Finding shortest gentle paths on polyhedral terrains by the method of multiple shooting

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Abstract

The problem of finding shortest θ-gentle paths can be stated as follows: given two points p,q on a polyhedral terrain and a slope parameter θ(0,π/2), the objective is to find a path joining p and q on the terrain which is shortest such that the slope of the path does not exceed θ. In this paper, we introduce some geometric and analysis properties of such paths and answer the question of whether known results of classical shortest paths hold for shortest θ-gentle paths. An algorithm for approximately computing such shortest θ-gentle paths on terrains is presented, where an approximate shortest θ-gentle path joining two points is a θ-gentle path whose length is the infimum of a sequence of that of θ-gentle paths in which they are decreasing. We also show that the sequence of lengths of paths obtained by the proposed algorithm is convergent. The algorithm is implemented in C++ using CGAL and Open GL in some specific circumstances.

Introduction

A variant of the shortest path problem is the shortest θ-gentle path problem (SGP problem for briefly): given a polyhedral terrain T, a source point p, a destination point q on T, and a slope value θ(0,π/2), find a shortest θ-gentle path joining p and q on T, in the sense that it is shortest such that its slope does not exceed θ. This problem can be seen as a generalization of the single source shortest path whose goal is to find classical shortest paths joining two points on a polyhedral terrain (θ=π/2), which is a well-studied problem in computational geometry and robotics. Applications of the slope-constrained shortest path problem appear in some fields. For example, when we design mobile robots for traversing on non-planar surfaces or when we ski down a mountain and avoid a too steep path, paths that are too steep should be replaced by “zig-zag lines” satisfying the slope constraint. In train transport, the railroads in Linz, Austria with a slope of 11.6%, and the one in Tram 28 in Lisbon, Portugal which has a 14.5% grade are considered to be the steepest railroads in the world.

When θ=π/2, the SGP problem becomes the single source shortest path problem. Several works [1], [2], [3] presented algorithms for solving the problem in this case. Unfortunately, when 0<θ<π/2, known methods for solving the single source shortest path problem such as using the star unfolding technique of Agarwal et al. [1], the modified Chen and Han’s sequence tree [2], [4], and the sequence of edges that the shortest path goes through [3] does not work. Some special cases of the SGP problem have been investigated by Ahmed, Lubiw, and Maheshwari in [5], [6]. Amed and Lubiw also showed that the problem of minimizing the total number of bends (of shortest θ-gentle paths) is NP-hard, and no polynomial time solution is known [5]. Thus, in the general case, the SGP problem is NP-hard.

To date, there are several approaches for finding such shortest θ-gentle paths. Nöllenburg and Sautter [7] presented an approximation approach based on determining the norm for finding shortest θ-gentle paths on a sequence of adjacent triangles. An algorithm proposed by Ahmed, Lubiw, Maheshwari [6] models the problem as a graph whose nodes are Steiner points added along the edges of the terrain. After discretizing, the required path is found on the whole terrain and therefore they take much memory of computers when the size of the problem is large. These two algorithms have not been implemented, and thus it is not clear how practical they are. Liu and Wong [8] proposed an algorithm to solve approximately the SGP problem using a technique of simplifying terrains and some graph tools. They concentrated on reducing the size of terrains rather than solving the SGP problem. To overcome this issue, we will present a new approach, namely the method of multiple shooting (MMS for short), for computing shortest θ-gentle paths on sequences of adjacent triangles of sub-terrains with a fewer number of triangles and implement it on computers therefore until now, there are only two methods (i.e., Liu and Wong’s method and our method of multiple shooting) having their implementations on computers. The method was proposed for solving the geometric shortest path problems in 2D and 3D [9], [10], [11]. We then use successfully MMS to deal with approximately the SGP problem but under the assumption of the connection given in Section 4.1 for terrains.

According to [6], [7], [8], solutions to the SGP problem are computed based on deducing the subproblem of finding a shortest θ-gentle path joining two given points along a sequence of adjacent triangles. Not many properties of such a path have been shown. A natural question is “whether the properties of classical shortest paths hold for shortest θ-gentle paths?”. Although the authors in [6], [8] proved that a shortest θ-gentle path is a polyline and not unique if it exists, the existence of such a path has been not stated.

In the paper, we answer the question of the existence of shortest θ-gentle paths joining two points along a sequence of adjacent triangles (Proposition 2). We show that shortest θ-gentle paths can go through convex vertices of polyhedral terrains (Example 1). The characterization of unreachable vertices by a θ-cone is presented (Proposition 1). An iterative algorithm based on MMS is given, and we prove that the sequence of lengths of paths obtained by the proposed algorithm is convergent (Proposition 5). Furthermore, the path obtained after some iterative steps of the algorithm is an approximate shortest θ-gentle path if the number of iterative steps is large enough (Theorem 1), where the notion of an approximate shortest θ-gentle path joining two points is presented in Definition 5.

The rest of the paper is organized as follows. Section 2 recalls preliminary notions. Section 3 presents some properties of shortest θ-gentle paths joining two points. Section 4 introduces an iterative algorithm using MMS for the SGP problem. The algorithm is implemented in C++ using CGAL and numerical results are given and visualized to describe how our method works in Section 5. We used MMS to compare with Liu and Wong’s algorithm [8] for solving the SGP problem. The lengths of final θ-gentle paths obtained by our algorithm are similar to that get by Liu and Wong’s one, while the running time of Liu and Wong’s algorithm is thousands of times higher than the proposed algorithm. Proofs of the correctness of the proposed algorithm are arranged in Appendix.

Section snippets

Preliminaries

We recall some definitions and properties. For any points p,q in R3, we denote [p,q]{(1λ)p+λq:0λ1},(p,q){(1λ)p+λq:0<λ<1}.

A polyhedral terrain or simply a terrain, denoted by T, is a polyhedral surface in R3 in which every vertical line intersects the surface at most once. This means that the projections of all faces of T on the xy-plane are pairwise non-overlap. For a point pT, let denote the triple x(p),y(p),z(p) its coordinates in R3. Then z(p) is also called the height of p. T is said

Some properties of shortest θ-gentle paths

In this section, we give some geometric and analysis properties of shortest gentle paths: characterization of a θ-unreachable vertex, the existence of shortest θ gentle paths along a sequence of adjacent triangles, and an example of shortest θ-gentle paths which can visit a convex vertex of a polyhedral terrain.

To characterize a θ-unreachable vertex, we introduce the concept of a θ-cone. Take aR3, a θ-cone whose vertex is a is constructed as follows: let Δ be the line passing through a and

Finding approximate θ-gentle paths on polyhedral terrains

To deal with the SGP problem, we use the method of multiple shooting applied for shortest path problems in 2D and 3D [9], [10], [11]. The iterative algorithm based on MMS gives a sequence of θ-gentle paths whose lengths are descending and convergent. Moreover, when the number of iterative steps is large enough, the path obtained is an approximate shortest θ-gentle path in the sense of Definition 5, which is similar to the definition of an approximate shortest path in [11].

According to

Numerical examples

In the section, we implement the proposed algorithm in C++ code using CGAL then compile and run the code on Ubuntu Linux platform Intel Core i5-7200U, CPU 568 2.50 GHz with 4 GB RAM. Our experiments are executed on convex polyhedral terrains to ensure the assumption of the connectedness of cutting slices (see Section 4.1). We run the algorithms several times to compute the lengths of obtained paths and corresponding running times. Average results are then reported. The results are visualized by

Concluding remarks

Generally, the problem of finding shortest θ-gentle path is quite hard to solve, and no polynomial time algorithm is known. Moreover, not many properties of shortest θ-gentle paths have been shown. In this paper, we give some geometric and analysis properties of shortest θ-gentle paths: characterization of a θ-unreachable vertex, the existence of shortest θ gentle paths along a sequence of adjacent triangles, and an example of shortest θ-gentle paths which can visit a convex vertex of a

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first author wishes to express his thanks to Dr. Johannes P. Schlöder for the useful discussions and his gracious hospitality during his visits in 2017–2018 to the Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, where a part of this paper was written. This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number T2022 - 20 - 01. The first and the third authors acknowledge Ho Chi Minh City University of Technology

Dr. Phan Thanh An is now an associate professor in Applied Mathematics at Faculty of Applied Science of the Ho Chi Minh City University of Technology (HCMUT), Vietnam. He also serves as director of the Institute of Mathematical and Computational Sciences – IMACS of HCMUT. He completed his Ph.D. in the field of optimization at the Vinh University in Vietnam in 1999 under the supervision of Professors Hoang Xuan Phu and Nguyen Nhuy. Then he had worked as a researcher at the Hanoi Institute of

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Dr. Phan Thanh An is now an associate professor in Applied Mathematics at Faculty of Applied Science of the Ho Chi Minh City University of Technology (HCMUT), Vietnam. He also serves as director of the Institute of Mathematical and Computational Sciences – IMACS of HCMUT. He completed his Ph.D. in the field of optimization at the Vinh University in Vietnam in 1999 under the supervision of Professors Hoang Xuan Phu and Nguyen Nhuy. Then he had worked as a researcher at the Hanoi Institute of Mathematics, Vietnam (as an associate professor of the institute since 2009), and has specialized in the use of optimization methods (the method of orienting curves and the method of multiple shooting) for solving computational geometry problems. During the years 2009–2014 he worked at the Center for Computational and Stochastic Mathematics (CEMAT) of the Instituto Superior Técnico, the University of Lisbon, Portugal and supervised successfully two Ph.D. theses for this university in the research trend “Optimization approaches for computational geometry”. After staying in the Institute of Mathematics and Computer Sciences (ICMC), University of São Paulo, Brasil in 3 years as a visiting professor in Computer Science from 2017 to 2019, he has moved to HCMUT. He regularly visits the Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Germany to collaborate with colleagues from the group “Simulation and Optimization” of the IWR on numerical optimization. He has also visited the Freie Universität Berlin, Germany since 2014 to collaborate with the Mathematical Geometry Processing working group on discrete geodesics. To date, Dr. An has supervised successfully 5 Ph.D. theses. He has authored or co-authored more than 40 scientific publications and two books. His research works and their applications can be found at his research website: https://sites.google.com/a/math.ac.vn/phanthanh-an-s-homepage/.

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