### 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 |

Pages | 197-206 |

Number of pages | 10 |

Volume | 2006 |

Publication status | Published - 2006 Jul 20 |

Externally published | Yes |

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

### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling*(Vol. 2006, pp. 197-206)

**A higher dimensional formulation for robust and interactive distance queries.** / Seong, Jun Kyung; Johnson, David E.; Cohen, Elaine.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling.*vol. 2006, pp. 197-206, SPM 2006 - ACM Symposium on Solid and Physical Modeling, Wales, United Kingdom, 05/6/6.

}

TY - GEN

T1 - A higher dimensional formulation for robust and interactive distance queries

AU - Seong, Jun Kyung

AU - Johnson, David E.

AU - Cohen, Elaine

PY - 2006/7/20

Y1 - 2006/7/20

N2 - 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.

AB - 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.

KW - Dimensionality lifting

KW - Minimum distance

KW - Problem reduction scheme

KW - Spline models

UR - http://www.scopus.com/inward/record.url?scp=33745968379&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745968379&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33745968379

SN - 1595933581

SN - 9781595933584

VL - 2006

SP - 197

EP - 206

BT - Proceedings SPM 2006 - ACM Symposium on Solid and Physical Modeling

ER -