NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes

Abstract

We present a subdivision framework that adds extraordinary vertices to NURBS of arbitrarily high degree. The surfaces can represent any odd degree NURBS patch exactly. Our rules handle non-uniform knot vectors, and are not restricted to midpoint knot insertion. In the absence of multiple knots at extraordinary points, the limit surfaces have bounded curvature.

References

1. Augsdörfer, U. H., Dodgson, N. A., and Sabin, M. A. 2006. Tuning Subdivision by Minimising Gaussian Curvature Variation Near Extraordinary Vertices. Comp. Graph. Forum 25, 3, 263--272.
2. Augsdörfer, U. H., Cashman, T. J., Dodgson, N. A., and Sabin, M. A. 2009. Numerical Checking of C¹ for Arbitrary Degree Quadrilateral Subdivision Schemes. In 13th IMA Conference on the Mathematics of Surfaces, Springer. To appear.
3. Barthe, L., and Kobbelt, L. 2004. Subdivision scheme tuning around extraordinary vertices. CAGD 21, 6, 561--583.
4. Boehm, W. 1980. Inserting new knots into B-spline curves. Computer-Aided Design 12, 4, 199--201.
5. Cashman, T. J., Dodgson, N. A., and Sabin, M. A. 2009. Selective knot insertion for symmetric, non-uniform refine and smooth B-spline subdivision. CAGD 26, 4, 472--479.
6. Catmull, E., and Clark, J. 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6, 350--355.
7. Cohen, E., Lyche, T., and Riesenfeld, R. 1980. Discrete B-splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics. Computer Graphics and Image Processing 14, 2, 87--111.
8. DeRose, T., Kass, M., and Truong, T. 1998. Subdivision surfaces in character animation. In Proc. SIGGRAPH 98, 85--94.
9. Farin, G. 2001. Curves and Surfaces for CAGD: A Practical Guide, 5th ed. Morgan Kaufmann.
10. Galil, Z., and Italiano, G. 1991. Data structures and algorithms for disjoint set union problems. ACM Computing Surveys 23, 3, 319--344.
11. Ginkel, I., and Umlauf, G. 2006. Loop subdivision with curvature control. In Eurographics Symposium on Geom. Proc., Eurographics, K. Polthier and A. Sheffer, Eds., 163--171.
12. Gonsor, D., and Neamtu, M. 2001. Subdivision Surfaces -- Can they be Useful for Geometric Modeling Applications? Tech. Rep. 01--011, The Boeing Company.
13. Holt, F. 1996. Toward a curvature-continuous stationary subdivision algorithm. Zeitschrift für angewandte Mathematik und Mechanik 76, 423--424.
14. Karciauskas, K., Peters, J., and Reif, U. 2004. Shape characterization of subdivision surfaces-case studies. CAGD 21, 6, 601--614.
15. Lane, J. M., and Riesenfeld, R. F. 1980. A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces. IEEE Trans. PAMI 2, 1, 35--46.
16. Levin, A. 2006. Modified subdivision surfaces with continuous curvature. ACM Trans. Graph. 25, 3, 1035--1040.
17. Loop, C. 2002. Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer 18, 316--325.
18. Ma, W. 2005. Subdivision surfaces for CAD---an overview. Computer-Aided Design 37, 7, 693--709.
19. Müller, K., Reusche, L., and Fellner, D. 2006. Extended subdivision surfaces: Building a bridge between NURBS and Catmull-Clark surfaces. ACM Trans. Graph. 25, 2, 268--292.
20. Peters, J., and Reif, U. 2008. Subdivision Surfaces. Springer.
21. Prautzsch, H. 1997. Freeform splines. CAGD 14, 3, 201--206.
22. Prautzsch, H. 1998. Smoothness of subdivision surfaces at extraordinary points. Adv. in Comp. Math. 9, 3, 377--389.
23. Ramshaw, L. 1989. Blossoms are polar forms. CAGD 6, 4, 323--358.
24. Reif, U. 1996. A Degree Estimate for Subdivision Surfaces of Higher Regularity. Proc. Amer. Math. Soc. 124, 7, 2167--2174.
25. Reif, U. 1998. TURBS---Topologically Unrestricted Rational B-Splines. Constructive Approximation 14, 1, 57--77.
26. Sabin, M. A., Dodgson, N. A., Hassan, M. F., and Ivrissimtzis, I. P. 2003. Curvature behaviours at extraordinary points of subdivision surfaces. Computer-Aided Design 35, 11, 1047--1051.
27. Sabin, M. 1991. Cubic recursive division with bounded curvature. In Curves and surfaces, Academic Press, 411--414.
28. Schaefer, S., and Goldman, R. 2009. Non-uniform Subdivision for B-splines of Arbitrary Degree. CAGD 26, 1, 75--81.
29. Sederberg, T. W., Zheng, J., Sewell, D., and Sabin, M. 1998. Non-Uniform Recursive Subdivision Surfaces. In Proc. SIGGRAPH 98, 387--394.
30. Sederberg, T. W., Zheng, J., Bakenov, A., and Nasri, A. 2003. T-splines and T-NURCCs. ACM Trans. Graph. 22, 3, 477--484.
31. Stam, J. 1998. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In Proc. SIGGRAPH 98, 395--404.
32. Stam, J. 2001. On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. CAGD 18, 5, 383--396.
33. Warren, J., and Weimer, H. 2001. Subdivision Methods for Geometric Design. Morgan Kaufmann.
34. Zorin, D., and Schröder, P. 2001. A unified framework for primal/dual quadrilateral subdivision schemes. CAGD 18, 5, 429--454.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models