A survey of geodesic paths on 3D surfaces

Finding shortest paths and shortest distances between points on a surface S in three-dimensional space is a well-studied problem in differential geometry and computational geometry. The shortest path between two points on S is denoted a geodesic path on the surface and the shortest distance between two points on S is denoted a geodesic distance. In this survey, we consider the case where a discrete surface representation of S is given. Namely, S is represented as a polyhedron P in three-dimensional space. Since discrete surfaces cannot be differentiated, methods from differential geometry to compute geodesic paths and distances cannot be applied in this case. However, algorithms from differential geometry can be discretized and extended. Furthermore, the discrete surface can be viewed as a graph in three-dimensional space. Therefore, methods from graph theory and computational geometry have been applied to find geodesic paths and distances on polyhedral surfaces. The general problem of computing a shortest path between polyhedral obstacles in 3D is shown to be NP hard by Canny and Reif using reduction from 3SAT [10]. Computing a geodesic path on a polyhedral surface is an easier problem and it is solvable in polynomial time. Computing geodesic paths and distances on polyhedral surfaces is applied in various areas such as robotics, geographic information systems (GIS), circuit design, and computer graphics. For example, geodesic path problems can be applied to finding the most efficient path a robotic arm can trace without hitting obstacles, analyzing water flow, studying traffic control, texture mapping and morphing, and face recognition. A survey related to geodesic paths in twoand higher-dimensional spaces can be found in the Handbook of Computational Geometry [31]. Note that the geodesic distance between any two points on P can be easily determined if the geodesic path is known by measuring the (weighted) length of the geodesic path. Hence, we will only consider the problem of computing geodesic paths on P . Problems on finding geodesic paths and distances depending on the number of source and destination points have been studied. The three most commonly studied problems are (a) finding the geodesic path from one source vertex s ∈ P to one destination vertex d ∈ P , (b) finding the geodesic paths from one source vertex s ∈ P to all destination vertices in P , or equivalently, finding the geodesic paths from all source vertices in P to one destination vertex d ∈ P , known as single source shortest path (SSSP) problem, and (c) finding the geodesic paths between all pairs of vertices in P , known as all-pairs shortest path (APSP) problem.