Geodesic Paths On 3D Surfaces: Survey and Open Problems Anil Maheshwari Stefanie Wuhrer This survey gives a brief overview of theoreti- SAT [10]. Computing a geodesic path on a polyhe- cally and practically relevant algorithms to compute dral surface is an easier problem and it is solvable in geodesic paths and distances on three-dimensional polynomial time. surfaces. The survey focuses on polyhedral three- Computing geodesic paths and distances on poly- dimensional surfaces. hedral surfaces is applied in various areas such as robotics, geographic information systems (GIS), cir- cuit design, and computer graphics. For example, 1 Introduction geodesic path problems can be applied to finding the most efficient path a robotic arm can trace without Finding shortest paths and shortest distances be- hitting obstacles, analyzing water flow, studying traf- tween points on a surface S in three-dimensional fic control, texture mapping and morphing, and face space is a well-studied problem in differential ge- recognition. A survey related to geodesic paths in ometry and computational geometry. The shortest two- and higher-dimensional spaces can be found in path between two points on S is denoted a geodesic the Handbook of Computational Geometry [31]. path on the surface and the shortest distance between Note that the geodesic distance between any two two points on S is denoted a geodesic distance. In points on P can be easily determined if the geodesic this survey, we consider the case where a discrete path is known by measuring the (weighted) length of surface representation of S is given. Namely, S is the geodesic path. Hence, we will only consider the represented as a polyhedron P in three-dimensional problem of computing geodesic paths on P . space. Since discrete surfaces cannot be differen- Problems on finding geodesic paths and distances tiated, methods from differential geometry to com- depending on the number of source and destination pute geodesic paths and distances cannot be applied arXiv:0904.2550v1 [cs.CG] 16 Apr 2009 points have been studied. The three most commonly in this case. However, algorithms from differential studied problems are (a) finding the geodesic path geometry can be discretized and extended. Further- from one source vertex s ∈ P to one destination ver- more, the discrete surface can be viewed as a graph in tex d ∈ P , (b) finding the geodesic paths from one three-dimensional space. Therefore, methods from source vertex s ∈ P to all destination vertices in P , graph theory and computational geometry have been or equivalently, finding the geodesic paths from all applied to find geodesic paths and distances on poly- source vertices in P to one destination vertex d ∈ P , hedral surfaces. known as single source shortest path (SSSP) prob- The general problem of computing a shortest path lem, and (c) finding the geodesic paths between all between polyhedral obstacles in 3D is shown to be pairs of vertices in P , known as all-pairs shortest NP hard by Canny and Reif using reduction from 3- path (APSP) problem. 1 The algorithms reviewed in this survey are com- graph theoretic algorithms. To obtain a good under- pared by means of the following five categories: standing of the reviewed algorithms, we first review • Accuracy of the computed geodesic path. some well-known graph theoretic algorithms. Dijkstra proposed an algorithm to solve the SSSP • Cost metric used to compute the geodesic path. problem on a directed weighted graph G(V, E) with The cost metric can be the Euclidean distance n vertices, m edges, and positive weights [13]. Dijk- or a weight function (for example when going stra’s algorithm proceeds by building a list of pro- uphill is more costly than going downhill). cessed vertices for which the shortest path to the • Space complexity of the algorithm. source point s is known. The algorithm iteratively decreases estimates on the shortest paths of non- • Time complexity of the algorithm. processed vertices, which are stored in a priority • Applicability of the algorithm by surveying if queue. In each iteration of the algorithm, the clos- the algorithm has been implemented and tested est unprocessed vertex from the source is extracted in practice. from the priority queue and processed by relaxing all its incident edges. The notion of relaxation under- Approximation algorithms are compared accord- lines the analogy between the length of the shortest ing to their approximation ratio (or approximation path and the length of an extended tension spring. factor) k. An algorithm that finds approximations to When the algorithm starts, the length of the short- a geodesic path with approximation ratio k returns est path is overestimated and can be compared to a path of length at most k times the exact geodesic an extended spring. In each iteration, a shorter path. path is found, which can be compared to relaxing To solve the problem of computing geodesic the spring. Although the original implementation paths on discrete surfaces, two different general ap- requires O(n2) time, the running time can be de- proaches can be used. First, the polyhedral surface creased to O(n log n+m) by using Fibonacci heaps. can be viewed as a graph and algorithms to com- Thorup [45] presented an O(m)-time algorithm in pute shortest paths on graphs can be extended to case where each edge is assigned a positive integer find geodesic paths on polyhedral surfaces. Algo- weight. The main idea is to use a hierarchical buck- rithms following this approach are reviewed in Sec- eting structure to avoid the bottleneck caused by sort- tion 2. Second, the polyhedral surface can be viewed ing the vertices in increasing order from s. as a discretized differentiable surface and algorithms The length of a path on S depends on the em- from differential geometry can be extended to find ployed cost metric. Hence, the shortest or geodesic geodesic paths on polyhedral surfaces. Algorithms path on S depends on this cost metric. In Section 2.1, following this approach are reviewed in Section 3. geodesic path algorithms with Euclidean cost metric At the end of each section, open problems related to are reviewed. Using the Euclidean cost metric im- the section are summarized. plies that the Euclidean length of the path is used to measure the length of the path. In Section 2.2, 2 Graph-Based Algorithms geodesic path algorithms on weighted surfaces are reviewed. Using a weighted cost metric implies This section reviews algorithms to compute geodesic that different faces of S can be weighted differently. shortest paths that can be viewed as extensions of Clearly, any algorithm that can solve a shortest path 2 problem using a weighted cost metric can also solve est path can be computed as the straight line joining the same problem using the Euclidean cost metric. s and p after unfolding the faces adjacent to the edge sequence to a plane. The authors aim to subdivide P 2.1 Euclidean Cost Metric with respect to a given source point s, such that the shortest path from s to any other point in P can be When using the Euclidean distance along a poly- found efficiently. They define ridge points x of P as hedron P as cost metric, shortest paths consist of points that have the property that there exists more straight line segments that cross faces of the polyhe- than one shortest path from s to x and prove that dron. An approach to compute shortest paths on P the ridge points can be represented by O(n2) straight aims to compute a superset of all the possible edges line segments. The algorithm partitions the boundary of shortest paths on P and to use this information to of P into at most n connected regions called peels compute shortest paths. Since all of the algorithms not containing any vertices or ridge points of P . The pursuing this strategy are mainly of theoretical in- boundaries of peels contain only ridge points, ver- terest to establish bounds on the number of possible tices, and edges of P . The algorithm to construct edges of shortest paths on P , they are not discussed the peels is similar to Dijkstra’s graph search algo- in detail in this survey. The algorithm’s pursuing this rithm. The peels are then iteratively unfolded to the approach are less efficient than the ones surveyed. plane. The algorithm first preprocesses P by con- A good overview of the algorithms finding edge se- structing the peels with respect to s in O(n3 log n) quences is given by Lanthier [25, p. 30–35]. time. The algorithm stores the computed peels in a We first review algorithms that only operate on the tree called slice tree that can then be used to deter- surface of a convex polyhedron. Second, we review mine the shortest path between an arbitrary point on algorithms that operate on the surface of any (convex P and s in O(n) time. The slice tree data structure or non-convex) polyhedron. uses O(n2) space. Mount [34] improves the algorithm by Sharir and Convex Polyhedra Schorr both in terms of time and space complexity. The main observation by Mount is that the peels de- This section discusses algorithms that operate on the fined by Sharir and Schorr can be viewed as Voronoi surface of a convex polyhedron P with n vertices. regions of a point set R containing the planar un- Shortest paths according to the Euclidean cost metric foldings of the source point s. Note that R contains are considered. at most n points per face of P because there are at Sharir and Schorr [42] proposed an algorithm that most n peels intersecting a face of P .