Abstract
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
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abhyankar, S.S., Algebraic Geometry for Scientists and Engineers, Mathematical Surveys and Monographs, 35, American Mathematical Society, Providence, RI, 1990.
Adams, C.C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W.H. Freeman, New York, 1994.
Bates, P.W., Wei, G.W., Zhao, S., “Minimal molecular surfaces and their applications,” J. Comput. Chem., 29, pp. 380–391, 2008.
Ben-Israel, A., Greville, T.N.E., Generalized Inverses: Theory and Applications, 2nd ed., Canadian Mathematical Society Books in Mathematics, Springer, New York, 2003.
Bishop, R., “There is more than one way to frame a curve,” Amer. Math. Month., 82, pp. 246–251, 1975.
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.
Bloomenthal, J., (ed.), Introduction to Implicit Surfaces, Morgan Kaufmann, San Francisco, 1997.
Bloomenthal, J., Shoemake, K., “Convolution surfaces,” Computer Graphics, 25, pp. 251–256, 1991 (Proc. SIGGRAPH'91).
Bottema, O., Roth, B., Theoretical Kinematics, Dover, New York, 1990.
Brakke, K.A., The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978.
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.
Chan, T.F., Vese, L.A., “Active contours without edges,” IEEE Trans. Image Process., 10, pp. 266–277, 2001.
Chazvini, M., “Reducing the inverse kinematics of manipulators to the solution of a generalized eigenproblem,” Computational Kinematics, pp. 15–26, 1993.
Chen, B.-Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.
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.
Chen, B.-Y., “On an inequality of mean curvature,” J. London Math. Soc., 4, pp. 647–650, 1972.
Chirikjian, G.S., “Closed-form primitives for generating locally volume preserving deformations,” ASME J. Mech. Des., 117, pp. 347–354, 1995.
Chopp, D.L., “Computing minimal surfaces via level set curvature flow,” J. Comput. Phys., 106, pp. 77–91, 1993.
do Carmo, M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.
Dombrowski, P., “Krummungsgrossen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten,” Math. Nachr., 38, pp. 133–180, 1968.
Evans, L.C., Spruck, J., “Motion of level sets by mean curvature,” J. Diff. Geom., 33, pp. 635–681, 1991.
Evans, L.C., Spruck, J., “Motion of level sets by mean curvature II,” Trans. Amer. Math. Soc., 330, pp. 321–332, 1992.
Farouki, R.T., Neff, C.A., “Analytical properties of plane offset curves,” Computer Aided Geometric Design, 7, pp. 83–99, 1990.
Farouki, R.T., Neff, C.A., “Algebraic properties of plane offset curves,” Computer Aided Geometric Design, 7, pp. 101–127, 1990.
Fary, I., “Sur la courbure totale d'une courbe gauche faisant un noeud,” Bull. Soc. Math. Fr., 77, pp. 128–138, 1949.
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.
Fenchel, W., “Uber Krümmung und Windung geschlossenen Raumkurven,” Math. Ann., 101, pp. 238–252, 1929.
Fox, R.H., “On the total curvature of some tame knots,” Ann. Math., 52, pp. 258–261, 1950.
Gage, M., Hamilton, R.S., “The heat equation shrinking convex plane curves,” J. Diff. Geom., 23, pp. 69–96, 1986.
Goldman, R., “Curvature formulas for implicit curves and surfaces,” Computer Aided Geometric Design, 22, pp. 632–658, 2005.
Gray, A., Abbena, E., Salamon, S., Modern Differential Geometry of Curves and Surfaces with MATHEMATICA, Chapman & Hall/CRC, Boca Raton, FL, 2006.
Gray, A., Tubes, 2nd ed., Birkhäuser, Boston, 2004.
Grayson, M., “The heat equation shrinks embedded plane curves to round points,” J. Diff. Geom., 26, pp. 285–314, 1987.
Grayson, M., “A short note on the evolution of a surface by its mean curvature,” Duke Math. J., 58, pp. 555–558, 1989.
Gromoll, D., Klingenberg, W., Meyer, W., Riemannsche Geometric im Grossen. Lecture Notes in Mathematics, Vol. 55. Springer, Berlin, 1975.
Guggenheimer, H.W., Differential Geometry, Dover, New York, 1977.
Hadwiger, H., Altes und Neues über Konvexe Körper, Birkhäuser Verlag, Basel, 1955.
Hodge, W.V.D., Pedoe, D., Methods of Algebraic Geometry, Vols. 1–3, Cambridge University Press, London, 1952, (reissued 1994).
Huisken, G., “Flow by mean curvature of convex surfaces into spheres,” J. Diff. Geom., 20, p. 237, 1984.
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.
Kass, M., Witkin, A., Terzopoulos, D., “Snakes: Active contour models,” Int. J. Comput. Vision, 1, pp. 321–331, 1988.
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.
Kimmel, R., Bruckstein, A.M., “Shape offsets via level sets,” Computer-Aided Design, 25, pp. 154–162, 1993.
Kohli, D., Osvatic, M., “Inverse kinematics of the general 6R and 5R; P serial manipulators,” ASME J. Mech. Des., 115, pp. 922–931, 1993.
Kuiper, N.H., Meeks, W.H., “The total curvature of a knotted torus,” J. Diff. Geom., 26, pp. 371–384, 1987.
Langevin, R., Rosenburg, H., “On curvature integrals and knots,” Topology, 15, pp. 405–416, 1976.
Lipschutz, M.M., Differential Geometry, Schaum's Outline Series, McGraw-Hill, New York, 1969.
Manocha, D., Canny, J., “Efficient inverse kinematics for general 6R manipulators,” IEEE Trans. Robot. Automat., 10, pp. 648–657, 1994.
Millman, R.S., Parker, G.D., Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1977.
Milnor, J., “On the total curvature of knots,” Ann. Math., 52, pp. 248–257, 1950.
Mumford, D., Shah, J., “Optimal approximations by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math., 42, p. 577, 1989.
Olver, P.J., Classical Invariant Theory, Cambridge University Press, London, 1999.
Oprea, J., Differential Geometry and Its Applications, 2nd ed., The Mathematical Association of America, Washington, DC, 2007.
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.
Osher, S.J., Fedkiw, R.P., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2002.
Osserman, R., “Curvature in the eighties,” Amer. Math. Month., 97, pp. 731–756, 1990.
Pham, B., “Offset curves and surfaces: A brief survey,” Computer-Aided Design, 24, pp. 223–229, 1992.
Raghavan, M., Roth, B., “Inverse kinematics of the general 6R manipulator and related linkages,” ASME J. Mech. Des., 115, pp. 502–508, 1993.
Rolfsen, D., Knots and Links, Publish or Perish Press, Wilmington, DE, 1976.
Ros, A., “Compact surfaces with constant scalar curvature and a congruence theorem,” J. Diff. Geom., 27, pp. 215–220, 1988.
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.
Schubert, H., “Über eine Numeriche Knoteninvariante,” Math. Z., 61, pp. 245–288, 1954.
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.
Shiohama, K., Takagi, R., “A characterization of a standard torus in E 3,” J. Diff. Geom., 4, pp. 477–485, 1970.
Sommese, A.J., Wampler, C.W., The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005.
Soner, H.M., Touzi, N., “A stochastic representation for mean curvature type geometric flows,” Ann. Prob., 31, pp. 1145–1165, 2003.
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.
Voss, K., “Eine Bemerkung über die Totalkrümmung geschlossener Raumkurven,” Arch. Math., 6, pp. 259–263, 1955.
Weyl, H., “On the volume of tubes,” Amer. J. Math., 61, pp. 461–472, 1939.
Willmore, T.J., “Mean curvature of Riemannian immersions,” J. London Math. Soc., 3, pp. 307–310, 1971.
Willmore, T.J., “Tight immersions and total absolute curvature,” Bull. London Math. Soc., 3, pp. 129–151, 1971.
Yip, N.K., “Stochastic motion by mean curvature,” Arch. Rational Mech. Anal., 144, pp. 331–355, 1998.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Birkhäuser Boston
About this chapter
Cite this chapter
Chirikjian, G.S. (2009). Geometry of Curves and Surfaces. In: Stochastic Models, Information Theory, and Lie Groups, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4803-9_5
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4803-9_5
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4802-2
Online ISBN: 978-0-8176-4803-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)