## Abstract

Geometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments. An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations. Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities. If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints. Otherwise, a diagnostic of inconsistency is expected. The three main approaches used for this problem are numerical, procedural or operational and mathematical. Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one. The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations. The common roots to this set of polynomials characterizes the solution space for such a problem. That work presents the use of Grobner basis techniques for verifying the consistency of the constraints. It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations. Although theoretically sound, these techniques require large amounts of computing resources. This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities. The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem. Cluster identification can be related to diagnostic of identifying short cycles in the Spatial Constraint graph for the GCS/SF problem. Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles. Besides improving the efficiency of the solution approach, the Divideand-Conquer techniques capture the physical essence of the problem. This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanisms.

Original language | English (US) |
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Title of host publication | 11th Biennial Conference on Reliability, Stress Analysis, and Failure Prevention; 7th International Conference on Design Theory and Methodology; JSME Symposium on Design and Production; Mechanical Design Education and History; Computer-Integrated Concurrent Design Conference |

Publisher | American Society of Mechanical Engineers (ASME) |

Pages | 939-952 |

Number of pages | 14 |

ISBN (Electronic) | 9780791817179 |

DOIs | |

State | Published - 1995 |

Event | ASME 1995 Design Engineering Technical Conferences, DETC 1995, collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium - Boston, United States Duration: Sep 17 1995 → Sep 20 1995 |

### Publication series

Name | Proceedings of the ASME Design Engineering Technical Conference |
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Volume | 2 |

### Conference

Conference | ASME 1995 Design Engineering Technical Conferences, DETC 1995, collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium |
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Country/Territory | United States |

City | Boston |

Period | 9/17/95 → 9/20/95 |

## ASJC Scopus subject areas

- Mechanical Engineering
- Computer Graphics and Computer-Aided Design
- Computer Science Applications
- Modeling and Simulation