New Numerical Solution for Two Parametric Surfaces Intersection Dragging Problem

Article Sidebar

PDF
Cite as:
Ramadhan A. M. Alsaidi, New Numerical Solution for Two Parametric Surfaces Intersection Dragging Problem, Int. J. Anal. Appl., 19 (5) (2021), 773-783.

Main Article Content

Ramadhan A. M. Alsaidi

Abstract

The problem of intersecting two parametric surfaces has been one of the main technical challenges in computer-aided design, computer graphics, solid modeling, and geometrics. This paper aims at reducing and minimizing time and space required for the computations process of parametric surface intersection. To do this, a new numerically accelerating method based on continuation technique was utilized first by calculating a starting point, and second by tracing sequential points along the intersection curve following Broyden's method. Two factors have been identified as influential in controlling component jumping: initial points and step size. Test examples of intersecting two parametric surfaces demonstrated that this method was highly efficient with high-speed parametric solution. The intersection results are often given as curve's points.

Article Details

References

  1. J. Hoschek, D. Lasser, Fundamentals of computer aided geometric design, AK Peters, Ltd., Wellesley, MA, 1993.
  2. N. M. Patrikalakis, Surface-to-surface intersections, IEEE Computer Graph. Appl. 13 (1) (1993), 89-95.
  3. R. E. Barnhill, Geometry processing for design and manufacturing, SIAM, Philadelphia, 1992.
  4. N. M. Patrikalakis, T. Maekawa, Shape interrogation for computer aided design and manufacturing, Vol. 15, Springer, 2002.
  5. N. M. Patrikalakis, T. Maekawa, Intersection problems, in: Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2010, pp. 109-160.
  6. J.-K. Seong, K.-J. Kim, M.-S. Kim, G. Elber, R. R. Martin, Intersecting a freeform surface with a general swept surface, Computer-Aided Design, 37 (5) (2005), 473-483.
  7. X.-M. Liu, C.-Y. Liu, J.-H. Yong, J.-C. Paul, Torus/torus intersection, Computer-Aided Design Appl. 8 (3) (2011), 465-477.
  8. K.-J. Kim, Circles in torus-torus intersections, J. Comput. Appl. Math. 236 (9) (2012), 2387-2397.
  9. Y. Park, S.-H. Son, M.-S. Kim, G. Elber, Surface-surface-intersection computation using a bounding volume hierarchy with osculating toroidal patches in the leaf nodes, Computer-Aided Design, 127 (2020), 102866.
  10. T. Nishita, T. W. Sederberg, M. Kakimoto, Ray tracing trimmed rational surface patches, in: Proceedings of the 17th annual conference on Computer graphics and interactive techniques, 1990, pp. 337-345.
  11. S. Campagna, P. Slusallek, H. P. Seidel, Ray tracing of parametric surfaces: Bezier clipping, chebyshev boxing and bounding volume hierarchy. a critical comparison with new results, Computer Graphics Group, University of Erlangen, Germany (1997).
  12. S.-W. Wang, Z.-C. Shih, R.-C. Chang, et al., An efficient and stable ray tracing algorithm for parametric surfaces, J. Inform. Sci. Eng. 18 (4) (2002), 541-561.
  13. S. Chau, M. Oberneder, A. Galligo, B. J ¨uttler, Intersecting biquadratic b ´ezier surface patches, in: Geometric Modeling and Algebraic Geometry, Springer, 2008, pp. 161-180.
  14. B. Bily, The method of finding points of intersection of two cubic bezier curves using the sylvester matrix, Silesian J. Pure Appl. Math. 6 (2016), 155-176.
  15. R. A. M. Alsaidi, A. Musleh, Two methods for surface/surface intersection problem comparative study, Int. J. Computer Appl. 92 (2014), 1-8.
  16. R. Burden, J. Faires, A. Burden, Numerical solutions of nonlinear systems of equations, In: Numerical analysis. Cengage Learning, Boston, (1997) pp 544-576.
  17. W. C. Rheinboldt, Numerical analysis of parametrized nonlinear equations, Wiley-Interscience, 1986.
  18. H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Springer Verlag, Heidelberg, Germany, 1987.
  19. K. Abdel-Malek, H.-J. Yeh, Determining intersection curves between surfaces of two solids, Computer-Aided Design 28 (6-7) (1996), 539-549.
  20. T. Dokken, Aspects of intersection algorithms and approximation, PhD Thesis, University of Oslo, (1997).