Geometry of the rolling ellipsoid

Krzysztof Andrzej Krakowski; Fátima Silva Leite

Kybernetika (2016)

  • Volume: 52, Issue: 2, page 209-223
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Krakowski, Krzysztof Andrzej, and Silva Leite, Fátima. "Geometry of the rolling ellipsoid." Kybernetika 52.2 (2016): 209-223. <http://eudml.org/doc/281552>.

@article{Krakowski2016,
abstract = {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.},
author = {Krakowski, Krzysztof Andrzej, Silva Leite, Fátima},
journal = {Kybernetika},
keywords = {ellipsoid; rolling maps; Gaussian curvature; geodesics; hypersurface; ellipsoid; rolling maps; Gaussian curvature; geodesics; hypersurface},
language = {eng},
number = {2},
pages = {209-223},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Geometry of the rolling ellipsoid},
url = {http://eudml.org/doc/281552},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Krakowski, Krzysztof Andrzej
AU - Silva Leite, Fátima
TI - Geometry of the rolling ellipsoid
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 2
SP - 209
EP - 223
AB - 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.
LA - eng
KW - ellipsoid; rolling maps; Gaussian curvature; geodesics; hypersurface; ellipsoid; rolling maps; Gaussian curvature; geodesics; hypersurface
UR - http://eudml.org/doc/281552
ER -

References

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  1. Bicchi, A., Sorrentino, R., Piaggio, C., 10.1109/robot.1995.525325, In: ICRA'95, IEEE Int. Conf. on Robotics and Automation 1995, pp. 452-457. DOI10.1109/robot.1995.525325
  2. Borisov, A. V., Mamaev, I. S., 10.1007/s11202-007-0004-6, Sibirsk. Mat. Zh. 48 (2007), 1, 33-45. Zbl1164.37342MR2304876DOI10.1007/s11202-007-0004-6
  3. Borisov, A. V., Mamaev, I. S., 10.1134/s1560354707020037, Regular and Chaotic Dynamics 12 (2007), 2, 153-159. Zbl1229.37081MR2350303DOI10.1134/s1560354707020037
  4. Borisov, A. V., Mamaev, I. S., 10.1134/s1560354709040030, Regular and Chaotic Dynamics 14 (2009), 4 - 5, 455-465. Zbl1229.37096MR2551869DOI10.1134/s1560354709040030
  5. Caseiro, R., Martins, P., Henriques, J. F., Leite, F. Silva, Batista, J., 10.1109/cvpr.2013.13, In: CVPR 2013, pp. 41-48. DOI10.1109/cvpr.2013.13
  6. Chavel, I., 10.1017/cbo9780511616822, Cambridge Studies in Advanced Mathematics, No. 98. Cambridge University Press, Cambridge 2006. MR2229062DOI10.1017/cbo9780511616822
  7. Crouch, P., Leite, F. Silva, 10.1109/cdc.2012.6426140, In: Proc. 51th IEEE Conference on Decision and Control, (Hawaii 2012). DOI10.1109/cdc.2012.6426140
  8. Fedorov, Y. N., Jovanović, B., 10.1007/s00332-004-0603-3, J. Nonlinear Science 14 (2004), 4, 341-381. Zbl1125.37045MR2076030DOI10.1007/s00332-004-0603-3
  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. MR2894254
  10. Hüper, K., Leite, F. Silva, 10.1007/s10883-007-9027-3, J. Dynam. Control Systems 13 (2007), 4, 467-502. MR2350231DOI10.1007/s10883-007-9027-3
  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., 10.1007/978-3-642-66282-9, Springer-Verlag, Classics in Mathematics 132, 1995. Zbl0836.47009MR1335452DOI10.1007/978-3-642-66282-9
  13. Knörrer, H., 10.1007/bf01390041, Inventiones Mathematicae 59 (1980), 119-144. Zbl0431.53003MR0577358DOI10.1007/bf01390041
  14. Knörrer, H., 10.1515/crll.1982.334.69, J. für die reine und angewandte Mathematik 334 (1982), 69-78. MR0667450DOI10.1515/crll.1982.334.69
  15. Korolko, A., Leite, F. Silva, 10.1109/cdc.2011.6160592, In: Proc. 50th IEEE Conference on Decision and Control and European Control Conference (IEEE CDC-ECC 2011), Orlando 2011, pp. 6522-6528. DOI10.1109/cdc.2011.6160592
  16. Krakowski, K., Leite, F. Silva, 10.14736/kyb-2014-4-0544, Kybernetika 50 (2014), 4, 544-562. MR3275084DOI10.14736/kyb-2014-4-0544
  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. MR1468735
  19. Moser, J., 10.1016/0001-8708(75)90151-6, Advances Math. 16 (1975), 2, 197-220. Zbl0303.34019MR0375869DOI10.1016/0001-8708(75)90151-6
  20. Moser, J., 10.1007/978-1-4613-8109-9_7, 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. Zbl0455.58018MR0609560DOI10.1007/978-1-4613-8109-9_7
  21. Nomizu, K., 10.2748/tmj/1178229921, Tôhoku Math. J. 30 (1978), 623-637. Zbl0395.53005MR0516894DOI10.2748/tmj/1178229921
  22. Okamura, A. M., Smaby, N., Cutkosky, M. R., 10.1109/robot.2000.844067, In: ICRA'00, IEEE Int Conf. on Robotics and Automation 2000, pp. 255–262. DOI:10.1109/robot.2000.844067 DOI10.1109/robot.2000.844067
  23. Raţiu, T., 10.1090/s0002-9947-1981-0603766-3, Trans. Amer. Math. Soc. 264 (1981), 2, 321-329. Zbl0475.58006MR0603766DOI10.1090/s0002-9947-1981-0603766-3
  24. Sharpe, R. W., Differential Geometry: Cartan's Generalization of Klein's Erlangen Program., Springer-Verlag, Graduate Texts in Mathematics 166, New York 1997. Zbl0876.53001MR1453120
  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. Zbl0381.58003
  27. Uhlenbeck, K., Minimal 2-spheres and tori in S k ., Preprint, 1975. 
  28. Veselov, A. P., 10.1134/s1560354708060038, Regular and Chaotic Dynamics 13 (2008), 6, 515-524. Zbl1229.37076MR2465721DOI10.1134/s1560354708060038
  29. Weintrit, A., Neumann, T., eds., 10.1201/b11344, CRC Press, 2011. DOI10.1201/b11344

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