### Abstract

We present an efficient and robust algorithm for computing the minimum distance between a point and freeform curve or surface by lifting the problem into a higher dimension. This higher dimensional formulation solves for all query points in the domain simultaneously, therefore providing opportunities to speed computation by applying coherency techniques. In this framework, minimum distance between a point and planar curve is solved using a single polynomial equation in three variables (two variables for a position of the point and one for the curve). This formulation yields two-manifold surfaces as a zero-set in a 3D parameter space. Given a particular query point, the solution space's remaining degrees-of-freedom are fixed and we can numerically compute the minimum distance in a very efficient way. We further recast the problem of analyzing the topological structure of the solution space to that of solving two polynomial equations in three variables. This topological information provides an elegant way to efficiently find a global minimum distance solution for spatially coherent queries. Additionally, we extend this approach to a 3D case. We formulate the problem for the surface case using two polynomial equations in five variables. The effectiveness of our approach is demonstrated with several experimental results.

Original language | English |
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Title of host publication | Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling |

Publisher | Association for Computing Machinery |

Pages | 197-206 |

Number of pages | 10 |

ISBN (Print) | 1595933581, 9781595933584 |

DOIs | |

Publication status | Published - 2006 |

Event | SPM 2006 - ACM Symposium on Solid and Physical Modeling - Wales, United Kingdom Duration: 2005 Jun 6 → 2005 Jun 8 |

### Publication series

Name | Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling |
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Volume | 2006 |

### Other

Other | SPM 2006 - ACM Symposium on Solid and Physical Modeling |
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Country | United Kingdom |

City | Wales |

Period | 05/6/6 → 05/6/8 |

### Keywords

- Dimensionality lifting
- Minimum distance
- Problem reduction scheme
- Spline models

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

*Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling*(pp. 197-206). (Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling; Vol. 2006). Association for Computing Machinery. https://doi.org/10.1145/1128888.1128916