De Casteljau algorithm on Lie groups and spheres

Journal of Dynamical and Control Systems

Volumn 5, Issue 3, 1999, Pages 397-429

  Crouch, P ^a   Kun, G ^b   Silva Leite, F ^c  

^a ARIZONA STATE UNIVERSITY   (United States)

^b RWTH AACHEN UNIVERSITY   (Germany)

^c UNIVERSITY OF COIMBRA   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BOUNDARY CONDITIONS; INTERPOLATION; POLYNOMIALS; SPHERES;

DE CASTELJAU ALGORITHM; LIE GROUP;

COMPUTATIONAL GEOMETRY;

EID: 0033155646     PISSN: 10792724     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021770717822     Document Type: Article

References (33)

1. A. Barr, B. Currin, S. Gabriel, and J. Hughes, Smooth interpolation of orientations with angular velocity constraints using quaternions. In: Proc. Computer Graphics (SIGRAPH 92) 26 (1992), No. 2, 313-320.
2. P. Bézier, The mathematical basis of the UNISURF CAD System. Butterworths, London, 1986.
3. W. M. Boothby, An introduction to differential manifolds and Riemannian geometry. Academic Press, Pure and Appl. Math. Series of Monographs and Textbooks, Vol. 63, New York, 1975.
4. k on non-Euclidean spaces. J. Math. Control and Inform. 12 (1995), 399-410.
5. _, Second order optimality conditions for a higher order variational problem on a Riemannian manifold. In: Proc. 35th IEEE CDC, Kobe-Japan, 11-13 December (1996), Vol. II, 1636-1641.
6. _, Sufficient conditions for an optimization problem on a Riemannian manifold. In: Proc. CONTROLO-96, Porto-Portugal, 11-13 September (1996), Vol. I, 127-131.
7. J. Cardoso and F. Silva Leite, Logarithms and exponentials for the Lie group of P-orthogonal matrices. Submitted in 1998.
8. Chao-Chi Chen, Interpolation of orientation matrices using sphere splines in computer animation. Master of Science Thesis, Arizona State University (1990).
9. P. Crouch and F. Silva Leite, The dynamic interpolation problem on riemannian manifolds, Lie groups and symmetric spaces, J. Dynam. Control Syst. 1 (1995), No. 2, 177-202.
10. _, Geometry and the dynamic interpolation problem, In: Proc. Am. Control Confer. Boston, 26-29 July (1991).
11. _, Closed forms for the exponential mapping on matrix Lie groups based on Putzer's method. To appear in: J. Math. Physics.
12. P. Crouch, G. Kun, and F. Silva Leite, De Casteljau algorithm for cubic polynomials on the rotation group. In: Proc. CONTROLO-96, 11-13 September (1996), Vol. II, 547-552.
13. _, Geometric splines. To appear in: Proc. 14th IFAC World Congress, 5-9 July (1999), Beijing, P. R. China.
14. P. Crouch and J. Jackson, A non-holonomic dynamic interpolation problem. In: Proc. Confer. Anal. Control Dynam. Syst. Lion-France, 1990, Birkhäuser, series Progress in Systems and Control, 1991, 156-166.
15. _, Dynamic interpolation and application to flight control. To appear in: Proc. IEEE Decision and Control Confer. Honolulu, Hawaii (1990).
16. _, Dynamic interpolation and application to flight control To appear in: J. Guidance, Control and Dynam.
17. J. Jackson, Dynamic interpolation and application to flight control. PhD thesis, Arizona State University, 1990.
18. P. De Casteljau, Outillages Méthodes Calcul. Technical Report, A. Citroen, Paris, 1959.
19. G. Farin, Curves and surfaces for CAGD. Academic Press, Third Edition, 1993.
20. Q. J. Ge and B. Ravani, Computer aided geometric design of motion interpolants. In: Proc. ASME Design Automation Conf. Miami, Fl, September, 1991, 33-41.
21. _, Computational geometry and mechanical design synthesis, In: Proc. 13th IMACS World Congress on Computation and Applied Mathematics, Dublin, Ireland, 1991, 1013-1015.
22. _, Geometric construction of Bezier motions. ASME J. Mechan. Design 116 (1994), 749-755.
23. R. Hirschorn, Curves in homogeneous spaces. Can. J. Math. 29 (1977), No. 1, 77-83.
24. B. Jutler, Visualization of moving objects using dual quaternion curves. Computers and Graphics 18 (1994), No. 3, 315-326.
25. M. J. Kim, M. S. Kim, and S. Y. Shin, A general construction scheme for unit quaternion curves with simple high order derivatives. In: Computer Graphics Proc., Annual Conf. Series, (SIGRAPH 95), Los Angeles, CA., 1995, 369-376.
26. J. Milnor, Morse theory. Ann. Math. Stud. 51 (1969), Princeton University Press.
27. G. Nielson and R. Heiland, Animated rotations using quaternions and splines on a 4D sphere. In: Programmirovanie (Russia) Springer Verlag, English edition. Programming and Computer Software, Plenum Pub. NY 17-27 (1992).
28. G. Nielson, Smooth interpolation of orientations. Models and Techniques in Computer Animation (N. Magnenat Thalmann and D. Thalmann Eds.), Springer Verlag, Tokyo, 1993, 75-93.
29. L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform. 6 (1989), 465-473.
30. K. Nomizu, Foundations of differential geometry. In: Interscience Tracts in Pure and Applied Mathematics, John Willey and Sons, Vol. I, II (1969).
31. F. Park and B. Ravani, Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mechan. Design 117 (1995), 36-40.
32. D. Sattinger and O. Weaver, Lie groups and algebras with applications to physics, geometry and mechanics. Appl. Math. Sci. 61, Springer Verlag, 1986.
33. K. Shoemake, Animating rotations with quaternion curves. ACM SI-GRAPH 85 19 245-254 (1985).