International Journal of CAD/CAM

Society for Computational Design and Engineering (한국CDE학회)
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Estimating Recursion Depth for Loop Subdivision

  • Wang Huawei (Dept. of Computer Science & Technology, Tsinghua University) ;
  • Sun Hanqiu (Dept. of Computer Science & Engineering, Chinese University of Hong Kong) ;
  • Qin Kaihuai (Dept. of Computer Science & Technology, Tsinghua University)
  • Published : 2004.12.01
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Abstract

In this paper, an exponential bound of the distance between a Loop subdivision surface and its control mesh is derived based on the topological structure of the control mesh. The exponential bound is independent of the process of recursive subdivisions and can be evaluated without subdividing the control mesh actually. Using the exponential bound, we can predict the depth of recursion within a user-specified tolerance as well as the error bound after n steps of subdivision. The error-estimating approach can be used in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and the like.

Keywords

References

  1. Cheng, F. (1992), Estimating subdivision depths for rational curves and surfaces, ACM Transaction on Graphics, 11(2), 140-151
  2. Cohen, E. and Schumaker, L.L. (1985), Rates of convergence of control polygons, Computer Aided Geometric Design, 2, 229-235
  3. Dai, J., Wang, H. and Qin, K. (2004), Parallelly generating NC tool paths for subdivision surfaces, International Journal of CAD/CAM, Vol. 4, No. 1
  4. Filip, D., Magedson, R. and Markot, R. (1986), Surface algorithms using bounds on derivatives, Computer Aided Geometric Design, 3(4), 295-311
  5. Halstead, M., Kass, M. and DeRose, T. (1993), Efficient, Fair Interpolation Using Catrnull-Clark Surfaces. Computer Graphics (Proceedings of SIGGRAPH '93), 27, 35-44
  6. Kobbelt, L., Daubert, K. and Seidel, H. (1998), Ray tracing of subdivision surfaces, Eurographics Rendering Workshop, Vienna, Austria, pp.69-80
  7. Loop, C.T. (1987), Smooth subdivision surfaces based on triangles, M.S. Thesis, Department of Mathematics, University of Utah
  8. Lutterkort, D. and Peters, J. (1998), The distance between a uniform B-spline and its control polygon, Technical Report TR-98-013, available from http://www.cise.ufl.edul research/tech-reports
  9. Nairn, D., Peters, J. and Lutterkort, D. (1999), Sharp, quantitative bounds on the distance between a polynomial piece and its Bezier control polygon, Computer Aided Geometric Design, 16, 613-631
  10. Prautzsch, H. and Kobbelt, L. (1994), Convergence of subdivision and degree elevation, Adv. Compo Math., 2, 143-154
  11. Reif, U. (2000), Best bounds on the approximation of polynomials and splines by their control structure, Computer Aided Geometric Design, 17,579-589
  12. Stam, J. (1998), Evaluation of Loop subdivision surfaces, Appearing as an appendix in the CD disc of SIGGRAPH'98
  13. Wang, G. (1984), The subdivision method for finding the intersection between two Bezier curves or surfaces, Zhejiang University J. Special Issue on Computational Geometry (in Chinese), 108-119
  14. Zorin, D. and Kristjansson, D. (2002), Evaluation of piecewise smooth subdivision surfaces, Visual Computer