# Computing approximately shortest descending paths on convex terrains via multiple shooting

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## Abstract

Given a polyhedral terrain and two points *p*, *q* on the terrain, a path from *p* to *q* on the terrain is descending if *z*-coordinate of a point *v* never increases while we move *v* along the path from *p* to *q*. The problem of finding shortest descending paths on polyhedral terrains was posed by de Berg and van Kreveld (Algorithmica 18:306–323, 1997). In this paper, the multiple shooting approach proposed by Hoai et al. (J Comp Appl Math 317:235–246, 2017) is applied for approximately computing shortest descending paths on convex polyhedral terrains. Three factors of the approach consisting of surface partition, straightness condition, and update of shooting points are presented. We also show that if the straightness condition is satisfied then a local shortest descending path is obtained. Proposed algorithm is implemented in C++. Numerical results indicate that once a local solution is obtained it is close to a global one.

## Keywords

Approximate shortest descending path Convex terrain Multiple shooting approach Straightest geodesic## Mathematics Subject Classification

68W25 68U05 52B55## Notes

### Acknowledgements

This research is funded mainly by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 102.01-2017.09. In addition, the financial support offered by TWAS Research Grants Programme 16-544 \(RG/MATHS/AS_G\) in Basic Sciences and Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.01-2017.321 is acknowledged. The authors also thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Simulation and Optimization Group, Interdisciplinary Center for Scientific Computing (IWR) University of Heidelberg, where a part of the paper was written. The authors would like to thank Dr. Johannes P. Schlöder, Prof. Nguyen Ngoc Hai for their invaluable comments and the anonymous reviewers and the editor for their helpful and constructive comments that greatly contributed to improving the final version of the paper.

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