Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T11:25:17.949Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of spline curves on smooth manifolds by action of Lie groups

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  01 February 2015

A. P. Pobegailo
A. P. Pobegailo*
Affiliation:
Belarusian State University, Faculty of Applied Mathematics and Computer Science, Pr. Nezavisimosti 4, 220030, Minsk, Belarus email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Polynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.

MSC classification

Primary: 65D05: Interpolation
Secondary: 65D07: Splines 65D17: Computer aided design (modeling of curves and surfaces)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 217 - 230
Copyright
© The Author 2015 

References

Camarinha, M., Silva Leite, F. and Crouch, P., ‘Splines of class C k on non-euclidean spaces’, Comput. Aided Geom. Design 12 (1995) 399–410.Google Scholar
Crouch, P., Kun, G. and Silva Leite, F., ‘The De Casteljau algorithm on Lie groups and spheres’, J. Dyn. Control Syst. 5 (1999) 397–429.CrossRefGoogle Scholar
Crouch, P. and Silva Leite, F., ‘The dynamic interpolation problem: on Riemannian manifolds, Lie groups and symmetric spaces’, J. Dyn. Control Syst. 1 (1995) 177–202.Google Scholar
Flöry, S. and Hofer, M., ‘Constrained curve fitting on manifolds’, Comput. Aided Des. 40 (2008) 25–34.Google Scholar
Hofer, M. and Pottmann, H., ‘Energy-minimizing splines in manifolds’, ACM Trans. Graph. 23 (2004) 284–293.CrossRefGoogle Scholar
Jakubiak, J., Silva Leite, F. and Rodrigues, R. C., ‘A two-step algorithm of smooth spline generation on Riemannian manifolds’, J. Comput. Appl. Math. 194 (2006) 177–191.Google Scholar
Marthinsen, A., ‘Interpolation in Lie groups’, SIAM J. Numer. Anal. 37 (1999) 269–285.CrossRefGoogle Scholar
Noakes, L., Heinzinger, G. and Paden, B., ‘Cubic splines on curved spaces’, IMA J. Math. Control Appl. 6 (1989) 465–473.CrossRefGoogle Scholar
Park, F. and Ravani, B., ‘Bézier curves on Riemannian manifolds and Lie groups with kinematic applications’, ASME J. Mech. Des. 117 (1995) 36–40.CrossRefGoogle Scholar
Pesenson, I., ‘Variational splines on Riemannian manifolds with applications to integral geometry’, Adv. Appl. Math. 33 (2004) 548–572.CrossRefGoogle Scholar
Pobegailo, A. P., ‘ C n interpolation on smooth manifolds with one-parameter transformations’, Comput. Aided Des. 28 (1996) 973–979.Google Scholar
Popiel, T. and Noakes, L., ‘Bézier curves and C 2 interpolation in Riemannian manifolds’, J. Approx. Theory 148 (2007) 111–127.Google Scholar
Rodrigues, R. C., Silva Leite, F. and Jakubiak, J., ‘A new geometric algorithm to generate smooth interpolating curves on Riemannian manifolds’, LMS J. Comput. Math. 8 (2005) 251–266.CrossRefGoogle Scholar
Rogers, D. F. and Adams, J. A., Mathematical elements for computer graphics (McGraw-Hill, New York, 1990).Google Scholar
Warner, F. W., Foundations of smooth manifolds and Lie groups (Springer, New York, 1983).Google Scholar