Efficiently Computing Shortest Paths on Curved Surfaces with Newton's Method

Geodesics are important in the study of metric geometry. Although Euler–Lagrange equations are used to formulate geodesics, closed-form solutions are not available except in a few cases. Therefore, researchers have to seek for numerical methods instead of finding geodesics in computer vision and graphics. In this paper, we first formulate the computation of geodesics on a parametric surface into an optimizationdriven problem and then propose an efficient solution to the optimization problem with a second-order Newton iteration method. The comparative study shows that our algorithm is an order of magnitude faster than the existing approaches for the same level of accuracy.