Communications in Information and Systems
Volume 18 (2018)
Algebraic equation of geodesics on the 2D Euclidean space with an exponential density function
Pages: 91 – 106
Given two points $s$ and $t$ on the two-dimensional Euclidean space, the straight-line segment connecting $s$ and $t$ gives the shortest path, or geodesic, between them. However, when the 2D plane is equipped with a non-uniform density function, generally one cannot find a closed-form solution to characterize the shortest path, assuming that the endpoints are given. We observe that when the 2D plane is equipped with an exponential-type density function, the algebraic equation of a geodesic path between two points, as well as the corresponding length, can be found explicitly by a variational approach. We systematically give both the theoretical and experimental results in this paper. We also extend the approach to compute the geodesic problem on a polyhedral surface with a pre-defined density function.
This work is partially supported by the NSF of China (61772016, 61772312), China 973 Program (2015CB352502), the NSF of Shandong (ZR2014FM015), and the key research and development project of Shandong province (2017GGX10110).
Published 17 October 2018