Communications in Information and Systems

Volume 18 (2018)

Number 2

Algebraic equation of geodesics on the 2D Euclidean space with an exponential density function

Pages: 91 – 106

DOI: https://dx.doi.org/10.4310/CIS.2018.v18.n2.a2

Authors

Shuangmin Chen (School of Information and Technology, Qingdao University of Science and Technology, Qingdao, China)

Guozhu Liu (School of Information and Technology, Qingdao University of Science and Technology, Qingdao, China)

Shiqing Xin (School of Computer Science and Technology, Shandong University, Qingdao, China)

Yuanfeng Zhou (School of Software, Shandong University, Jinan, China)

Ying He (School of Computer Engineering, Nanyang Techonological University, Singapore)

Changhe Tu (School of Computer Science and Technology, Shandong University, Qingdao, China)

Abstract

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.

Published 17 October 2018