### Abstract

Given a set of three circles in a plane, we want to find a circumcircle to these given circles called generators. This problem is well known as Apollonius Tenth Problem and is often encountered in geometric computations for CAD systems. This problem is also a core part of an algorithm to compute the Voronoi diagram of circles. We show that the problem can be reduced to a simple point-location problem among the regions bounded by two lines and two transformed circles. The transformed circles are produced from the generators via linear fractional transformations in a complex space. Then, some of the lines tangent to these transformed circles corresponds to the desired circumcircle to the generators. The presented algorithm is very simple yet fast. In addition, several degenerate cases are all incorporated into one single general framework.

Original language | English |
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Title of host publication | Computational Science - ICCS 2001 - International Conference, 2001, Proceedings |

Editors | Vassil N. Alexandrov, Jack J. Dongarra, Benjoe A. Juliano, René S. Renner, C.J. Kenneth Tan |

Publisher | Springer Verlag |

Pages | 728-737 |

Number of pages | 10 |

ISBN (Print) | 3540422323, 9783540422327 |

DOIs | |

State | Published - 2001 Jan 1 |

Event | International Conference on Computational Science, ICCS 2001 - San Francisco, United States Duration: 2001 May 28 → 2001 May 30 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2073 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | International Conference on Computational Science, ICCS 2001 |
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Country | United States |

City | San Francisco |

Period | 01/05/28 → 01/05/30 |

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## Cite this

*Computational Science - ICCS 2001 - International Conference, 2001, Proceedings*(pp. 728-737). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2073). Springer Verlag. https://doi.org/10.1007/3-540-45545-0_84