## Computing Minimum Length Paths of a Given Homotopy Class (1991)

Venue: | Comput. Geom. Theory Appl |

Citations: | 74 - 7 self |

### BibTeX

@ARTICLE{Hershberger91computingminimum,

author = {John Hershberger and Jack Snoeyink},

title = {Computing Minimum Length Paths of a Given Homotopy Class},

journal = {Comput. Geom. Theory Appl},

year = {1991},

volume = {4},

pages = {331--342}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified linear-time algorithms for shortest path trees, for minimum-link paths in simple polygons, and for paths restricted to c given orientations. 1 Introduction If a wire, a pipe, or a robot must traverse a path among obstacles in the plane, then one might ask what is the best route to take. For the wire, perhaps the shortest distance is best; for the pipe, perhaps the fewest straight-line segments. For the robot, either might be best depending on the relative costs of turning and moving. In this paper, we find shortest paths and shortest closed curve...

### Citations

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Citation Context ...ons ff and fi are homotopic if there is a continuous function \Gamma: X \Theta [0; 1] ! Y such that \Gamma(x; 0) = ff(x) and \Gamma(x; 1) = fi(x). One can see that homotopy is an equivalence relation =-=[3, 39]-=-. In this paper, the range set Y is always a boundary-triangulated 2-manifold M under the subspace topology. We specify the set X in two different ways. First and most importantly, we consider paths j... |

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Citation Context ...orithms. 1.2 Link shortest paths Researchers have also looked at finding minimum paths in simple polygons under the link metric, in which the length of a path is the number of its line segments. Suri =-=[48]-=- developed a linear time algorithm for computing the minimum path between two points in a simple polygon. Ghosh [19] recently gave a linear time algorithm as a consequence of his work on computing the... |

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Citation Context ...ng rectilinear barriers under the Manhattan, or L 1 , metric, researchers have developed algorithms that work in simple polygons (e.g. [46]) and in the presence of obstacles [9, 15, 32]. Mark de Berg =-=[13]-=- has given an algorithm that finds a path that is both a minimum-link and an L 1 shortest path in a simple polygon. He and others [14] give a quadratic algorithm for a combined link and L 1 metric for... |

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