Fast Distance Computation Using Quadratically Supported Surfaces

  • Margot Rabl
  • Bert Jüttler


We use the class of surfaces with quadratic polynomial support functions in order to define bounding geometric primitives for shortest distance computation. The common normals of two such surfaces can be computed by solving a single polynomial equation of degree six. Based on this observation, we formulate an algorithm for computing the shortest distance between enclosures of two moving or static objects by surfaces of this type. It is demonstrated that the performance of this algorithm compares favourably with methods for computing the distance between two ellipsoids, which can also be used as bounding primitives for distance computation and collision detection.


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  1. 1.
    Chen X-D, Yong J-H, Zheng G-Q, Paul J-C, Sun J-G (2006) Computing minimum distance between two implicit algebraic surfaces. Computer-Aided Design 38: 1053–1061.Google Scholar
  2. 2.
    Choi Y-K, Wang W, Kim M-S (2003) Exact Collision Detection of Two Moving Ellipsoids under Rational Motions. In: Proc. Int. Conf. Robotics and Automation, IEEE Press, 349–354.Google Scholar
  3. 3.
    Chou W, Xiao J (2006) Real-time and Accurate Multiple Contact Detection between General Curved Objects. In: Proc. Int. Conf. on Intelligent Robots and Systems, IEEE Press, 556–561.Google Scholar
  4. 4.
    Cohen J, Lin M, Manocha D, Ponamgi M (1995) I-COLLIDE: An interactive and exact collision detection system for large-scale environments. In: Proc. ACM Interactive 3D Graphics Conference. ACM Press, 189–196.Google Scholar
  5. 5.
    Eberly DH (2001) Game Engine Design. Academic Press.Google Scholar
  6. 6.
    Jenkins MA (1975) Algorithm 493 - Zeros of a Real Polynomial [C2], ACM Trans. Math. Software 1: 178–189.Google Scholar
  7. 7.
    Ju M, Liu J, Shiang S, Chien Y (2001) A Novel Collision Detection Method Based on Enclosed Ellipsoid. In: Proc. Int. Conf. Robotics and Automation, IEEE Press, 2897–2902.Google Scholar
  8. 8.
    Klosowski J, Held M, Mitchell JSB, Sowizral H, Zikan K (1998) Efficient collision detection using bounding volume hierarchies of k-DOPs. IEEE Transactions on Visualization and Computer Graphics 4: 21–36.Google Scholar
  9. 9.
    Latombe J-C (1991) Robot Motion Planning. Kluwer Academic Publishers, Boston.Google Scholar
  10. 10.
    Lennerz, C and Schömer, E (2002) Efficient Distance Computation for Quadratic Curves and Surfaces. In: Proc. Geometric Modeling and Processing, IEEE Press, 60–69.Google Scholar
  11. 11.
    Lin MC, Gottschalk S (1998) Collision detection between geometric models: A survey. In: Cripps R (Ed.), Proc. IMA Conf. on Mathematics of Surfaces, Information Geometers: 37–56.Google Scholar
  12. 12.
    Lu L, Choi YK, Wang W and Kim M-S (2007) Variational 3D Shape Segmentation for Bounding Volume Computation, Computer Graphics Forum 26: 329–338.Google Scholar
  13. 13.
    Oberneder M, Jüttler B, Gonzalez-Vega L (2008) Exact envelope computation for moving surfaces with quadratic support functions. In Lenarčič J, Wenger P, Advances in Robot Kinematics - Analysis and Design, Springer, 283–290.Google Scholar
  14. 14.
    Rabl M, Gonzalez-Vega L, Jüttler B, Schröcker H-P (2009) Oriented Bounding Surfaces with at most Six Common Normals. In: Proc. Int. Conf. Robotics and Automation, to appear. Available as FSP report no. 77 at
  15. 15.
    Sabin M (1974) A Class of Surfaces Closed under Five Important Geometric Operations. Technical Report VTO/MS/207, British Aircraft Corporation. Available at
  16. 16.
    Šír Z, Gravesen J, Jüttler B (2008) Curves and surfaces represented by polynomial support functions. Theoretical Computer Science 392: 141–157.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Margot Rabl
    • 1
  • Bert Jüttler
    • 1
  1. 1.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria

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