Global Curve Analysis via a Dimensionality Lifting Scheme

  • Gershon Elber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)


Freeform rational parametric curves and surfaces have been playing a major role in computer aided design for several decades. The ability to analyze local (differential) properties of parametric curves is well established and extensively exploited. In this work, we explore a different lifting approach to global analysis of freeform geometry, mostly curves, in IR 2 and IR 3. In this lifting scheme, we promote the problem into a higher dimension, where we find that in the higher dimension, the solution is simplified.


Bottom Boundary Planar Curve Projection Direction Lift Scheme Local Visibility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahn, H.-K., de Berg, M., Bose, P., Cheng, S., Halperin, D., Matoušek, J., Cheong, O.: Separating an object from its cast. Computer-Aided Design 34, 547–559 (2002)CrossRefGoogle Scholar
  2. 2.
    Appel, A.: The notion of quantitative invisibility and the machine rendering of solids. In: Proc. ACM National Conference, Washington, DC, pp. 387–393 (1967)Google Scholar
  3. 3.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry, Algorithms, and Applications, 2nd edn. Springer, Berlin (2000)zbMATHGoogle Scholar
  4. 4.
    Bloomenthal, M.: Approximation of sweep surfaces by tensor product B-splines, Tech Reports UUCS-88-008, University of Utah (1988)Google Scholar
  5. 5.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press and McGraw-Hill (1990)Google Scholar
  6. 6.
    do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  7. 7.
    Elber, G., Cohen, E.: Hidden curve removal for free form surfaces, Computer Graphics. In: Proc. SIGGRAPH, vol. 24, pp. 95–104 (1990)Google Scholar
  8. 8.
    Elber, G.: Symbolic and numeric computation in curve interrogation. Computer Graphics Forum 14, 25–34 (1995)CrossRefGoogle Scholar
  9. 9.
    Elber, G.: Multiresolution curve editing with linear constraints. The Journal of Computing & Information Science in Engineering 1(4), 347–355 (2001)CrossRefGoogle Scholar
  10. 10.
    Elber, G.: Trimming local and global self-intersections in offset curves using distance maps. In: Proc. of the 10th IMA Conference on the Mathematics of Surfaces, Leeds, UK, pp. 213–222 (2003)Google Scholar
  11. 11.
    Elber, G.: Distance separation measures between parametric curves and surfaces toward intersection and collision detection applications. In: Proceedings of COMPASS 2003, Schloss Weinberg, Austria (October 2003)Google Scholar
  12. 12.
    Elber, G., Chen, X., Cohen, E.: Mold accessibility via Gauss map analysis. In: Shape Modeling International 2004, Genova, Italy, pp. 263–274 (2004)Google Scholar
  13. 13.
    Elber, G., Sayegh, R., Barequet, G., Martin, R.R.: Two-dimensional visibility charts for continuous curves. In: Shape Modeling International 2005, Boston, USA (June 2005) (to appear)Google Scholar
  14. 14.
    Gonzales-Ochoa, C., Mccamnon, S., Peters, J.: Computing moments of objects enclosed by piecewise polynomial surfaces. ACM Transactions on Graphics 17(3), 143–157 (1998)CrossRefGoogle Scholar
  15. 15.
    Hahmann, S., Bonneau, G.-P., Sauvage, B.: Area preserving deformation of multiresolution curves (submitted)Google Scholar
  16. 16.
    Keyser, J., Culver, T., Manocha, D., Krishnan, S.: Efficient and exact manipulation of algebraic points and curves. Computer-Aided Design 32(11), 649–662 (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Klok, F.: Two moving coordinate frames for sweeping along a 3D trajectory. Computer Aided Geometric Design 3(3), 217–229 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rappaport, A., Sheffer, A., Bercovier, M.: Volume-preserving free-form solids. IEEE Transactions on Visualization and Computer Graphics 2(1), 19–27 (1996)CrossRefGoogle Scholar
  19. 19.
    Woo, T.: Visibility maps and spherical algorithms. Computer-Aided Design 26, 6–16 (1994)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Gershon Elber
    • 1
  1. 1.Israel Institute of TechnologyTechnionHaifaIsrael

Personalised recommendations