Adams-Moulton Method Approach to Geodesic Paths on 2D Surfaces

Document Type : Original Article

Authors
Esa Sharahi 1
Esmaiel Peyghan 1
Amir Baghban 2
1 Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.
2 Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, Tabriz 53751 71379, Iran.
10.29252/maco.1.2.8
Abstract
Our aim in this paper, is applying Adams-Moulton algorithm to find the geodesics as the answers of the classical system of ordinary differential equations on a 2-dimensional surface for which a Riemannian metric is defined. 

Keywords

[1] D. Bao, S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, SpringerVerlag, 2000.
[2] M. Berger, A Panoramic View of Riemannian Geometry, Springer-Verlag, 2003.
[3] K. Burns, V. Matveev, Open Problems and Questions About Geodesics, arXiv:1308.5417v2.
[4] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley, 2003.
[5] S. Fiori, Quasi-Geodesic neural learning algorithms over the orthogonal group: A tutorial, Journal of Machine Learning Research, 6, 2005, 743-781.
[6] D. Gabay, Minimizing a differentiable function over a differential manifold, Journal of Optimization Theory and Applications, 37(2), 1982, 177-219.
[7] M. M. Postnikov, Geometry VI: Riemannian Geometry, Springer-Verlag, 2001.
[8] C. Scheffer, J. Vahrenhold, Approximating geodesic distances on 2-manifolds in R 2, Computational Geometry, 47(2A), 2014, 125-140.
[9] C. Udrişte, Convex functions and optimization methods in Riemannian manifolds, Mathematics and Its Applications, Vol. 297, Kluwer Academic, Dordrecht, 1994.
[10] S. Vacaru, Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces, Math. Sci., 6(18), 2012, doi:10.1186/2251-7456-6-18, arXiv: 0705.0032.
[11] D. Zosso, X, Bresson, JP. Thiran, Geodesic active fields-a geometric framework for image registration, IEEE Trans Image Process, 20(5), 2011, 1300-1312.
Mathematical Analysis and Convex Optimization
Volume 1, Issue 2
December 2020
Pages 81-93