### Abstract

An open question in Exact Geometric Computation is whether there are transcendental computations that can be made "geometrically exact". Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p, q in the plane, where the obstacles are a collection of n discs. This problem can be solved in O(n ^{2} log n) time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first show the Turing-computability of this problem, provided the radii of the discs are rationally related. We make the usual assumption that the numerical input data are real algebraic numbers. By appealing to effective bounds from transcendental number theory, we further show a single-exponential time upper bound when the input numbers are rational. Our result appears to be the first example of a non-algebraic combinatorial problem which is shown computable. It is also a rare example of transcendental number theory yielding positive computational results.

Original language | English (US) |
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Title of host publication | Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05 |

Pages | 116-125 |

Number of pages | 10 |

State | Published - 2005 |

Event | 21st Annual Symposium on Computational Geometry, SCG'05 - Pisa, Italy Duration: Jun 6 2005 → Jun 8 2005 |

### Other

Other | 21st Annual Symposium on Computational Geometry, SCG'05 |
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Country | Italy |

City | Pisa |

Period | 6/6/05 → 6/8/05 |

### Keywords

- Disc obstacles
- Exact geometric computation
- Exponential complexity
- Guaranteed precision computation
- Real RAM model
- Robust numerical algorithms
- Shortest path

### ASJC Scopus subject areas

- Software
- Geometry and Topology
- Safety, Risk, Reliability and Quality
- Chemical Health and Safety

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## Cite this

*Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05*(pp. 116-125)