### Abstract

In geometric constraint solving, the constraints are represented with an equation system F(U, X) = 0, where X denotes the unknowns and U denotes a set of parameters. The target solution for X is noted X_{T}. A witness is a couple (U_{W}, X_{W}) such that F(U_{W}, X_{W}) = 0. The witness is not the target solution, but they share the same combinatorial features, even when the witness and the target lie on two distinct connected components of the solution set of F(U, X) = 0. Thus a witness enables the qualitative study of the system: the detection of over- and under-constrained systems, the decomposition into irreducible subsystems, the computation of subsystems boundaries. This paper investigates the witness computation in various configurations. The witness computation will be studied under several numerical methods: Newton iterations from random seeds either in R and C, the Broyden-Fletcher-Goldfarb-Shanno method, the Nelder-Mead simplex method. The robustness and performances of these methods will be analyzed and compared. The paper also presents the numerical probabilistic method from which the witness method was originated, and shows how the witness can be used for detecting dependent parameters within systems of geometric constraints.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of 2014 Science and Information Conference, SAI 2014 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 759-770 |

Number of pages | 12 |

ISBN (Electronic) | 9780989319317 |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 Science and Information Conference, SAI 2014 - London, United Kingdom Duration: Aug 27 2014 → Aug 29 2014 |

### Other

Other | 2014 Science and Information Conference, SAI 2014 |
---|---|

Country | United Kingdom |

City | London |

Period | 8/27/14 → 8/29/14 |

### Fingerprint

### Keywords

- Geometric constraint solving
- Numerical algorithms
- Witness computation

### ASJC Scopus subject areas

- Information Systems

### Cite this

*Proceedings of 2014 Science and Information Conference, SAI 2014*(pp. 759-770). [6918272] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/SAI.2014.6918272

**Witness computation for solving geometric constraint systems.** / Kubicki, Arnaud; Michelucci, Dominique; Foufou, Sebti.

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

*Proceedings of 2014 Science and Information Conference, SAI 2014.*, 6918272, Institute of Electrical and Electronics Engineers Inc., pp. 759-770, 2014 Science and Information Conference, SAI 2014, London, United Kingdom, 8/27/14. https://doi.org/10.1109/SAI.2014.6918272

}

TY - GEN

T1 - Witness computation for solving geometric constraint systems

AU - Kubicki, Arnaud

AU - Michelucci, Dominique

AU - Foufou, Sebti

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In geometric constraint solving, the constraints are represented with an equation system F(U, X) = 0, where X denotes the unknowns and U denotes a set of parameters. The target solution for X is noted XT. A witness is a couple (UW, XW) such that F(UW, XW) = 0. The witness is not the target solution, but they share the same combinatorial features, even when the witness and the target lie on two distinct connected components of the solution set of F(U, X) = 0. Thus a witness enables the qualitative study of the system: the detection of over- and under-constrained systems, the decomposition into irreducible subsystems, the computation of subsystems boundaries. This paper investigates the witness computation in various configurations. The witness computation will be studied under several numerical methods: Newton iterations from random seeds either in R and C, the Broyden-Fletcher-Goldfarb-Shanno method, the Nelder-Mead simplex method. The robustness and performances of these methods will be analyzed and compared. The paper also presents the numerical probabilistic method from which the witness method was originated, and shows how the witness can be used for detecting dependent parameters within systems of geometric constraints.

AB - In geometric constraint solving, the constraints are represented with an equation system F(U, X) = 0, where X denotes the unknowns and U denotes a set of parameters. The target solution for X is noted XT. A witness is a couple (UW, XW) such that F(UW, XW) = 0. The witness is not the target solution, but they share the same combinatorial features, even when the witness and the target lie on two distinct connected components of the solution set of F(U, X) = 0. Thus a witness enables the qualitative study of the system: the detection of over- and under-constrained systems, the decomposition into irreducible subsystems, the computation of subsystems boundaries. This paper investigates the witness computation in various configurations. The witness computation will be studied under several numerical methods: Newton iterations from random seeds either in R and C, the Broyden-Fletcher-Goldfarb-Shanno method, the Nelder-Mead simplex method. The robustness and performances of these methods will be analyzed and compared. The paper also presents the numerical probabilistic method from which the witness method was originated, and shows how the witness can be used for detecting dependent parameters within systems of geometric constraints.

KW - Geometric constraint solving

KW - Numerical algorithms

KW - Witness computation

UR - http://www.scopus.com/inward/record.url?scp=84909606204&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84909606204&partnerID=8YFLogxK

U2 - 10.1109/SAI.2014.6918272

DO - 10.1109/SAI.2014.6918272

M3 - Conference contribution

SP - 759

EP - 770

BT - Proceedings of 2014 Science and Information Conference, SAI 2014

PB - Institute of Electrical and Electronics Engineers Inc.

ER -