. The NURBS Book. Springer Verlag, Berlin/Heidelberg, second edition, 1997. [100] E. Praun and H. Hoppe. Spherical parametrization and remeshing. Computer Graphics Proceedings (SIGGRAPH 2003), pages 340 349, 2003. [101] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C. Cambridge University Press, second edition, 2002. [102] P. Prusinkiewicz and A. Lindenmayer. The Algorithmic Beauty of Plants. Springer Verlag, New York, 1996. [103] N. Ray, W. C. Li, B. Lévy, A. Sheffer, and P. Alliez. Periodic global parameterization. ACM Trans. Graph., 25(4):1460 1485, 2006. [104] A. Razdan and G. Farin. Determination of end conditions for NURB surface interpolation. Computer Aided Geometric Design, 15(7):757 768, 1998. [105] U. Reif. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design, 12(2):153 174, 1995. [106] S. M. Roth, P. Diezi, and M. H. Gross. Ray tracing triangular Bézier patches. Computer graphics forum, 20(3), 2001. [107] P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe. Texture mapping progressive meshes. In SIGGRAPH 01: Proceedings of the 28th annual conference on Computer graphics and interactive techniques, pages 409 416, New York, NY, USA, 2001. ACM Press. [108] P. Schröder. Wavelets in computer graphics. Proceedings of the IEEE, 84(4):615 625, 1996. [109] W. Schuster. A closed algebraic interpolation curve. Computer Aided Geometric Design, 17(7):631 642, 2000.

468 Bibliography [110] W. Schuster. Erratum to: a closed algebraic interpolation curve. Computer Aided Geometric Design, 18(1):73 76, 2001. [111] W. Schuster. Harmonische interpolation. In Math. Semesterber., volume 48, pages 1 27. Springer-Verlag, 2001. [112] D. Schweitzer and E. S. Cobb. Scanline rendering of parametric surfaces. In Proceedings of the 9th annual conference on Computer graphics and interactive techniques, pages 265 271. ACM Press, 1982. [113] T. W. Sederberg and D. C. Anderson. Ray tracing of Steiner patches. Computer Graphics Proceedings (SIGGRAPH 1984), 18(3):159 164, 1984. [114] A. Sheffer, C. Gotsman, and N. Dyn. Robust spherical parameterization of triangular meshes. Computing, 72(1-2):185 193, 2004. [115] A. Sheffer and J. C. Hart. Seamster: Inconspicuous low-distortion texture seam layout. In VIS 02: Proceedings of the conference on Visualization 02, Washington, DC, USA, 2002. IEEE Computer Society. [116] A. Sheffer, B. Lévy, M. Mogilnitsky, and A. Bogomyakov. ABF++: fast and robust angle based flattening. ACM Trans. Graph., 24(2):311 330, 2005. [117] L. A. Shirman and C. H. Séquin. Local surface interpolation with Bézier patches. Computer Aided Geometric Design, 4(4):279 295, 1987. [118] L. A. Shirman and C. H. Séquin. Local surface interpolation with Bézier patches: errata and improvements. Computer Aided Geometric Design, 8(3):217 221, 1991. [119] K. Shoemake. Animating rotation with quaternion curves. In SIG- GRAPH 85: Proceedings of the 12th annual conference on Computer graphics and interactive techniques, pages 245 254, New York, NY, USA, 1985. ACM Press. [120] J. M. Snyder. Interval analysis for computer graphics. In Proceedings of the 19th annual conference on Computer graphics and interactive techniques, pages 121 130. ACM Press, 1992. [121] J. Stam. Stable fluids. In SIGGRAPH 99: Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pages 121 128, New York, NY, USA, 1999. ACM Press/Addison- Wesley Publishing Co.

Bibliography 469 [122] W.-H. Steeb. Matrix Calculus and Kronecker Product with Applications and C++ Programs. World Scientific, Singapore, 1997. [123] W.-H. Steeb. Problems and Solutions in Theoretical and Mathematical Physics. World Scientific, Singapore, 2002. [124] W.-H. Steeb. Mathematical Tools in Signal Processing with C++ and Java Simulations. World Scientific, Singapore, 2005. [125] W.-H. Steeb. The Nonlinear Workbook. World Scientific, Singapore, third edition, 2005. [126] W.-H. Steeb. Problems and Solutions in Introductory and Advanced Matrix Calculus. World Scientific, Singapore, 2006. [127] W.-H. Steeb, Y. Hardy, A. Hardy, and R. Stoop. Problems and Solutions in Scientific Computing. World Scientific, Singapore, 2004. [128] E. J. Stollnitz, T. D. DeRose, and D. H. Salesin. Wavelets for computer graphics: A primer, part 1. IEEE Computer Graphics and Applications, 15(3):76 84, 1995. [129] E. J. Stollnitz, T. D. DeRose, and D. H. Salesin. Wavelets for computer graphics: A primer, part 2. IEEE Computer Graphics and Applications, 15(4):75 85, 1995. [130] G. Strang. Wavelets. American Scientist, 82:250 255, 1994. [131] W. Stürzlinger. Ray tracing triangular trimmed free-form surfaces. IEEE Transactions on Visualization and Computer Graphics, 4(3):202 214, 1998. [132] W. Sweldens and P. Schröder. Building your own wavelets at home. In Wavelets in Computer Graphics, pages 15 87. ACM SIGGRAPH Course notes, 1996. [133] D. L. Toth. On ray tracing parametric surfaces. In Proceedings of the 12th annual conference on Computer graphics and interactive techniques, pages 171 179. ACM Press, 1985. [134] A. Tucker. Applied Combinatorics. John Wiley & Sons, New York, third edition, 1995. [135] A. Vlachos, J. Peters, C. Boyd, and J. L. Mitchell. Curved PN triangles. In ACM Symposium on Interactive 3D Graphics 2001, pages 159 166, 2001.

470 Bibliography [136] S.-W. Wang, Z.-C. Shih, and R.-C. Chang. An efficient and stable ray tracing algorithm for parametric surfaces. Journal of Information Science and Engineering, 18(4):541 561, 2002. [137] J. Warren and H. Weimer. Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann Publishers, 2001. [138] A. Watt and M. Watt. Advanced Animation and Rendering Techniques. Addison Wesley, New York, 1992. [139] D. F. Wiley, M. Bertram, B. Hamann, K. I. Joy, N. Max, and G. Scheuermann. Hierarchical spline approximation. In G. Farin, B. Hamann, and H. Hagen, editors, Hierarchical and Geometrical Methods in Scientific Visualization, pages 63 88, Heidelberg, Germany, 2003. Springer-Verlag. [140] S. Yoshizawa, A. G. Belyaev,, and H.-P. Seidel. A fast and simple stretch-minimizing mesh parameterization. International Conference on Shape Modeling and Applications, pages 200 208, 2004. [141] S. Zhukov, A. Iones, and G. Kronin. An ambient light illumination model. Rendering Techniques 98, pages 45 56, 1998. [142] D. Zorin. A method for analysis of C 1 -continuity of subdivision surfaces. SIAM Journal on Numerical Analysis, 37(5):1677 1708, 2000. [143] D. Zorin, P. Schröder, T. DeRose, L. Kobbelt, A. Levin, and W. Sweldens. Subdivision for modelling and animation. In Course Notes at SIGGRAPH 2000, 2000. [144] D. Zorin, P. Schröder, and W. Sweldens. Interpolating subdivision for meshes with arbitrary topology. Computer Graphics Proceedings (SIGGRAPH 96), pages 189 192, 1996.

Index Symbols σ(t), 121, 125 1-neighborhood, 183 1-ring, 183 2-neighborhood, 184, 187 2-ring, 184, 187 A Absorbance, 39 Adini s method, 177 Advection, 438 Affine combination, 78 Affine invariance, 76, 83, 98, 138 Algebraic curves, 75 Algebraic distance, 369 Aliasing, 386 Ambient, 26 Ambient occlusion, 389 Anisotropic surfaces, 32 Anti-aliasing, 386 Arc length, 113, 200 Attenuation, 29 Attractor, 64 B B-splines, 94, 100 affine invariance, 98 convex hull, 98 interpolation with, 100 non-uniform, 99 periodic interpolation with, 102 uniform, 94 Bézier, 83 Bézier curves, 83 affine invariance, 83 convex hull, 84 piecewise continuous, 86 rational, 91 subdivision, 88 Bézier surfaces, 169 Bézier patch, 170 tensor product surface, 170 triangular, 171 Barycentric coordinates, 172 Barycentric mapping, 223 Basic vectors, 200 Beckmann distribution, 33 Beer-Lambert law, 38, 337 Bent normal, 392 Bernstein polynomials, 83, 172 Bessel-Overhauser splines, 93 Binomial theorem, 84 Binormal, 115 Blending functions, 97 BRIDF, 31 C Cantor set, 71 Capacity, 59 Catmull-Rom splines, 92 Chaikin s scheme, 179 Chebyshev polynomial, 142 first kind, 142 second kind, 142 471

472 Index Chord length, 175 Cone, 372 Conformal mapping, 226 Conic sections, 91 Continuous, 86 C k, 86 G k, 86 Convex hull, 77, 84, 98 Convex polygons, 53 Cook, 31 Coordinate curves, 200 Cox-de Boor formula, 98 Cramer s rule, 343 Cross product, 3 CSG, 376 Curvature, 112 curvature plot, 113 inflection, 114 radius, 115 Curvature vector, 115 D de Boor algorithm, 98 de Casteljau, 87 Differencing, 238 central, 240 forward, 239 Diffuse, 26 Diffusion, 438 Direct matrix product, 69 Dirichlet kernel, 122 Discrete Fourier transform, 106 Distance surfaces, 369 Divergence, 439 E Emissive, 26 Euclidean distance, 2, 369 Euclidean norm, 2 Euler, 8 Euler transform, 8, 9 F Fair, 114, 183 Fairness, 256 Filter bank, 155 First fundamental form, 201 Flat shading, 54 Fluids Navier-Stokes equations, 438 Fourier, 106 Fourier transform, variation, 149 Fourier matrix, 105 Fractal, 59 Frenet frame, 116 Fresnel equation, 38 G Gaussian curvature, 203 Gaussian distribution, 33 Geometric distance, 369 Geometry image, 219 Geometry images, 219 Gimbal lock, 9 Gouraud shading, 23 Gradient, 438 Graph, 222 Grassmann space, 5 H Hadamard matrix, 156 Harmonic interpolation, 118 affine invariance, 138 even case, 122 non-uniform, 144 odd case, 120 stability, 136 Harmonic subdivision, 207 Harmonic surfaces, 237 tensor product surface, 206, 229 Hausdorff dimension, 59 Helicoid, 203 Helmholtz Hodge decomposition, 439 Hermite polynomials

Index 473 cubic, 429 Hermitian, 149 Hilbert curve, 66 Homogeneous coordinates, 4 Hooke s law, 434 Hypersurface, 167 I Implicit surfaces, 368 Inclusion monotonic, 381 Infinite cylinder, 371 Interpolation, 100 Intersection ray-plane, 342 ray-sphere, 342 ray-triangle, 343 Interval, 380 centered forms, 384 mean value form, 385 midpoint, 383 Newton iteration, 385 root finding, 381, 382 width, 383 Interval arithmetic, 380 operations, 380 transcendental functions, 381 Interval extension, 381 Isocurve, 168, 169 Iterated function system, 64 J Jacobi identity, 3 Jacobian, 433 Jacobian matrix, 382 Jitter, 388 Julia set, 62 K κ, 113 Klein bottle, 248 Koch snowflake, 67 Kronecker delta, 79 Kronecker product, 69, 71 multiple, 70 L Lagrange, 78 Lagrange interpolation, 78, 146 trigonometric, 107, 122 λ, 32, 37 Lambert, 27 Lambert s Law, 27 Laplacian, 439 Lebesgue covering dimension, 60 Lighting, 23 ambient term, 26 Cook-Torrance model, 31 diffuse term, 26 emissive term, 26 specular, 27 spot lights, 30 Lighting model Phong, 25 Lindenmayer systems, 65 Line segments, 4 Linear combination, 78 M Möbius band, 248 Mandelbrot set, 60 Matrix class, 17 Max norm, 383 Mean curvature, 203 Mesh parameterization, 219 METAPOST, 108 Meusnier s theorem, 202 Mock curvature, 109 Moments, 145 Monte Carlo methods, 396 N Navier-Stokes equations, 438 Newton-Raphson, 379 Normal, 170 infinite plane, 338 sphere, 338

474 Index triangle, 339 Normalized, 3 NURBS, 104 Nyquist limit, 387 O Obscurance, 389 Orthogonal, 201 Orthogonal parameterization, 201 Osculating circle, 115 Osculating plane, 115 Overhand knot, 248 P Parameterization, 222 mean value coordinates, 227 shape-preserving, 223 stretch minimization, 225 Parameterized curve, 112 Parametric curves, 76 Parametric surface, 167 Partition of unity, 76, 83, 138 Pauli matrices, 15 Periodic interpolation, 102 Permutation matrix, 118 Perpendicular, 3 Perspective projection matrix, 12 Phong, 25 Phong shading, 23 Planar, 222 Poisson equation, 439 Principal directions, 202 Principal normal, 115 Q Quasi-interpolation, 200 Quaternion class, 20 Quaternions, 13 R Rational Bézier curves, 91 Raytracing, 333 Reflection, 338 Refraction, 339 Rendering, 238 Root finding interval bisection, 381 interval Newton-Raphson, 382 Rotation, 7, 9 generalized, 9 quaternions, 13 S Scalar product, 2 Scaling, 6 Second fundamental form, 202 Shading, 23 Gouraud, 23 Phong, 23 Sierpinski carpet, 71 triangle, 73 Signed distance bound, 369 Snell s law, 37, 339 Solomon s Seal knot, 249 Spectral representation, 118 Specular, 27 Sphere parametric, 168 Sphere tracing, 369 Stochastic supersampling, 387 Stretch, 225 Subdivision 3, 185 Catmull-Clark, 191 curves, 177 Doo-Sabin, 192 Loop, 180 modified butterfly, 183 stationary, 179 Superquadric, 372 Supersampling, 386 stochastic, 387 Surface, 167 Surface normal, 170

Index 475 Surface of revolution, 248 T Tension-Continuity-Bias splines, 93 Tensor product, 69 Tensor product surfaces, 168 Toeplitz, 149 Toeplitz matrix, 149 Topological dimension, 60 Torrance, 31 Torsion, 115 Torus, 372 parametric, 168 Transforms affine, 5, 6 concatenation, 10 identity, 6 linear, 6 rotation, 7 scaling, 6 translation, 7 Translation, 7 Transport notation, 394 Transpose, 2 Tridiagonal, 102 Trigonometric interpolation, 104 Twist vector, 176 U Umbilical point, 202 USFFT, 148 V Valence, 179 Vandermonde matrix, 145 Vector class, 15 Vector field, 437 Vector identity, 3 Vector product, 3 Vector space, 1 Viscosity, 438 W Wavelets, 153 father wavelet, 153 Haar, 157 Haar wavelet transform, 154 lifting scheme, 158 mother wavelet, 154 nonstandard construction, 159 scaling functions, 156 Weighted average, 78, 84 World coordinates, 436