APPROXIMATION OF QUADRIC SURFACES USING SPLINES

  • Received : 2009.07.26
  • Accepted : 2009.09.10
  • Published : 2009.09.25

Abstract

In this paper we present an approximation method of quadric surface using quartic spline. Our method is based on the approximation of quadratic rational B$\acute{e}$zier patch using quartic B$\acute{e}$zier patch. We show that our approximation method yields $G^1$ (tangent plane) continuous quartic spline surface. We illustrate our results by the approximation of helicoid-like surface.

Acknowledgement

Supported by : Korea Research Foundation

References

  1. Y.J. Ahn. Conic approximation of planar curves. Comp. Aided Desi. 33 (2001) 867-872. https://doi.org/10.1016/S0010-4485(00)00110-X
  2. Y.J. Ahn. Helix approximations with conic and quadratic Bezier curves. Comp. Aided Geom. Desi. 22 (2005) 551-565. https://doi.org/10.1016/j.cagd.2005.02.003
  3. Y.J. Ahn. Approximation of conic sections by curvature continuous quartic splines. submitted.
  4. Y.J. Ahn, H.O. Kim. Approximation of circular arcs by Bezier curves. J. Comp. Appl. Math. 81 (1997) 145-163. https://doi.org/10.1016/S0377-0427(97)00037-X
  5. Y.J. Ahn, H.O. Kim. Curvatures of the quadratic rational Bezier curves. Comp. Math. Appl. 36 (1998) 71–83.
  6. Y.J. Ahn, Y. S. Kim, Y. S. Shin. Approximation of circular arcs and offset curves by Bezier curves of high degree. J. Comp. Appl. Math. 167 (2004) 181–191.
  7. C. de Boor, K. Hollig, M. Sabin. High accuracy geometric Hermite interpolation. Comp. Aided Geom. Desi. 4 (1987) 169-178.
  8. T. Dokken, M. Daehlen, T. Lyche, K. Morken. Good approximation of circles by curvature-continuous Bezier curves. Comp. Aided Geom. Desi. 7 (1990) 33–41. https://doi.org/10.1016/0167-8396(90)90019-N
  9. J. D. Emery. The definition and computation of a metric on plane curves. Comp. Aided Desi. 18 (1986) 25-28. https://doi.org/10.1016/S0010-4485(86)80006-9
  10. L. Fang. Circular arc approximation by quintic polynomial curves. Comp. Aided Geom. Desi. 15 (1998) 843- 861. https://doi.org/10.1016/S0167-8396(98)00019-3
  11. L. Fang. $G^3$ approximation of conic sections by quintic polynomial curves. Comp. Aided Geom. Desi. 16 (1999) 755-766 https://doi.org/10.1016/S0167-8396(99)00017-5
  12. L. Fang. A rational quartic Bezier representation for conics. Comp. Aided Geom. Desi. 19 (2002) 297–312. https://doi.org/10.1016/S0167-8396(02)00096-1
  13. G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Morgan-Kaufmann, San Francisco, 2002.
  14. M. Floater. High order approximation of conic sectons by quadratic splines. Comp. Aided Geom. Desi. 12 (1995) 617–637. https://doi.org/10.1016/0167-8396(94)00037-S
  15. M. Floater. An $O(h^{2n})$ Hermite approximation for conic sectons. Comp. Aided Geom. Desi. 14 (1997) 135– 151. https://doi.org/10.1016/S0167-8396(96)00025-8
  16. M. Floater. High order approximation of rational curves by polynomial curves. Comp. Aided Geom. Desi. 23 (2006) 621-628. https://doi.org/10.1016/j.cagd.2006.06.003
  17. M. Goldapp. Approximation of circular arcs by cubic polynomials. Comp. Aided Geom. Desi. 8 (1991) 227–238. https://doi.org/10.1016/0167-8396(91)90007-X
  18. J. A. Gregory. 1989. Geometric continuity. In T. Lyche and L.L. Schumaker, editors, Mathematical Methods in CAGD, pages 353–371, Nashville, Academic Press.
  19. Q. Q. Hu, G. J.Wang. Necessary and sufficient conditions for rational quartic representation of conic sections. J. Comput. Appl. Math. 203 (2007) 190-208. https://doi.org/10.1016/j.cam.2006.03.024
  20. S. H. Kim, Y. J. Ahn. Approximation of circular arcs by quartic Bezier curves. Comp. Aided Desi. 39 (2007) 490–493. https://doi.org/10.1016/j.cad.2007.01.004
  21. L. Lu. Approximating tensor product Bezier surfaces with tangent plane continuity. J. Comp. Appl. Math. 231 (2009) 412–422.
  22. K. Morken. 1990. Best approximation of circle segments by quadratic Bezier curves. In P.J. Laurent, A. Le Mehaute, and L.L. Schumaker, editors, Curves and Surfaces, New York, Academic Press.