On shortest paths in polyhedral spaces
SIAM Journal on Computing
SIAM Journal on Computing
Shortest paths on a polyhedron
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
A Note on the Complexity of Dijkstra's Algorithm for Graphs with Weighted Vertices
IEEE Transactions on Computers
Constructing Approximate Shortest Path Maps in Three Dimensions
SIAM Journal on Computing
Practical methods for approximating shortest paths on a convex polytope in R3
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
SIAM Review
Computing approximate shortest paths on convex polytopes
Proceedings of the sixteenth annual symposium on Computational geometry
Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
Computational Geometry in C
Approximating Shortest Paths on a Nonconvex Polyhedron
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
New results on shortest paths in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Fast exact and approximate geodesics on meshes
ACM SIGGRAPH 2005 Papers
Geodesic Remeshing Using Front Propagation
International Journal of Computer Vision
Efficiently determining a locally exact shortest path on polyhedral surfaces
Computer-Aided Design
Improving Chen and Han's algorithm on the discrete geodesic problem
ACM Transactions on Graphics (TOG)
Isotropic remeshing with fast and exact computation of Restricted Voronoi Diagram
SGP '09 Proceedings of the Symposium on Geometry Processing
Constant-time all-pairs geodesic distance query on triangle meshes
I3D '12 Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
Approximating generalized distance functions on weighted triangulated surfaces with applications
Journal of Computational and Applied Mathematics
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The computation of shortest paths on a polyhedral surface is a common operation in many computer graphics applications. There are two best known exact algorithms for the ''single source, any destination'' shortest path problem. One is proposed by Mitchell et al. (1987) [1]. The other is by Chen and Han (1990) [11]. Recently, Xin and Wang (2009) [9] improved the CH algorithm by exploiting a filtering theorem and achieved a practical method that outperforms both the CH algorithm and the MMP algorithm whether in time or in space. In this paper, we apply the improved CH algorithm to different versions of shortest path problems. The contributions of this paper include: (1) For a surface point p@?@?v"1v"2v"3, we present an unfolding technique for estimating the distance value at p using the distances at v"1,v"2 and v"3. (2) We show that the improved CH algorithm can be naturally extended to the ''multiple sources, any destination'' version. Also, introducing a well-chosen heuristic factor into the improved CH algorithm will induce an exact solution to the ''single source, single destination'' version. (3) At the conclusion of multi-source shortest path algorithms, we can use the distance values at vertices to approximately compute the geodesic-distance-based offsets, the Voronoi diagram and the Delaunay triangulation in O(n) time. (4) By importing a precision parameter @l, we obtain a precision controlled approximant which varies from the improved CH algorithm to Dijkstra's algorithm as @l increases from 0 to 1. Thus, an interesting relationship between them can be naturally established.