Kybernetika 52 no. 2, 209-223, 2016

Geometry of the rolling ellipsoid

Krzysztof Andrzej Krakowski and Fátima Silva LeiteDOI: 10.14736/kyb-2016-2-0209


We study rolling maps of the Euclidean ellipsoid, rolling upon its affine tangent space at a point. Driven by the geometry of rolling maps, we find a simple formula for the angular velocity of the rolling ellipsoid along any piecewise smooth curve in terms of the Gauss map. This result is then generalised to rolling any smooth hyper-surface. On the way, we derive a formula for the Gaussian curvature of an ellipsoid which has an elementary proof and has been previously known only for dimension two.


ellipsoid, rolling maps, Gaussian curvature, geodesics, hypersurface


53B21, 53A05, 58E10, 70B10, 35B06


  1. A. Bicchi, R. Sorrentino and C. Piaggio: Dexterous manipulation through rolling. In: ICRA'95, IEEE Int. Conf. on Robotics and Automation 1995, pp. 452-457.   DOI:10.1109/robot.1995.525325
  2. A. V. Borisov and I. S. Mamaev: Isomorphism and {H}amilton representation of some nonholonomic systems. Sibirsk. Mat. Zh. 48 (2007), 1, 33-45.   DOI:10.1007/s11202-007-0004-6
  3. A. V. Borisov and I. S. Mamaev: Rolling of a non-homogeneous ball over a sphere without slipping and twisting. Regular and Chaotic Dynamics 12 (2007), 2, 153-159.   DOI:10.1134/s1560354707020037
  4. A. V. Borisov and I. S. Mamaev: Isomorphisms of geodesic flows on quadrics. Regular and Chaotic Dynamics 14 (2009), 4 - 5, 455-465.   DOI:10.1134/s1560354709040030
  5. R. Caseiro, P. Martins, J. F. Henriques, F. Silva Leite and J. Batista: Rolling Riemannian manifolds to solve the multi-class classification problem. In: CVPR 2013, pp. 41-48.   DOI:10.1109/cvpr.2013.13
  6. I. Chavel: Riemannian Geometry -- A Modern Introduction. Second edition. Cambridge Studies in Advanced Mathematics, No. 98. Cambridge University Press, Cambridge 2006.   DOI:10.1017/cbo9780511616822
  7. P. Crouch and F. Silva Leite: Rolling maps for pseudo-Riemannian manifolds. In: Proc. 51th IEEE Conference on Decision and Control, (Hawaii 2012).   DOI:10.1109/cdc.2012.6426140
  8. Y. N. Fedorov and B. Jovanović: Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces. J. Nonlinear Science 14 (2004), 4, 341-381.   DOI:10.1007/s00332-004-0603-3
  9. K. H{ü}per, K. A. Krakowski. and F. Silva Leite: Rolling Maps in a Riemannian Framework. Textos de Matemática 43, Department of Mathematics, University of Coimbra 2011, pp. 15-30.   CrossRef
  10. K. Hüper and F. Silva Leite: On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Gra\ss mann manifolds. J. Dynam. Control Systems 13 (2007), 4, 467-502.   DOI:10.1007/s10883-007-9027-3
  11. N. M. Justin Carpentier J.-P. L. Andrea Del Prete: An analytical model of rolling contact and its application to the modeling of bipedal locomotion. In: Proc. IMA Conference on Mathematics of Robotics 2015, pp. 452-457.   CrossRef
  12. T. Kato: Perturbation Theory for Linear Operators. Springer-Verlag, Classics in Mathematics 132, 1995.   DOI:10.1007/978-3-642-66282-9
  13. H. Knörrer: Geodesics on the ellipsoid. Inventiones Mathematicae 59 (1980), 119-144.   DOI:10.1007/bf01390041
  14. H. Knörrer: Geodesics on quadrics and a mechanical problem of C. Neumann. J. für die reine und angewandte Mathematik 334 (1982), 69-78.   DOI:10.1515/crll.1982.334.69
  15. A. Korolko and F. Silva Leite: Kinematics for rolling a {L}orentzian sphere. In: Proc. 50th IEEE Conference on Decision and Control and European Control Conference (IEEE CDC-ECC 2011), Orlando 2011, pp. 6522-6528.   DOI:10.1109/cdc.2011.6160592
  16. K. Krakowski and F. Silva Leite: An algorithm based on rolling to generate smooth interpolating curves on ellipsoids. Kybernetika 50 (2014), 4, 544-562.   DOI:10.14736/kyb-2014-4-0544
  17. K. A. Krakowski and F. Silva Leite: Why controllability of rolling may fail: a few illustrative examples. In: Pré-Publicações do Departamento de Matemática, no. 12-26. University of Coimbra 2012, pp. 1-30.   CrossRef
  18. J. M. Lee and J: Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag, Graduate Texts in Mathematics 176, New York 1997.   CrossRef
  19. J. Moser: Three integrable {Hamiltonian} systems connected with isospectral deformations. Advances Math. 16 (1975), 2, 197-220.   DOI:10.1016/0001-8708(75)90151-6
  20. J. Moser: Geometry of quadrics and spectral theory. In: The Chern Symposium 1979 (W.-Y. Hsiang, S. Kobayashi, I. Singer, J. Wolf, H.-H. Wu, and A. Weinstein, eds.), Springer, New York 1980, pp. 147-188.   DOI:10.1007/978-1-4613-8109-9_7
  21. K. Nomizu: Kinematics and differential geometry of submanifolds. T{\^o}hoku Math. J. 30 (1978), 623-637.   DOI:10.2748/tmj/1178229921
  22. A. M. Okamura, N. Smaby and M. R. Cutkosky:     CrossRef
  23. T. Ra{ţ}iu: The {C.} {Neumann} problem as a completely integrable system on an adjoint orbit. Trans. Amer. Math. Soc. 264 (1981), 2, 321-329.   DOI:10.1090/s0002-9947-1981-0603766-3
  24. R. W. Sharpe: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, Graduate Texts in Mathematics 166, New York 1997.   CrossRef
  25. F. Silva Leite and K. A. Krakowski: Covariant differentiation under rolling maps. In: Pr{é}-Publicações do Departamento de Matemática, No. 08-22, University of Coimbra 2008, pp. 1-8.   CrossRef
  26. M. Spivak: Calculus on Manifolds. Mathematics Monograph Series, Addison-Wesley, New York 1965.   CrossRef
  27. K. Uhlenbeck: Minimal 2-spheres and tori in {$S^k$}. Preprint, 1975.   CrossRef
  28. A. P. Veselov: A few things {I} learnt from {J}{ü}rgen {M}oser. Regular and Chaotic Dynamics 13 (2008), 6, 515-524.   DOI:10.1134/s1560354708060038
  29. A. Weintrit, T. Neumann and eds.: Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation. CRC Press, 2011.   DOI:10.1201/b11344