Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Two-point Euclidean shortest path queries in the plane
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Ununfoldable polyhedra with convex faces
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Improving Chen and Han's algorithm on the discrete geodesic problem
ACM Transactions on Graphics (TOG)
Shortest Path Problems on a Polyhedral Surface
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Shortest path queries between geometric objects on surfaces
ICCSA'07 Proceedings of the 2007 international conference on Computational science and its applications - Volume Part I
The geodesic diameter of polygonal domains
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Approximate shortest path queries on weighted polyhedral surfaces
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Short-Cuts on star, source and planar unfoldings
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Constant-time all-pairs geodesic distance query on triangle meshes
I3D '12 Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
Querying two boundary points for shortest paths in a polygonal domain
Computational Geometry: Theory and Applications
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We introduce the notion of a star unfolding of the surface ${\cal P}$ of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on ${\cal P}$.The first algorithm computes the edge sequences traversed by shortest paths on ${\cal P}$ in time $O(n^6 \beta (n) \log n)$, where $\beta (n)$ is an extremely slowly growing function. A much simpler $O(n^6)$ time algorithm that finds a small superset of all such edge sequences is also sketched.The second algorithm is an $O(n^{8}\log n)$ time procedure for computing the geodesic diameter of ${\cal P}$: the maximum possible separation of two points on ${\cal P}$ with the distance measured along ${\cal P}$. Finally, we describe an algorithm that preprocesses ${\cal P}$ into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter $1 \le m \le n^2$, it can preprocess ${\cal P}$ in time $O(n^6 m^{1+\delta})$, for any $\delta 0$, into a data structure of size $O(n^6m^{1+\delta})$, so that a query can be answered in time $O((\sqrt{n}/m^{1/4}) \log n)$. If one query point always lies on an edge of ${\cal P}$, the algorithm can be improved to use $O(n^5 m^{1+\delta})$ preprocessing time and storage and guarantee $O((n/m)^{1/3} \log n)$ query time for any choice of $m$ between 1 and $n$.