Algebraic Equation Of Geodesics On The 2d Euclidean Space With An Exponential Density Function
COMMUNICATIONS IN INFORMATION AND SYSTEMS(2018)
摘要
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.
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