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Elastic curves on the sphere

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Abstract

This paper deals with the derivation of equations suitable for the computation of elastic curves on the sphere. To this end, equations for the main invariants of spherical elastic curves are given. A new method for solving geometrically constraint differential equations is used to compute the curves for given initial values. A classification of the fundamental forms of the curves is presented.

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References

  1. O. Bolza,Vorlesungen über Variationsrechnung (Koehler und Amelang, Leipzig, 1949).

    Google Scholar 

  2. A.M. Bruckstein and A.N. Netravali, On minimal energy trajectories, Comp. Vision, Graphics, and Image Proc. 49(1990)283–296.

    Article  Google Scholar 

  3. G. Brunnett, Properties of minimal energy splines, in:Curve and Surface Design, ed. H. Hagen (SIAM, 1992), pp. 3–22.

  4. G. Brunnett, A new characterization of plane elastica, in:Mathematical Methods in Computer Aided Design II, ed. T. Lyche and L. Schumaker (Academic Press, 1992) pp. 43–56.

  5. G. Brunnet and J. Kiefer, Interpolation with minimal energy splines, to be published in CAD.

  6. M.P. do Carmo,Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976).

  7. P.E. Crouch, R. Grossman and Y. Yan, A third order Runge-Kutta algorithm on a manifold, submitted to BIT (1992).

  8. P.E. Crouch, Y. Yan and R. Grossman, On the numerical integration of the synamic attitude equations,Proc. IEEE CDC Conf., Tucson, AZ (1992), to appear.

  9. P.E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlin. Sci. (1991), to appear.

  10. L. Euler,Additamentum De Curvis Elasticis, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Ser. 1, Vol. 24, Lausanne (1744).

  11. J. Hoschek and G. Seemann, Spherical splines, Math. Mod. Numer. Anal. 26(1992)1–22.

    MATH  MathSciNet  Google Scholar 

  12. J. Jackson and P.E. Crouch, Dynamic interpolation and application to flight control. J. Guidance, Control and Dynamics 14(1991)814–822.

    Article  Google Scholar 

  13. E. Jou and W. Han, Minimal energy splines with various end constraints, in:Curve and Surface Design, ed. H. Hagen (SIAM, 1992), pp. 23–30.

  14. A.E.H. Love,A Treatise on the Mathematical Theory of Elasticity, 4th ed. (Cambridge University Press, 1927).

  15. H. Moreton and C. Sequin, Surface design with minimum energy networks, ACM Comp. Graphics,Proc. SIGGRAPH (1991).

  16. G. Nielson, Bernstein/Bézier curves and splines on spheres based upon a spherical de Casteljau algorithm, Technical Report TR-88-028, Arizona State University (1988).

  17. L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inf. 6(1989)465–473.

    MATH  MathSciNet  Google Scholar 

  18. K. Shoemaker, Animating rotation with quaternion curves, ACM Comp. Graphics 10,Proc. SIGGRAPH’85 (1985) pp. 245–254.

  19. K. Strubecker,Differentialgeometrie I–III, Sammlung Göschen (de Gruyter, Berlin, 1969).

    MATH  Google Scholar 

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Authors and Affiliations

  1. FB Informatik, Universität Kaiserslautern, Postfach 3049, 6750, Kaiserslautern, Germany

    Guido Brunnett

  2. Center for Systems Science and Engineering, Arizona State University, 85287, Tempe, AZ, USA

    Peter E. Crouch

Authors
  1. Guido Brunnett
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  2. Peter E. Crouch
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Brunnett, G., Crouch, P.E. Elastic curves on the sphere. Adv Comput Math 2, 23–40 (1994). https://doi.org/10.1007/BF02519034

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