Beyond Catmull-Clark? A survey of advances in subdivision surface methods

Computer Graphics Forum

Volumn 31, Issue 1, 2012, Pages 42-61

  Cashman, Thomas J ^a  

^a UNIVERSITY OF LUGANO   (Switzerland)

Author keywords

Evaluation; Polar subdivision; PTER framework; Rendering approximation; Smoothness analysis

Indexed keywords

COMPUTER GRAPHICS;

EVALUATION; POLAR SUBDIVISION; PTER FRAMEWORK; RENDERING APPROXIMATION; SMOOTHNESS ANALYSIS;

SURVEYS;

EID: 84878126961     PISSN: 01677055     EISSN: 14678659     Source Type: Journal    
DOI: 10.1111/j.1467-8659.2011.02083.x     Document Type: Review

References (137)

1. ALEXA M., BOUBEKEUR T.: Subdivision shading. ACM Transactions on Graphics 27, 5 (2008), 142:1-142:4.
2. AUGSDÖRFER U. H., DODGSON N. A., SABIN M. A.: Tuning subdivision by minimising Gaussian curvature variation near extraordinary vertices. Computer Graphics Forum 25, 3 (2006), 263-272.
3. [ADS11a] AUGSDÖRFER U. H., DODGSON N. A., SABIN M. A.: Artifact analysis on B-splines, box-splines and other surfaces defined by quadrilateral polyhedra. Computer Aided Geometric Design 28, 3 (2011), 177-197.
4. [ADS11b] AUGSDÖRFER U. H., DODGSON N. A., SABIN,M. A.: Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra. Computer Aided Geometric Design 28, 3 (2011), 198-211.
5. BARTHE L., KOBBELT L.: Subdivision scheme tuning around extraordinary vertices. Computer AidedGeometric Design 21, 6 (2004), 561-583.
6. BIERMANN H.,KRISTJANSSON D., ZORIN,D.: Approximate Boolean operations on free-form solids. In Proceedings of SIGGRAPH (Los Angeles, CA, USA, 2001), E. Fiume (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 185-194.
7. BIERMANN H., LEVIN A., ZORIN, D.: Piecewise smooth subdivision surfaces with normal control. In Proceedings of SIGGRAPH (New Orleans, LA, USA, 2000), K. Akeley (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 113-120.
8. BOIER-MARTIN I., ZORIN D.: Differentiable parameterization of Catmull-Clark subdivision surfaces. In Proceedings of the Symposium on Geometry Processing (Nice, France, 2004), Eurographics, pp. 155-164.
9. BOLZ J., SCHRÖDER P.: Rapid evaluation of Catmull- Clark subdivision surfaces. In Proceedings of the Web3D Symposium (Tempe, AZ, USA, 2002), ACM, pp. 11-17.
10. BUNNELL M.: Adaptive tessellation of subdivision surfaces with displacement mapping. In GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation. M. Pharr and R. Fernando (Eds.). Addison-Wesley (Boston, MA, USA, 2005).
11. CASHMAN T. J., AUGSDÖRFER U. H., DODGSON N. A., SABIN M. A.: NURBS with extraordinary points: Highdegree, non-uniform, rational subdivision schemes. ACM Transactions on Graphics, 28, 3 (2009), 46:1-46:9.
12. CASHMAN T. J.: NURBS-Compatible Subdivision Surfaces. PhD thesis, University of Cambridge, 2010. Technical Report UCAM-CL-TR-773.
13. CATMULL E., CLARK J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6 (1978), 350-355.
14. CHEN G., CHENG F.: Matrix based subdivision depth computation for extra-ordinary Catmull-Clark subdivision surface patches. In Geometric Modeling and Processing - GMP 2006, M.-S. Kim and K. Shimada (Eds.), Vol. 4077 of Lecture Notes in Computer Science. Springer (2006), pp. 545-552.
15. CHENG F., CHEN G., YONG J.: Subdivision depth computation for extra-ordinary Catmull-Clark subdivision surface patches. In Advances in Computer Graphics - CCI 2006, T. Nishita, Q. Peng and H.-P. Seidel (Eds.), Vol. 4035 of Lecture Notes in Computer Science. Springer (2006), pp. 404-416.
16. CHENG F., YONG J.: Subdivision depth computation for Catmull-Clark subdivision surfaces. Computer Aided Design & Applications 3, 1-4 (2006), 485-494.
17. DEROSE T., KASS M., TRUONG T.: Subdivision surfaces in character animation. In Proceedings of SIGGRAPH (Orlando, FL, USA, 1998), M. Cohen (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 85-94.
18. DYN N., LEVIN D.: Subdivision schemes in geometric modelling. Acta Numerica 11 (2002), 73-144.
19. DYN N., LEVIN D., GREGORY J. A.: A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4, 4 (1987), 257-268.
20. DYN N., LEVIN D., GREGORY J. A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 2 (1990), 160-169.
21. DYN N., LEVIN D., LIU D.: Interpolatory convexitypreserving subdivision schemes for curves and surfaces Computer-Aided Design 24, 4 (1992), 211-216.
22. DOO D.W. H.:Asubdivision algorithm for smoothing down irregular shaped polyhedrons. In Proceedings: Interactive Techniques in Computer Aided Design (Bologna, Italy, 1978), IEEE, pp. 157-165.
23. DOO D., SABIN M. A.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10, 6 (1978), 356-360.
24. DODGSON N., SABIN M., SOUTHERN R.: Preconditions on Geometrically Sensitive Subdivision Schemes. Tech. Rep.UCAM-CL-TR-691, University of Cambridge, 2007.
25. GRINSPUN E., SCHRÖDER P.: Normal bounds for subdivision-surface interference detection. In Proceedings of the Conference on Visualization '01 (San Diego, CA, USA, 2001), IEEE, pp. 333-340.
26. GINKEL I., UMLAUF G.: Loop subdivision with curvature control. In Proceedings of the Symposium on Geometry Processing (Cagliari, Sardinia, Italy, 2006),K. Polthier and A. Sheffer (Eds.), Eurographics, pp. 163-171.
27. GINKEL I., UMLAUF G.: Symmetry of shape charts. Computer Aided Geometric Design 25, 3 (2008), 131-136.
28. HAN B.: Classification and construction of bivariate subdivision schemes. In Curve and Surface Fitting: Saint-Malo 2002. A. Cohen, J.-L. Merrien and L. L. Schumaker (Eds.). Nashboro Press (Saint-Malo, France, 2003), pp. 187-197.
29. HUANG Z., DENG J., WANG G.: A bound on the approximation of a Catmull-Clark subdivision surface by its limit mesh. Computer Aided Geometric Design 25, 7 (2008), 457-469.
30. HALSTEAD M., KASS M., DEROSE T.: Efficient, fair interpolation using Catmull-Clark surfaces. In Proceedings of SIGGRAPH (Anaheim, CA, USA, 1993), J. T. Kajiya (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 35-44.
31. HOLT F.: Toward a curvature-continuous stationary subdivision algorithm. Zeitschrift f?ur angewandte Mathematik und Mechanik 76 (1996), 423-424.
32. HE L., SCHAEFER S., HORMANN K.: Parameterizing subdivision surfaces. ACM Transactions on Graphics 29, 4 (2010), 120:1-120:6.
33. HUANG Z.,WANG G.: Distance between a Catmull- Clark subdivision surface and its limit mesh. In Proceedings of the ACM symposium on Solid and Physical Modeling (Tsinghua University, Beijing, China, 2007), pp. 233-240.
34. HUANG Z., WANG G.: Improved error estimate for extraordinary Catmull-Clark subdivision surface patches. The Visual Computer 23, 12 (2007), 1005- 1014.
35. HUANG Z., WANG G.: Bounding the distance between a Loop subdivision surface and its limit mesh. In Advances in Geometric Modeling and Processing. F. Chen and B. Jüttler (Eds.). Vol. 4975 of Lecture Notes in Computer Science. Springer, 2008, pp. 33-46.
36. IVRISSIMTZIS I., DODGSON N., SABIN M.: A generative classification of mesh refinement rules with lattice transformations. Computer Aided Geometric Design 21, 1 (2004), 99-109.
37. IGLESIAS A., GÁLVEZ A., PUIG-PEY J.: Computational methods for geometric processing. Applications to industry. In Proceedings of the International Conference on Computational Science - ICCS (San Francisco, CA, USA, 2001), pp. 698-707.
38. IVRISSIMTZIS I. P., SABIN M. A., DODGSON N. A.: On the support of recursive subdivision. ACM Transactions on Graphics 23, 4 (2004), 1043-1060.
39. JIANG D., STEWART N. F.: Robustness of Boolean operations on subdivision-surface models. In Numerical Validation in Current Hardware Architectures. A. Cuyt, W. Krämer, W. Luther and P. Markstein (Eds.). Springer-Verlag (Berlin, Heidelberg, 2009), pp. 161- 174.
40. KAZAKOV M.: Catmull-Clark subdivision for geometry shaders. In Proceedings of AFRIGRAPH (Grahamstown, South Africa, 2007), H. Slay, S. N. Spencer and S. Bangay (Eds.), ACM, pp. 77-84.
41. KOVACS D., MITCHELL J., DRONE S., ZORIN D.: Real-time creased approximate subdivision surfaces. In Proceedings of the Symposium on Interactive 3D Graphics and Games (Boston, MA, USA, 2009), pp. 155- 160.
42. KOBBELT L.: Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15, 3 (1996), 409-420.
43. KOBBELT L.: √3-subdivision. In Proceedings of SIGGRAPH (New Orleans, LA, USA, 2000), K. Akeley (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 103-112.
44. KARČIAUSKAS K., PETERS J.: Bicubic polar subdivision. ACM Transactions on Graphics 26, 4 (2007), 14:1-14:6.
45. KARČIAUSKAS K., PETERS J.: Concentric tessellation maps and curvature continuous guided surfaces. Computer Aided Geometric Design 24, 2 (2007), 99-111.
46. KARČIAUSKAS K., PETERS J.: Surfaces with polar structure. Computing 79, 2 (2007), 309-315.
47. KARČIAUSKAS K., PETERS J.: On the curvature of guided surfaces. Computer Aided Geometric Design 25, 2 (2008), 69-79.
48. KARČIAUSKAS K., PETERS J.: Adjustable speed surface subdivision. Computer Aided Geometric Design 26, 9 (2009), 962-969.
49. KARČIAUSKAS K., PETERS J.: Guided spline surfaces. Computer Aided Geometric Design 26, 1 (2009), 105- 116.
50. KARČIAUSKAS K., PETERS J., REIF U.: Shape characterization of subdivision surfaces - case studies. Computer Aided Geometric Design 21, 6 (2004), 601- 614.
51. KARBACHER S., SEEGER S., HÄUSLER G.: A nonlinear subdivision scheme for trianglemeshes. In Proceedings of Vision, Modeling and Visualization (Saarbrücken, Germany, 2000), pp. 163-170.
52. LAI S., CHENG F.: Robust and error controllable Boolean operations on free-form solids represented by Catmull-Clark subdivision surfaces. Computer Aided Design and Applications 4, 1-4 (2007), 487-496.
53. LEVIN A.: Combined subdivision schemes for the design of surfaces satisfying boundary conditions. Computer Aided Geometric Design 16, 5 (1999), 345-354.
54. LEVIN A.: Modified subdivision surfaces with continuous curvature. ACM Transactions on Graphics 25, 3 (2006), 1035-1040.
55. LANQUETIN S., FOUFOU S., KHEDDOUCI H., NEVEU M.: A graph based algorithm for intersection of subdivision surfaces. In Proceedings of Computational Science and Its Applications - ICCSA (Montreal, Canada, 2003), pp. 387-396.
56. LABSIK U., GREINER G.: Interpolatory √3-subdivision. Computer Graphics Forum 19, 3 (2000), 131-138.
57. LEVIN A., LEVIN D.: Analysis of quasi-uniform subdivision. Applied and Computational Harmonic Analysis 15, 1 (2003), 18-32.
58. LITKE N., LEVIN A., SCHRÖDER P.: Trimming for subdivision surfaces. Computer Aided Geometric Design 18, 5 (2001), 463-482.
59. LI G., MA W.: A method for constructing interpolatory subdivision schemes and blending subdivisions. Computer Graphics Forum 26, 2 (2007), 185-201.
60. LEE A., MORETON H., HOPPE H.: Displaced subdivision surfaces. In Proceedings of SIGGRAPH (New Orleans, LA, USA, 2000), K. Akeley (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 85-94.
61. LOOP C. T.: Smooth Subdivision Surfaces Based on Triangles.Master's thesis, University of Utah,Department of Mathematics, 1987.
62. LOOP C.: Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer 18 (2002), 316-325.
63. LOOP C.: Second order smoothness over extraordinary vertices. In Proceedings of the Symposium on Geometry Processing (Nice, France, 2004), pp. 165-174.
64. LANE J. M., RIESENFELD R. F.: A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 2, 1 (1980), 35-46.
65. LOOP C., SCHAEFER S.: Approximating Catmull- Clark subdivision surfaces with bicubic patches. ACM Transactions on Graphics 27, 1 (2008), 8:1-8:11.
66. LOOP C., SCHAEFER S.: G2 tensor product splines over extraordinary vertices. In Proceedings of the Symposium on Geometry Processing (Copenhagen, Denmark, 2008), pp. 1373-1382.
67. LOOP C., SCHAEFER S., NI T., CASTA?NO I.: Approximating subdivision surfaces with Gregory patches for hardware tessellation. ACM Transactions on Graphics 28 (2009), 151:1-151:9.
68. MÜLLER K., FÜNFZIG C., REUSCHE L., HANSFORD D., FARIN G. E., HAGEN H.: Dinus: Double insertion, nonuniform, stationary subdivision surfaces. ACM Transactions on Graphics 29 (2010), 25:1-25:21.
69. MAEKAWA T., MATSUMOTO Y., NAMIKI K.: Interpolation by geometric algorithm. Computer-Aided Design 39, 4 (2007), 313-323.
70. MYLES A., NI T., PETERS J.: Fast parallel construction of smooth surfaces from meshes with tri/quad/pent facets. Computer Graphics Forum 27, 5 (2008), 1365-1372.
71. MYLES A., PETERS J.: Bi-3 C2 polar subdivision. ACM Transactions on Graphics 28, 3 (2009), 48:1- 48:12.
72. MÜLLER K., REUSCHE L., FELLNER D.: Extended subdivision surfaces: Building a bridge between NURBS andCatmull-Clark surfaces. ACMTransactions on Graphics 25, 2 (2006), 268-292.
73. MORIN G., WARREN J., WEIMER H.: A subdivision scheme for surfaces of revolution. Computer Aided Geometric Design 18, 5 (2001), 483-502.
74. MYLES A., YEO Y. I., PETERS J.: GPU conversion of quad meshes to smooth surfaces. In Proceedings of the ACM Symposium on Solid and Physical Modeling (Stony Brook, NY, USA, 2008), pp. 321-326.
75. NI T., CASTA?NO I., PETERS J., MITCHELL J., SCHNEIDER P., VERMA V.: Efficient substitutes for subdivision surfaces. In ACM SIGGRAPH Courses (2009), pp. 13:1-13:107.
76. NI T., YEO Y., MYLES A., GOEL V., PETERS J.: GPU smoothing of quad meshes. In Proceedings of the International Conference on Shape Modeling and Applications (Stony Brook, NY, USA, 2008), IEEE, pp. 3-9.
77. OSWALD P., SCHRÖDER P.:Composite primal/dual√3- subdivision schemes. Computer Aided Geometric Design 20, 3 (2003), 135-164.
78. PRAUTZSCH H., CHEN Q.: Analyzing midpoint subdivision. Computer Aided Geometric Design 28, 7 (2011), 407-419.
79. PETERS J.: Patching Catmull-Clark meshes. In Proceedings of SIGGRAPH (New Orleans, LA, USA, 2000), K. Akeley (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 255-258.
80. PETERS J.: Smoothness, fairness and the need for better multi-sided patches. In Topics in Algebraic Geometry and Geometric Modeling: Workshop on Algebraic Geometry and Geometric Modeling, (Vilnius University, Lithuania, 2002), vol. 334, p. 55.
81. PETERS J., KARČIAUSKAS K.: An introduction to guided and polar surfacing. In Mathematical Methods for Curves and Surfaces. M. Dæhlen, M. Floater, T. Lyche, J.-L.Merrien,K.Mørken and L. Schumaker (Eds.). Vol. 5862 of Lecture Notes in Computer Science. Springer (2010), pp. 299-315.
82. PETERS J., REIF U.: The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics 16, 4 (1997), 420-431.
83. PETERS J., REIF U.: Analysis of algorithms generalizing B-spline subdivision. SIAM Journal on Numerical Analysis 35, 2 (1998), 728-748.
84. PRAUTZSCH H., REIF U.: Degree estimates for Ckpiecewise polynomial subdivision surfaces. Advances in Computational Mathematics 10, 2 (1999), 209-217.
85. PETERS J., REIF U.: Shape characterization of subdivision surfaces - basic principles. Computer Aided Geometric Design 21, 6 (2004), 585-599.
86. PETERS J., REIF U.: Subdivision Surfaces. Vol. 3 of Geometry and Computing. Springer, Berlin, Heidelberg, 2008.
87. PRAUTZSCH H.: Freeform splines. Computer Aided Geometric Design 14, 3 (1997), 201-206.
88. PRAUTZSCH H.: Smoothness of subdivision surfaces at extraordinary points. Advances in Computational Mathematics 9, 3 (1998), 377-389.
89. PULLI K., SEGAL M.: Fast rendering of subdivision surfaces. In Proceedings of the EurographicsWorkshop on Rendering Techniques (Porto, Portugal, 1996), Springer- Verlag, pp. 61-70.
90. PETERS J., SHIUE L. J.: Combining 4- and 3-direction subdivision. ACMTransactions on Graphics 23, 4 (2004), 980-1003.
91. PIEGL L., TILLERW.: Curve and surface constructions using rational B-splines. Computer-Aided Design 19, 9 (1987), 485-498.
92. PRAUTZSCH H., UMLAUF G.: A G2-subdivision algorithm. In Geometric Modelling, Dagstuhl, 1996. G. Farin, H. Bieri, G. Brunnett and T. DeRose (Eds.). Vol. 13 of Computing Supplement, Springer (1998), pp. 217-224.
93. PETERS J., UMLAUF G.: Computing curvature bounds for bounded curvature subdivision. Computer Aided Geometric Design 18, 5 (2001), 455-461.
94. PETERS J.,WU X.: The distance of a subdivision surface to its control polyhedron. Journal of Approximation Theory 161, 2 (2009), 491-507.
95. QIN K., WANG H.: Eigenanalysis and continuity of non-uniform Doo-Sabin surfaces. In Proceedings of Pacific Conference on Computer Graphics and Applications (Seoul, Korea, 1999), IEEE, pp. 179-186.
96. REIF U.: A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design 12, 2 (1995), 153-174.
97. REIF U.: A degree estimate for subdivision surfaces of higher regularity. Proceedings of the American Mathematical Society 124, 7 (1996), 2167-2174.
98. REIF U.: TURBS - Topologically unrestricted rational B-splines. Constructive Approximation 14, 1 (1998), 57-77.
99. REIF U.: An appropriate geometric invariant for the C2-analysis of subdivision surfaces. In 12th IMA Conference on the Mathematics of Surfaces (Sheffield, UK, 2007), R. R.Martin,M. A. Sabin and J. R.Winkler (Eds.), Vol. 4647 of Lecture Notes in Computer Science, Springer, pp. 364-377.
100. REIF U., SCHRÖDER P.: Curvature integrability of subdivision surfaces. Advances in Computational Mathematics 14, 2 (2001), 157-174.
101. SABIN M. A.: Cubic recursive division with bounded curvature. In Curves and Surfaces. P. J. Laurent, A. Le MÉhautÉ and L. L. Schumaker (Eds.). Academic Press Professional, Inc. (1991), pp. 411-414.
102. SABIN M. A.: Recent progress in subdivision: A survey. In Advances in Multiresolution for Geometric Modelling. N. A. DODGSON, M. S. Floater and M. A. Sabin (Eds.). Springer (2005), pp. 203-230.
103. SABIN M.: Analysis and Design of Univariate Subdivision Schemes, vol. 6 of Geometry and Computing. Springer, Berlin, Heidelberg, 2010.
104. SKARIA S., AKLEMAN E., PARKE F.:Modeling subdivision control meshes for creating cartoon faces. In Proceedings of the International Conference on Shape Modeling and Applications (Genoa, Italy, 2001), IEEE, pp. 216-225.
105. SABIN M. A., BARTHE L.: Artifacts in recursive subdivision surfaces. In Curve and Surface Fitting: Saint-Malo 2002.A.Cohen, J.-L.Merrien and L. L. Schumaker (Eds.). Nashboro Press (Saint-Malo, France, 2003), pp. 353- 362.
106. SCHAEFER S.: A Factored Interpolatory Subdivision Scheme for Surfaces of Revolution. Master's thesis, Rice University, 2003.
107. SABIN M. A., DODGSON N. A., HASSAN M. F., IVRISSIMTZIS I. P.: Curvature behaviours at extraordinary points of subdivision surfaces. Computer-Aided Design 35, 11 (2003), 1047-1051.
108. STEWART I. F., FOISY A. R.: Arbitrary-degree subdivision with creases and attributes. Journal of Graphics Tools 9, 4 (2004), 3-17.
109. SEDERBERG T. W., FINNIGAN G. T., LI X., LIN H., IPSON H.:Watertight trimmed NURBS. ACMTransactions on Graphics 27, 3 (2008), 79:1-79:8.
110. SHIUE L. J., JONES I., PETERS J.: A realtime GPU subdivision kernel. ACM Transactions on Graphics 24, 3 (2005), 1010-1015.
111. STAM J., LOOP C.: Quad/triangle subdivision. Computer Graphics Forum 22, 1 (2003), 79-85.
112. SEVERN A., SAMAVATI F.: Fast intersections for subdivision surfaces. In Computational Science and Its Applications - ICCSA (Glasgow, UK, 2006), pp. 91-100.
113. STAM J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In Proceedings of SIGGRAPH (Orlando, FL, USA, 1998),M. Cohen (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 395-404.
114. STAM J.: On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Computer Aided Geometric Design 18, 5 (2001), 383-396.
115. SCHAEFER S., WARREN J.: A factored interpolatory subdivision scheme for quadrilateral surfaces. In Curve and Surface Fitting: Saint-Malo 2002. A. Cohen, J.-L. Merrien and L. L. Schumaker (Eds.). Nashboro Press (Saint-Malo, France, 2003), pp. 373-382.
116. SCHAEFER S.,WARREN J.:On C2 triangle/quad subdivision. ACM Transactions on Graphics 24 (2005), 28-36.
117. SCHAEFER S., WARREN J.: Exact evaluation of nonpolynomial subdivision schemes at rational parameter values. In Proceedings of Pacific Conference on Computer Graphics and Applications (Maui, HI, USA, 2007), IEEE, pp. 321-330.
118. SCHAEFER S., WARREN J.: Exact evaluation of limits and tangents for non-polynomial subdivision schemes. Computer AidedGeometric Design 25, 8 (2008), 607-620.
119. SEDERBERG T. W., ZHENG J., BAKENOV A., NASRI A.: T-splines and T-NURCCs. ACM Transactions on Graphics 22, 3 (2003), 477-484.
120. SEDERBERG T. W., ZHENG J., SEWELL D., SABIN M.: Non-uniform recursive subdivision surfaces. In Proceedings of SIGGRAPH (Orlando, FL, USA, 1998),M. Cohen (Ed.), Computer Graphics Proceedings, Annual Conference Series, ACM, pp. 387-394.
121. TSITSIKLIS J. N., BLONDEL V. D.: The Lyapunov exponent and joint spectral radius of pairs of matrices are hard - when not impossible - to compute and to approximate. Mathematics of Control, Signals, and Systems 10, 1 (1997), 31-40.
122. UMLAUF G.: Analyzing the characteristic map of triangular subdivision schemes. Constructive Approximation 16, 1 (2000), 145-155.
123. VLACHOS A., PETERS J., BOYD C., MITCHELL J.: Curved PN triangles. In Proceedings of the Symposium on Interactive 3D Graphics (Chapel Hill, NC, USA, 2001), ACM, pp. 159-166.
124. VELHO L., ZORIN D.: 4-8 Subdivision. Computer Aided Geometric Design 18, 5 (2001), 397-427.
125. WALLNER J., DYN N.: Convergence and C1 analysis of subdivision schemes on manifolds by proximity. Computer Aided Geometric Design 22, 7 (2005), 593-622.
126. WU X., PETERS J.: Interference detection for subdivision surfaces. Computer Graphics Forum 23, 3 (2004), 577-584.
127. WARREN J., WEIMER H.: Subdivision Methods for Geometric Design. Morgan Kaufmann, San Francisco, CA, USA, 2001.
128. YANG X.: Surface interpolation of meshes by geometric subdivision. Computer-Aided Design 37, 5 (2005), 497-508.
129. ZENG X., CHEN X.: Computational formula of depth for Catmull-Clark subdivision surfaces. Journal of Computational and Applied Mathematics 195, 1-2 (2006), 252-262.
130. ZORIN D., KRISTJANSSON D.: Evaluation of piecewise smooth subdivision surfaces. The Visual Computer 18, 5 (2002), 299-315.
131. ZULTI A., LEVIN A., LEVIN D., TEICHER M.: C2 subdivision over triangulations with one extraordinary point. Computer Aided GeometricDesign 23, 2 (2006), 157-178.
132. ZORIN D.: A method for analysis of C1-continuity of subdivision surfaces. SIAM Journal on Numerical Analysis 37, 5 (2000), 1677-1708.
133. ZORIN D.: Smoothness of stationary subdivision on irregular meshes. Constructive Approximation 16, 3 (2000), 359-398.
134. ZORIN D.: Constructing curvature-continuous surfaces by blending. In Proceedings of the Symposium on Geometry Processing (Cagliari, Sardinia, Italy, 2006), K. Polthier and A. Sheffer (Eds.), Eurographics, pp. 31-40.
135. ZORIN D., SCHRÖDER P.: A unified framework for pryimal/ dual quadrilateral subdivision schemes. Computer Aided Geometric Design 18, 5 (2001), 429-454.
136. ZORIN D., SCHRÖDER P., DEROSE T., KOBBELT L., LEVIN A., SWELDENS W.: Subdivision for modeling and animation. In ACM SIGGRAPH Courses (2000), pp. 23:1-23:194.
137. ZORIN D., SCHRÖDER P., SWELDENSW.: Interpolating subdivision for meshes with arbitrary topology. In Proceedings of SIGGRAPH (New Orleans, LA, USA, 1996), H. Rushmeier (Ed.),Computer Graphics Proceedings,Annual Conference Series, ACM, pp. 189-192.