Geometry of Curves and Surfaces

  • Gregory S. Chirikjian
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter consists of a variety of topics in geometry. The approach to geometry that is taken in this chapter and throughout this book is one in which the objects of interest are described as being embedded1 in Euclidean space. There are two natural ways to describe such embedded objects: (1) parametrically and (2) implicitly. The vector-valued functions x = x(t) and x = x(u, v) are respectively parametric descriptions of curves and surfaces when \(X \varepsilon\mathbb{R}^3\). For example, \(x(\Psi ) = [\cos \Psi , \sin \Psi , 0]^T\) for ψ ∈ [0, 2π) is a parametric description of a unit circle in \(\mathbb{R}^3\), and \(x(\phi ,\theta ) = [\cos \phi \sin {\rm \theta },\sin \phi \sin \theta {\rm ,}\cos {\rm \theta }]^T\) for φ ∈ [0, 2π) and θ ∈ [0, π] is a parametric description of a unit sphere in \(\mathbb{R}^3\). Parametric descriptions are not unique. For example, \(x(t) = [{\rm 2t/(1 + t}^{\rm 2} {\rm ), (1 } - {\rm t}^{\rm 2} {\rm )/(1 + t}^{\rm 2} {\rm ), 0]}^{\rm T}\) for \(t \varepsilon\mathbb{R}\) describes the same unit circle as the one mentioned above.2


Fundamental Form Gaussian Curvature Euler Characteristic Inverse Kinematic Total Curvature 
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.
    Abhyankar, S.S., Algebraic Geometry for Scientists and Engineers, Mathematical Surveys and Monographs, 35, American Mathematical Society, Providence, RI, 1990.Google Scholar
  2. 2.
    Adams, C.C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W.H. Freeman, New York, 1994.MATHGoogle Scholar
  3. 3.
    Bates, P.W., Wei, G.W., Zhao, S., “Minimal molecular surfaces and their applications,” J. Comput. Chem., 29, pp. 380–391, 2008.CrossRefGoogle Scholar
  4. 4.
    Ben-Israel, A., Greville, T.N.E., Generalized Inverses: Theory and Applications, 2nd ed., Canadian Mathematical Society Books in Mathematics, Springer, New York, 2003.MATHGoogle Scholar
  5. 5.
    Bishop, R., “There is more than one way to frame a curve,” Amer. Math. Month., 82, pp. 246–251, 1975.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blackmore, D., Leu, M.C., Wang, L.P., “The sweep-envelope differential equation algorithm and its application to NC machining verification,” Computer-Aided Design, 29, pp. 629–637, 1997.CrossRefGoogle Scholar
  7. 7.
    Bloomenthal, J., (ed.), Introduction to Implicit Surfaces, Morgan Kaufmann, San Francisco, 1997.MATHGoogle Scholar
  8. 8.
    Bloomenthal, J., Shoemake, K., “Convolution surfaces,” Computer Graphics, 25, pp. 251–256, 1991 (Proc. SIGGRAPH'91).CrossRefGoogle Scholar
  9. 9.
    Bottema, O., Roth, B., Theoretical Kinematics, Dover, New York, 1990.MATHGoogle Scholar
  10. 10.
    Brakke, K.A., The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978.MATHGoogle Scholar
  11. 11.
    Buttazzo, G., Visintin, A., eds.,Motion by Mean Curvature and Related Topics, Proceedings of the international conference held at Trento, July 20–24, 1992. de Gruyter, Berlin, 1994.Google Scholar
  12. 12.
    Chan, T.F., Vese, L.A., “Active contours without edges,” IEEE Trans. Image Process., 10, pp. 266–277, 2001.MATHCrossRefGoogle Scholar
  13. 13.
    Chazvini, M., “Reducing the inverse kinematics of manipulators to the solution of a generalized eigenproblem,” Computational Kinematics, pp. 15–26, 1993.Google Scholar
  14. 14.
    Chen, B.-Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.MATHGoogle Scholar
  15. 15.
    Chen, B.-Y., “On the total curvature of immersed manifolds,” Amer. J. Math., 93, pp. 148–162, 1971; 94, pp. 899–907, 1972; 95, pp. 636–642, 1973.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chen, B.-Y., “On an inequality of mean curvature,” J. London Math. Soc., 4, pp. 647–650, 1972.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Chirikjian, G.S., “Closed-form primitives for generating locally volume preserving deformations,” ASME J. Mech. Des., 117, pp. 347–354, 1995.CrossRefGoogle Scholar
  18. 18.
    Chopp, D.L., “Computing minimal surfaces via level set curvature flow,” J. Comput. Phys., 106, pp. 77–91, 1993.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    do Carmo, M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.MATHGoogle Scholar
  20. 20.
    Dombrowski, P., “Krummungsgrossen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten,” Math. Nachr., 38, pp. 133–180, 1968.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Evans, L.C., Spruck, J., “Motion of level sets by mean curvature,” J. Diff. Geom., 33, pp. 635–681, 1991.MATHMathSciNetGoogle Scholar
  22. 22.
    Evans, L.C., Spruck, J., “Motion of level sets by mean curvature II,” Trans. Amer. Math. Soc., 330, pp. 321–332, 1992.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Farouki, R.T., Neff, C.A., “Analytical properties of plane offset curves,” Computer Aided Geometric Design, 7, pp. 83–99, 1990.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Farouki, R.T., Neff, C.A., “Algebraic properties of plane offset curves,” Computer Aided Geometric Design, 7, pp. 101–127, 1990.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Fary, I., “Sur la courbure totale d'une courbe gauche faisant un noeud,” Bull. Soc. Math. Fr., 77, pp. 128–138, 1949.MATHMathSciNetGoogle Scholar
  26. 26.
    Faugeras, O., Keriven, R., “Variational principles, surface evolution, PDE's, level set methods and the stereo problem,” IEEE Trans. Image Process., 7, pp. 336–344, 1998.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Fenchel, W., “Uber Krümmung und Windung geschlossenen Raumkurven,” Math. Ann., 101, pp. 238–252, 1929.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Fox, R.H., “On the total curvature of some tame knots,” Ann. Math., 52, pp. 258–261, 1950.CrossRefGoogle Scholar
  29. 29.
    Gage, M., Hamilton, R.S., “The heat equation shrinking convex plane curves,” J. Diff. Geom., 23, pp. 69–96, 1986.MATHMathSciNetGoogle Scholar
  30. 30.
    Goldman, R., “Curvature formulas for implicit curves and surfaces,” Computer Aided Geometric Design, 22, pp. 632–658, 2005.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Gray, A., Abbena, E., Salamon, S., Modern Differential Geometry of Curves and Surfaces with MATHEMATICA, Chapman & Hall/CRC, Boca Raton, FL, 2006.MATHGoogle Scholar
  32. 32.
    Gray, A., Tubes, 2nd ed., Birkhäuser, Boston, 2004.Google Scholar
  33. 33.
    Grayson, M., “The heat equation shrinks embedded plane curves to round points,” J. Diff. Geom., 26, pp. 285–314, 1987.MATHMathSciNetGoogle Scholar
  34. 34.
    Grayson, M., “A short note on the evolution of a surface by its mean curvature,” Duke Math. J., 58, pp. 555–558, 1989.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Gromoll, D., Klingenberg, W., Meyer, W., Riemannsche Geometric im Grossen. Lecture Notes in Mathematics, Vol. 55. Springer, Berlin, 1975.Google Scholar
  36. 36.
    Guggenheimer, H.W., Differential Geometry, Dover, New York, 1977.MATHGoogle Scholar
  37. 37.
    Hadwiger, H., Altes und Neues über Konvexe Körper, Birkhäuser Verlag, Basel, 1955.MATHGoogle Scholar
  38. 38.
    Hodge, W.V.D., Pedoe, D., Methods of Algebraic Geometry, Vols. 1–3, Cambridge University Press, London, 1952, (reissued 1994).MATHGoogle Scholar
  39. 39.
    Huisken, G., “Flow by mean curvature of convex surfaces into spheres,” J. Diff. Geom., 20, p. 237, 1984.MATHMathSciNetGoogle Scholar
  40. 40.
    Juan, O., Keriven, R., Postelnicu, G., “Stochastic motion and the level set method in computer vision: Stochastic active contours,” Int. J. Comput. Vision, 69, pp. 7–25, 2006.CrossRefGoogle Scholar
  41. 41.
    Kass, M., Witkin, A., Terzopoulos, D., “Snakes: Active contour models,” Int. J. Comput. Vision, 1, pp. 321–331, 1988.CrossRefGoogle Scholar
  42. 42.
    Katsoulakis, M.A., Kho, A.T., “Stochastic curvature flows: Asymptotic derivation, level set formulation and numerical experiments,” J. Interfaces Free Boundaries, 3, pp. 265–290, 2001.MATHMathSciNetGoogle Scholar
  43. 43.
    Kimmel, R., Bruckstein, A.M., “Shape offsets via level sets,” Computer-Aided Design, 25, pp. 154–162, 1993.MATHCrossRefGoogle Scholar
  44. 44.
    Kohli, D., Osvatic, M., “Inverse kinematics of the general 6R and 5R; P serial manipulators,” ASME J. Mech. Des., 115, pp. 922–931, 1993.CrossRefGoogle Scholar
  45. 45.
    Kuiper, N.H., Meeks, W.H., “The total curvature of a knotted torus,” J. Diff. Geom., 26, pp. 371–384, 1987.MATHMathSciNetGoogle Scholar
  46. 46.
    Langevin, R., Rosenburg, H., “On curvature integrals and knots,” Topology, 15, pp. 405–416, 1976.MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Lipschutz, M.M., Differential Geometry, Schaum's Outline Series, McGraw-Hill, New York, 1969.MATHGoogle Scholar
  48. 48.
    Manocha, D., Canny, J., “Efficient inverse kinematics for general 6R manipulators,” IEEE Trans. Robot. Automat., 10, pp. 648–657, 1994.CrossRefGoogle Scholar
  49. 49.
    Millman, R.S., Parker, G.D., Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1977.MATHGoogle Scholar
  50. 50.
    Milnor, J., “On the total curvature of knots,” Ann. Math., 52, pp. 248–257, 1950.CrossRefMathSciNetGoogle Scholar
  51. 51.
    Mumford, D., Shah, J., “Optimal approximations by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math., 42, p. 577, 1989.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Olver, P.J., Classical Invariant Theory, Cambridge University Press, London, 1999.MATHGoogle Scholar
  53. 53.
    Oprea, J., Differential Geometry and Its Applications, 2nd ed., The Mathematical Association of America, Washington, DC, 2007.MATHGoogle Scholar
  54. 54.
    Osher, S., Sethian, J.A., “Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys., 79, pp. 12–49, 1988.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Osher, S.J., Fedkiw, R.P., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2002.Google Scholar
  56. 56.
    Osserman, R., “Curvature in the eighties,” Amer. Math. Month., 97, pp. 731–756, 1990.MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Pham, B., “Offset curves and surfaces: A brief survey,” Computer-Aided Design, 24, pp. 223–229, 1992.CrossRefGoogle Scholar
  58. 58.
    Raghavan, M., Roth, B., “Inverse kinematics of the general 6R manipulator and related linkages,” ASME J. Mech. Des., 115, pp. 502–508, 1993.CrossRefGoogle Scholar
  59. 59.
    Rolfsen, D., Knots and Links, Publish or Perish Press, Wilmington, DE, 1976.MATHGoogle Scholar
  60. 60.
    Ros, A., “Compact surfaces with constant scalar curvature and a congruence theorem,” J. Diff. Geom., 27, pp. 215–220, 1988.MATHMathSciNetGoogle Scholar
  61. 61.
    San Jose Estepar, R., Haker, S., Westin, C.F., “Riemannian mean curvature flow,” in Lecture Notes in Computer Science: ISVC05, 3804, pp. 613–620, Springer, 2005.Google Scholar
  62. 62.
    Schubert, H., “Über eine Numeriche Knoteninvariante,” Math. Z., 61, pp. 245–288, 1954.MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Sethian, J.A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics. Computer Vision, and Materials Science, 2nd ed., Cambridge University Press, London, 1999.MATHGoogle Scholar
  64. 64.
    Shiohama, K., Takagi, R., “A characterization of a standard torus in E 3,” J. Diff. Geom., 4, pp. 477–485, 1970.MATHMathSciNetGoogle Scholar
  65. 65.
    Sommese, A.J., Wampler, C.W., The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005.MATHGoogle Scholar
  66. 66.
    Soner, H.M., Touzi, N., “A stochastic representation for mean curvature type geometric flows,” Ann. Prob., 31, pp. 1145–1165, 2003.MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Sullivan, J.M., “Curvatures of smooth and discrete surfaces,” in Discrete Differential Geometry, A.I. Bobenko, P. Schröder, J.M. Sullivan, and G.M. Ziegler, eds., Oberwolfach Seminars, Vol. 38, pp. 175–188, Birkhäuser, Basel, 2008.CrossRefGoogle Scholar
  68. 68.
    Voss, K., “Eine Bemerkung über die Totalkrümmung geschlossener Raumkurven,” Arch. Math., 6, pp. 259–263, 1955.MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Weyl, H., “On the volume of tubes,” Amer. J. Math., 61, pp. 461–472, 1939.CrossRefMathSciNetGoogle Scholar
  70. 70.
    Willmore, T.J., “Mean curvature of Riemannian immersions,” J. London Math. Soc., 3, pp. 307–310, 1971.MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Willmore, T.J., “Tight immersions and total absolute curvature,” Bull. London Math. Soc., 3, pp. 129–151, 1971.MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    Yip, N.K., “Stochastic motion by mean curvature,” Arch. Rational Mech. Anal., 144, pp. 331–355, 1998.CrossRefMathSciNetGoogle Scholar
  73. 73.
    Zhang, S., Younes, L., Zweck, J., Ratnanather, J.T., “Diffeomorphic surface flow: A novel method of surface evolution,” SIAM J. Appl. Math., 68, pp. 806–824, 2008.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations