Previous |  Up |  Next


ellipsoid; rolling maps; Gaussian curvature; geodesics; hypersurface
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.
[1] Bicchi, A., Sorrentino, R., Piaggio, C.: 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] Borisov, A. V., Mamaev, I. S.: Isomorphism and Hamilton representation of some nonholonomic systems. Sibirsk. Mat. Zh. 48 (2007), 1, 33-45. DOI 10.1007/s11202-007-0004-6 | MR 2304876 | Zbl 1164.37342
[3] Borisov, A. V., Mamaev, I. S.: 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 | MR 2350303 | Zbl 1229.37081
[4] Borisov, A. V., Mamaev, I. S.: Isomorphisms of geodesic flows on quadrics. Regular and Chaotic Dynamics 14 (2009), 4 - 5, 455-465. DOI 10.1134/s1560354709040030 | MR 2551869 | Zbl 1229.37096
[5] Caseiro, R., Martins, P., Henriques, J. F., Leite, F. Silva, Batista, J.: Rolling Riemannian manifolds to solve the multi-class classification problem. In: CVPR 2013, pp. 41-48. DOI 10.1109/cvpr.2013.13
[6] Chavel, I.: Riemannian Geometry - A Modern Introduction. Second edition. Cambridge Studies in Advanced Mathematics, No. 98. Cambridge University Press, Cambridge 2006. DOI 10.1017/cbo9780511616822 | MR 2229062
[7] Crouch, P., Leite, F. Silva: Rolling maps for pseudo-Riemannian manifolds. In: Proc. 51th IEEE Conference on Decision and Control, (Hawaii 2012). DOI 10.1109/cdc.2012.6426140
[8] Fedorov, Y. N., Jovanović, B.: 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 | MR 2076030 | Zbl 1125.37045
[9] Hüper, K., Krakowski., K. A., Leite, F. Silva: Rolling Maps in a Riemannian Framework. Textos de Matemática 43, Department of Mathematics, University of Coimbra 2011, pp. 15-30. MR 2894254
[10] Hüper, K., Leite, F. Silva: On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Graßmann manifolds. J. Dynam. Control Systems 13 (2007), 4, 467-502. DOI 10.1007/s10883-007-9027-3 | MR 2350231
[11] Prete, N. M. Justin Carpentier J.-P. L. Andrea Del: 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.
[12] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Classics in Mathematics 132, 1995. DOI 10.1007/978-3-642-66282-9 | MR 1335452 | Zbl 0836.47009
[13] Knörrer, H.: Geodesics on the ellipsoid. Inventiones Mathematicae 59 (1980), 119-144. DOI 10.1007/bf01390041 | MR 0577358 | Zbl 0431.53003
[14] Knörrer, H.: 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 | MR 0667450
[15] Korolko, A., Leite, F. Silva: Kinematics for rolling a Lorentzian 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] Krakowski, K., Leite, F. Silva: 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 | MR 3275084
[17] Krakowski, K. A., Leite, F. Silva: 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.
[18] Lee, J. M., J: Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag, Graduate Texts in Mathematics 176, New York 1997. MR 1468735
[19] Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Advances Math. 16 (1975), 2, 197-220. DOI 10.1016/0001-8708(75)90151-6 | MR 0375869 | Zbl 0303.34019
[20] Moser, J.: 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 | MR 0609560 | Zbl 0455.58018
[21] Nomizu, K.: Kinematics and differential geometry of submanifolds. Tôhoku Math. J. 30 (1978), 623-637. DOI 10.2748/tmj/1178229921 | MR 0516894 | Zbl 0395.53005
[22] Okamura, A. M., Smaby, N., Cutkosky, M. R.: An overview of dexterous manipulation. In: ICRA'00, IEEE Int Conf. on Robotics and Automation 2000, pp. 255–262. DOI:10.1109/robot.2000.844067 DOI 10.1109/robot.2000.844067
[23] Raţiu, T.: 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 | MR 0603766 | Zbl 0475.58006
[24] Sharpe, R. W.: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, Graduate Texts in Mathematics 166, New York 1997. MR 1453120 | Zbl 0876.53001
[25] Leite, F. Silva, Krakowski, K. A.: Covariant differentiation under rolling maps. In: Pré-Publicações do Departamento de Matemática, No. 08-22, University of Coimbra 2008, pp. 1-8.
[26] Spivak, M.: Calculus on Manifolds. Mathematics Monograph Series, Addison-Wesley, New York 1965. Zbl 0381.58003
[27] Uhlenbeck, K.: Minimal 2-spheres and tori in $S^k$. Preprint, 1975.
[28] Veselov, A. P.: A few things I learnt from Jürgen Moser. Regular and Chaotic Dynamics 13 (2008), 6, 515-524. DOI 10.1134/s1560354708060038 | MR 2465721 | Zbl 1229.37076
[29] Weintrit, A., Neumann, T., eds.: Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation. CRC Press, 2011. DOI 10.1201/b11344
Partner of
EuDML logo