## Approximate shortest path on a polyhedral surface and its applications (2000)

Venue: | Computer-Aided Design |

Citations: | 28 - 1 self |

### BibTeX

@INPROCEEDINGS{Kanai00approximateshortest,

author = {Takashi Kanai},

title = {Approximate shortest path on a polyhedral surface and its applications},

booktitle = {Computer-Aided Design},

year = {2000},

pages = {241--250}

}

### Years of Citing Articles

### OpenURL

### Abstract

A new algorithm is proposed for calculating the approximate shortest path on a polyhedral surface. The method mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of a polyhedron. Although the algorithm is an approximation, it has the significant advantages of being fast, easy to implement, high approximation accuracy, and numerically robust. The approximation accuracy and computation time are compared between this approximation algorithm and the extended Chen & Han (ECH) algorithm that can calculate the exact shortest path for non-convex polyhedra. The approximation algorithm can calculate shortest paths within 0.4 % accuracy to roughly 100-1000 times faster than the ECH algorithm in our examples. Two applications are discussed of the approximation algorithm to geometric modeling.

### Citations

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Citation Context ...d computation time. We used the simplified model of a “bunny” shown in Figure 7. To simplify the original model, we used the quadric error metric (QEM) based approach proposed by Garland and Heckbert =-=[4]-=-. It was difficult to use the original model because our naive implementation of the ECH algorithm needed quite a large amount of space to store the shortest path information. Figure 7(b) shows a simp... |

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Citation Context ... are needed for modeling, rendering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface [9], local deformation of a polyhedral surface =-=[12, 10, 13]-=- and 3D morphing [7, 8]. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of local regions. It is desirable for the boundaries of thes... |

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Citation Context ..., especially those generated from range images, are needed for modeling, rendering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface =-=[9]-=-, local deformation of a polyhedral surface [12, 10, 13] and 3D morphing [7, 8]. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of l... |

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Citation Context ...hms for finding the shortest path on 2D polygons and 3D surfaces, and in 3D spaces. However, the algorithms for finding the exact shortest path on a polyhedral surface (including the non-convex case) =-=[17, 2]-=- usually involve high time and space costs. It is therefore not practical to apply these algorithms to a dense polyhedral surface in most cases. Instead, we have focused on finding the approximate sho... |

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Citation Context ...exact. 2. Shortest Path Problem on a Polyhedral Surface A survey of the shortest path problem concerning a two or higher dimensional geometric object (a surface, space, network, etc.) can be found in =-=[16]-=-. We mainly discuss here about finding the shortest path between two points on a polyhedral surface. An important property of the shortest path on a polyhedron is its local optimality called unfolding... |

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Citation Context ...fter picking up the faces under the calculated shortest path of G (called the sleeve), the O(log k) algorithm (k is the number of edges passed by the shortest path) proposed by Guibas and Hershberger =-=[5]-=- is used to refine the graph. However, [5] uses a rather special data structure called the hourglass, and its algorithm is difficult to implement. Mata and Mitchell [15] have proposed another approach... |

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Citation Context ...hms for finding the shortest path on 2D polygons and 3D surfaces, and in 3D spaces. However, the algorithms for finding the exact shortest path on a polyhedral surface (including the non-convex case) =-=[17, 2]-=- usually involve high time and space costs. It is therefore not practical to apply these algorithms to a dense polyhedral surface in most cases. Instead, we have focused on finding the approximate sho... |

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Citation Context ...t. The computation time largely depends on a Dijkstra’s algorithm. Our implementation executes in O(n log n)-time. Thorup has recently proposed an O(n)-time method for processing Dijkstra’s algorithm =-=[20]-=-. We need more consideration about this method. • It provides high approximation accuracy. In our examples, an approximation accuracy within 0.4% was established. • It is numerically robust. Our appro... |

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Citation Context ...volve high time and space costs. It is therefore not practical to apply these algorithms to a dense polyhedral surface in most cases. Instead, we have focused on finding the approximate shortest path =-=[11, 15]-=-. The algorithm proposed in this paper for a polyhedral surface (possibly including the non-convex case) mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of ... |

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Citation Context ...dering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface [9], local deformation of a polyhedral surface [12, 10, 13] and 3D morphing =-=[7, 8]-=-. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of local regions. It is desirable for the boundaries of these regions to be smooth ... |

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Citation Context ...volve high time and space costs. It is therefore not practical to apply these algorithms to a dense polyhedral surface in most cases. Instead, we have focused on finding the approximate shortest path =-=[11, 15]-=-. The algorithm proposed in this paper for a polyhedral surface (possibly including the non-convex case) mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of ... |

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Citation Context ...s evaluation, smaller ɛ results in better accuracy. Algorithms theoretically exist that are fast and require less space if limited to convex polyhedra. For example, the approach proposed by Har-Peled =-=[6]-=- is based on the construction of a tight bounding volume covering a polyhedron. By performing O(n) pre-processing, the O((log n)/ɛ 3/2 + 1/ɛ 3 )-time computation of a (1 + ɛ)-approx. shortest path is ... |

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Citation Context ...dering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface [9], local deformation of a polyhedral surface [12, 10, 13] and 3D morphing =-=[7, 8]-=-. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of local regions. It is desirable for the boundaries of these regions to be smooth ... |

21 | Approximating shortest paths on a nonconvex polyhedron, in Proc. 38th Annu
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Citation Context ...polyhedron. By performing O(n) pre-processing, the O((log n)/ɛ 3/2 + 1/ɛ 3 )-time computation of a (1 + ɛ)-approx. shortest path is possible. In the case of general polyhedra, Varadarajan and Agarwal =-=[21]-=- have proposed algorithms that compute a 13-approx. (ɛ =12) path in O(n 5/3 log 5/3 n)-time or a 15approx. (ɛ =14) path in O(n 8/5 log 8/5 n)-time. These are based on partitioning the surface into O(n... |

12 | Interactive multiresolution editing of arbitrary meshes
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Citation Context ... are needed for modeling, rendering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface [9], local deformation of a polyhedral surface =-=[12, 10, 13]-=- and 3D morphing [7, 8]. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of local regions. It is desirable for the boundaries of thes... |

4 | T.: Discrete parametrization for deforming arbitrary meshes
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(Show Context)
Citation Context ... are needed for modeling, rendering and animation. Such examples are the addition of attributes such as texture [14], conversion to a parametric surface [9], local deformation of a polyhedral surface =-=[12, 10, 13]-=- and 3D morphing [7, 8]. We discuss here about the use of our approximation algorithm as a tool for interactively specifying the boundaries of local regions. It is desirable for the boundaries of thes... |