Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Algebraic conditions for classifying the positional relationships between two conics and their applications

  • 43 Accesses

  • 10 Citations

Abstract

In many fields of computer science such as computer animation, computer graphics, computer aided geometric design and robotics, it is a common problem to detect the positional relationships of several entities. Based on generalized characteristic polynomials and projective transformations, algebraic conditions are derived for detecting the various positional relationships between two planar conics, namely, outer separation, exterior contact, intersection, interior contact and inclusion. Then the results are applied to detecting the positional relationships between a cylinder (or a cone) and a quadric. The criteria is very effective and easier to use than other known methods.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Farouki R T, Neff C A, O'Connor M A. Automatic parsing of degenerate quadric-surface intersections.ACM Trans. Graph., 1989, 8(3): 174–203.

  2. [2]

    Levin J Z. A parametric algorithm for drawing pictures of solid objects composed of quadrics.Communications of the ACM, 1976, 19(10): 555–563.

  3. [3]

    Levin J Z. Mathematical models for determining the intersecitons of quadric surfaces.Comput. Graph. Image Process., 11: 73–87.

  4. [4]

    James R Miller. Geometric approaches to nonplanar quadric surface intersection curves.ACM Trans. Graph., October 1987, 6(4): 274–307.

  5. [5]

    Abhyankar S S, Bajaj C L. Automatic parameterization of rational curves and surface IV: Algebraic space curves.ACM Trans. Graph., October 1989, 8(4): 325–334.

  6. [6]

    Ching-Kuang Shene, John K Johnstone. On the planar intersection of natural quadrics. InProc. the First ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, Austin. Texas, USA, 1991, pp. 233–242.

  7. [7]

    Ching-Kuang Shene, John K Johnstone. On the lower degree intersections of two natural quadrics.ACM Trans. Graphics (TOG), October 1994, 13(4): 400–424.

  8. [8]

    Wenping Wang, Barry Joe, Ronald-Goldman. Computing quadric surface intersections bases on an analysis of plane cubic curves. CSIS Tech Report TR-2002-13, Hong Kong University.

  9. [9]

    Laurent Dupont, Daniel Lazard, Sylvain Lazard. Sylvain Petitjean. Towards the robust intersection of implicit quadrics. In Uncertainty in Geometric Computations, Winkler J, Niranjan M (eds.), Kluwer Academic Publishers, 2002, pp. 59–68.

  10. [10]

    Christian Lennerz, Elmar Schömer. Efficient distance computation for quadratic curves and surfaces. InProc. the Geometric Modeling and Processing Theory and Applications, Wako, Saitama, Japan, 2002, pp. 1–10.

  11. [11]

    Kyung-Ah Sohn, Bert Jütter, Myung-Soo Kim, Wenping Wang. Computing distances between surfaces using line geometry. InProc. 10th Pacific Conference on Computer Graphics and Applications, Beijing, China, 2002, pp. 236–245.

  12. [12]

    Wenping Wang, Jiaye Wang, Myung-Soo Kim. An algebraic condition for the separation of two ellipsoids.Computer Aided Geometric Design, 2001, 18(6): 531–539.

  13. [13]

    Yang Liu, Liyong Shen. An algebraic condition for classifying the positional relationship of two planar ellipses.Journal of Computer-Aided Design and Computer Graphics, 2003, 15: 555–560. (in Chinese)

  14. [14]

    Liyong Shen, Yang Liu. An algebraic condition for the positional relationship of an ellipse and a parabola (Hyperbola).Journal of System Simulation, 2002, 14(9): 1208–1211. (in Chinese)

  15. [15]

    Semple J G, Kneebone G K. Algebraic Projective Geometry. Oxford University Press, London, 1952.

Download references

Author information

Correspondence to Yang Liu.

Additional information

Supported by the Outstanding Youth Grant of the National Natural Science Foundation of China (Grant No.60225002), the TRAPOYT and the Doctoral Program of MOE of China (Grant No.20010358003).

Yang Liu is a Ph.D. candidate in the Computer Science Department at the University of Hong Kong. He received his B.S. (2000) and M.S. (2003) degrees in mathematics from the University of Science and Technology of China. His research interests include computer-aided design, computer graphics and computational algebraic geometry.

Fa-Lai Chen is currently a professor in the Department of Mathematics at the University of Science and Technology of China. He received his B.S., M.S. and Ph.D. degrees in mathematics in 1997. 1989 and 1994 respectively, all from the University of Science and Technology of China. His research interests include computer aided geometric design and computer graphics.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liu, Y., Chen, F. Algebraic conditions for classifying the positional relationships between two conics and their applications. J. Comput. Sci. & Technol. 19, 665–673 (2004). https://doi.org/10.1007/BF02945593

Download citation

Keywords

  • collision detection
  • projective transformation
  • generalized characteristic polynomial
  • positional relationship