International Journal of Computer Vision

, Volume 4, Issue 1, pp 79–100 | Cite as

Straight homogeneous generalized cylinders: Differential geometry and uniqueness results

  • Jean Ponce


It is well known that certain objects, like ellipsoids or spheres, admit several, sometimes many, different parameterizations by generalized cylinders. Under what conditions does a given surface admit at most one description by a straight homogeneous generalized cylinder (SHGC)? Under what other conditions does a surface admit at most a few such descriptions? To answer these questions, it is first necessary to understand the geometry and intrinsic properties of SHGC's. In this paper, a necessary and sufficient condition for the regularity of the surface described by an SHGC is given, the Gaussian curvature of this surface is calculated, and it is proved that its parabolic lines are either meridians or parallels of the associated SHGC. These results are used in the second part of the paper to prove several new uniqueness results. It is first proved that if a surface is described by two SHGC's with the same cross-section plane and axis, then these SHGC's are necessarily deduced from each other through inverse scalings of their cross-sections and sweeping rule curve, and that the surface associated with an SHGC with at least two parabolic meridians and two parabolic parallels cannot be described by a different SHGC. Shafer's pivot and slant theorems are then extended to prove that the surface associated with a non-linear SHGC does not have any different SHGC parameterization with the same cross-section plane or the same axis. Finally, it is shown that a surface with at least two parabolic lines admits at most three different SHGC descriptions, and that a surface with at least four parabolic lines admits at most one SHGC description. In particular, a closed surface with at least one concave point admits at most three different SHGC descriptions.


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  1. 1.
    G.J. Agin, “Representation and description of curved objects,” Ph.D. dissertation, AIM-273, Stanford AI Lab., October 1972.Google Scholar
  2. 2.
    N. Ayache and O.D. Faugeras, “HYPER: A new approach for the recognition and positioning of two-dimensional objects,” IEEE Trans. PAMI 8 (1): 44–54, January 1986.Google Scholar
  3. 3.
    T.O. Binford, “Visual perception by computer,” Proc. IEEE Conf. Systems and Control, Miami, December 1971.Google Scholar
  4. 4.
    T.O. Binford, T. Levitt, and W. Mann, “Bayesian inference in model-based machine vision,” Proc. Workshop on Uncertainty in AI, 1987.Google Scholar
  5. 5.
    R.C. Bolles and R.A. Cain, “Recognizing and locating partially visible objects: The local-feature-focus method,” Intern. J. Robotics Res. 1 (3) 57–82, 1982.Google Scholar
  6. 6.
    J.M. Brady, “Criteria for representation of shape,” In Human and Machine Vision, Beck, Hope, and Rosenfeld (eds.), Academic Press, 1983.Google Scholar
  7. 7.
    J.M. Brady, J. Ponce, A. Yuille and H. Asada, “Describing surfaces,” Proc. 2nd Intern. Symp. Robotics Res. (Harasuja and Inoue, eds.), MIT Press, 1985.Google Scholar
  8. 8.
    R.A. Brooks, “Symbolic reasoning among 3D models and 2D images,” Artificial Intelligence 17: 285–348, 1981.Google Scholar
  9. 9.
    M.P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.Google Scholar
  10. 10.
    O.D. Faugeras, and M. Hebert, “A 3-D recognition and positioning algorithm using geometrical matching between primitive surfaces,” Proc. 8th Intern. Joint Conf. Arti. Intell., Karlsruhe, pp. 996–1002, 1983.Google Scholar
  11. 11.
    W.E.L. Grimson, and T. Lozano-Pérez, “Model-based recognition and localization from sparse range and tactile data,” Intern. J. Robotics Res. 3 (3): 3–35, 1984.Google Scholar
  12. 12.
    J.M. Hollerbach, “Hierarchical shape description of objects by selection and modification of prototypes,” MIT AI Technical Report TR-346, 1975.Google Scholar
  13. 13.
    R. Horaud, “New methods for matching 3-d objects with single perspective views,” IEEE Trans PAMI 9 (3): 401–412, 1987.Google Scholar
  14. 14.
    R. Horaud, and J.M. Brady, “On the geometric interpretation of image contours,” Proc. 1st Intern. Conf. Comput. Vision, London, June 1987.Google Scholar
  15. 15.
    D.P. Huttenlocher and S. Ullman, “Object recognition using alignment,” Proc. 1st Intern. Conf. Compu. Vision, London, June 1987.Google Scholar
  16. 16.
    J.J. Koenderink, “What does the occluding contour tell us about solid shape,” Perception 13, 1984.Google Scholar
  17. 17.
    D.J. Kriegman, and T.O. Binford, “Generic models for robot navigation,” Proc. IEEE Conf. Robotics Automation, Philadelphia, April 1988.Google Scholar
  18. 18.
    D. Lowe, “The viewpoint consistency constraint,” Intern. J. Comput. Vision 1 (1) 1987.Google Scholar
  19. 19.
    D. Marr, “Analysis of occluding contour,” Proc. Roy. Soc. London B-197: 441–475, 1977.Google Scholar
  20. 20.
    D. Marr and K. Nishihara, “Representation and recognition of the spatial organization of three dimensional shapes,” Proc Roy. Soc. London B-200: 269–294, 1978.Google Scholar
  21. 21.
    R. Nevatia and T.O. Binford, “Description and recognition of complex curved objects,” Artificial Intelligence 8: 77–98, 1977.Google Scholar
  22. 22.
    R. Nevatia, “Machine perception,” Prentice-Hall, Englewood Cliffs, NJ, 1982.Google Scholar
  23. 23.
    J. Ponce, and D. Chelberg, “Finding the limbs and cusps of generalized cylinders,” Intern. J. Comput. Vision 1 (3), 1987.Google Scholar
  24. 24.
    J. Ponce, D. Chelberg, D.J. Kriegman and W. Mann, “Geometric modeling with generalized cylinders,” Proc IEEE Workshop Comput. Vision, Miami, December 1987.Google Scholar
  25. 25.
    J. Ponce and G. Healey, “Using generic geometric and physical models for representing solids,” Proc. DARPA Image Understanding Workshop, Boston, April 1988.Google Scholar
  26. 26.
    J. Ponce, “Straight homogeneous generalized cylinders: differential geometry and uniqueness results,” Proc. IEEE Conf. Comput. Vision and Pattern Recog., June 1988.Google Scholar
  27. 27.
    J. Ponce, D. Chelberg, and W. Mann, “Invariant properties of straight homogeneous generalized cylinders and their contours,” IEEE Trans. PAMI 11 (9): 951–966.Google Scholar
  28. 28.
    K. Rao, and R. Nevatia, “From sparse 3-D data directly to volumetric shape descriptions,” Proc. DARPA Image Understanding Workshop, Los Angeles, February 1987.Google Scholar
  29. 29.
    K. Rao, and G. Médioni,, “Useful geometric properties of the generalized cone,” Proc. IEEE Conf. Comput. Vision Pattern Recog., June 1988.Google Scholar
  30. 30.
    K.R. Roberts, “Equivalent descriptions of generalized cylinders,” Proc. DARPA Image Understanding Workshop, Miami, December 1985.Google Scholar
  31. 31.
    Z. Rubinstein, A Course in Ordinary Differential Equations, Academic Press: New York, 1969.Google Scholar
  32. 32.
    W. Rudin, Principles of Mathematical Analysis, McGraw-Hill: 1976.Google Scholar
  33. 33.
    S.A. Shafer, Shadows and Silhouettes in Computer Vision, Kluwer Academic 1985.Google Scholar
  34. 34.
    U. Shani and D.H. Ballard, “Splines as embeddings for generalized cylinders,” Comput. Graphics and Image Process. 27: 129–156, 1984.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Jean Ponce
    • 1
  1. 1.Robotics LaboratoryStanford University

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