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

Venue: | Computer-Aided Design |

Citations: | 29 - 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

1019 | Surface simplification using quadric error metrics
- Garland, Heckbert
- 1997
(Show Context)
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... |

838 | A volumetric Method for Building Complex Models from Range Images
- Curless, Levoy
- 1996
(Show Context)
Citation Context ...ces with a large number of faces (called dense) can now be generated more easily due to the recent development of range scanners, GPS, and so on. For example, surface reconstruction from range images =-=[3]-=- and the use of such dense polyhedral surfaces have been active research topics for computer graphics applications. Geographic information systems (GIS) usually use triangular irregular networks (TINs... |

771 |
Data Structures and Algorithms
- Aho, Hopcroft, et al.
- 1983
(Show Context)
Citation Context ...r each graph by our algorithm STEP 1 by using Dijkstra’s algorithm. We used a simple method for this problem based on a priority queue of partialorder trees that can be processed with O(n log n)-time =-=[1]-=-. As this method evaluates the length of each edge, its processing time can be decided by the number of edges in G 0 and thus by γ. To implement our algorithm, we prepared a simple graph structure for... |

243 | D.: Maps: Multiresolution adaptive parameterization of surfaces
- LEE, SWELDENS, et al.
- 1998
(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... |

224 | Fitting Smooth Surfaces to Dense Polygon Meshes. Computer Graphics
- Krishnamurthy, Levoy
(Show Context)
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... |

160 | Geometric shortest paths and network optimization
- Mitchell
- 2000
(Show Context)
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... |

158 | The discrete geodesic problem
- Mitchell, Mount, et al.
- 1987
(Show Context)
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... |

143 | Interactive texture mapping
- MAILLOT, YAHIA, et al.
- 1993
(Show Context)
Citation Context ...ify a region of dense polyhedral surfaces, 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 inte... |

101 |
On shortest paths in polyhedral spaces
- Sharir, Schorr
- 1986
(Show Context)
Citation Context ...locally optimal sub-path joining two consecutive obstacle vertices can be unfolded at each edge along its edge sequence, thus obtaining a straight segment. This property was used by Sharir and Schorr =-=[19]-=- who first proposed an O(n 3 log n)-time algorithm (n is the number of edges) to find the exact shortest path on a convex surface. Mitchell et al. [17] have obtained an O(n 2 log n)time algorithm for ... |

84 |
Optimal shortest path queries in a simple polygon
- Guibas, Hershberger
- 1987
(Show Context)
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... |

73 |
Shortest paths on a polyhedron
- Chen, Han
- 1996
(Show Context)
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... |

50 | Approximating weighted shortest paths on polyhedral surfaces
- Lanthier, Maheshwari, et al.
- 1997
(Show Context)
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 ... |

49 | Undirected single source shortest path in linear time
- Thorup
- 1997
(Show Context)
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... |

41 | Metamorphosis of arbitrary triangular meshes
- Kanai, Suzuki, et al.
(Show Context)
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 ... |

35 | A new algorithm for computing shortest paths in weighted planar subdivisions
- Mata, Mitchell
- 1997
(Show Context)
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 ... |

27 | Approximate shortest-path and geodesic diameter on convex polytopes
- Har-Peled
- 1999
(Show Context)
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 ... |

24 | Interactive mesh fusion based on local 3D metamorphosis
- Kanai, Suzuki, et al.
- 1999
(Show Context)
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 ... |

20 | Approximating shortest paths on a nonconvex polyhedron
- Varadarajan, Agarwal
- 1997
(Show Context)
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... |

13 | Interactive multiresolution editing of arbitrary meshes
- Lee
- 1999
(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... |

5 | Discrete parametrization for deforming arbitrary meshes
- Kuriyama, Kaneko
- 1999
(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... |