Re-parameterization reduces irreducible geometric constraint systems

A formalization of geometric constraint systems and their decomposition

Formal Aspects of Computing ◽

10.1007/s00165-009-0117-8 ◽

2009 ◽

Vol 22 (2) ◽

pp. 129-151 ◽

Cited By ~ 8

Author(s):

Pascal Mathis ◽

Simon E. B. Thierry

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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GPDOF — A FAST ALGORITHM TO DECOMPOSE UNDER-CONSTRAINED GEOMETRIC CONSTRAINT SYSTEMS: APPLICATION TO 3D MODELING

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195906002154 ◽

2006 ◽

Vol 16 (05n06) ◽

pp. 479-511 ◽

Cited By ~ 3

Author(s):

GILLES TROMBETTONI ◽

MARTA WILCZKOWIAK

Keyword(s):

Equation System ◽

3D Models ◽

General Purpose ◽

Constrained Systems ◽

Geometric Constraints ◽

Geometric Constraint ◽

Maximum Matching ◽

Constraint System ◽

Constraint Systems ◽

Geometrical Constraints

Our approach exploits a general-purpose decomposition algorithm, called GPDOF, and a dictionary of very efficient solving procedures, called r-methods, based on theorems of geometry. GPDOF decomposes an equation system into a sequence of small subsystems solved by r-methods, and produces a set of input parameters.1. Recursive assembly methods (decomposition-recombination), maximum matching based algorithms, and other famous propagation schema are not well-suited or cannot be easily extended to tackle geometric constraint systems that are under-constrained. In this paper, we show experimentally that, provided that redundant constraints have been removed from the system, GPDOF can quickly decompose large under-constrained systems of geometrical constraints. We have validated our approach by reconstructing, from images, 3D models of buildings using interactively introduced geometrical constraints. Models satisfying the set of linear, bilinear and quadratic geometric constraints are optimized to fit the image information. Our models contain several hundreds of equations. The constraint system is decomposed in a few seconds, and can then be solved in hundredths of seconds.

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Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems

Computer-Aided Design ◽

10.1016/j.cad.2011.06.018 ◽

2011 ◽

Vol 43 (10) ◽

pp. 1234-1249 ◽

Cited By ~ 6

Author(s):

Simon E.B. Thierry ◽

Pascal Schreck ◽

Dominique Michelucci ◽

Christoph Fünfzig ◽

Jean-David Génevaux

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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Body-and-cad geometric constraint systems

Computational Geometry ◽

10.1016/j.comgeo.2010.06.003 ◽

2012 ◽

Vol 45 (8) ◽

pp. 385-405 ◽

Cited By ~ 8

Author(s):

Kirk Haller ◽

Audrey Lee-St.John ◽

Meera Sitharam ◽

Ileana Streinu ◽

Neil White

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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Using invariance under the similarity group to solve geometric constraint systems

Computer-Aided Design ◽

10.1016/j.cad.2006.01.002 ◽

2006 ◽

Vol 38 (5) ◽

pp. 475-484 ◽

Cited By ~ 7

Author(s):

Pascal Schreck ◽

Étienne Schramm

Keyword(s):

Geometric Constraint ◽

Constraint Systems ◽

Similarity Group

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Solving geometric constraint systems. I. A global propagation approach

Computer-Aided Design ◽

10.1016/s0010-4485(97)00052-3 ◽

1998 ◽

Vol 30 (1) ◽

pp. 47-54 ◽

Cited By ~ 39

Author(s):

Xiao-Shan Gao ◽

Shang-Ching Chou

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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Elimination in generically rigid 3D geometric constraint systems

Algebraic Geometry and Geometric Modeling - Mathematics and Visualization ◽

10.1007/978-3-540-33275-6_13 ◽

2006 ◽

pp. 205-216

Author(s):

Jörg Peters ◽

Meera Sitharam ◽

Yong Zhou ◽

JianHua Fan

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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Variational B-rep model analysis for direct modeling using geometric perturbation

Journal of Computational Design and Engineering ◽

10.1016/j.jcde.2019.03.002 ◽

2019 ◽

Vol 6 (4) ◽

pp. 606-616

Author(s):

Qiang Zou ◽

Hsi-Yung Feng

Keyword(s):

Rigid Body ◽

Boundary Representation ◽

Geometric Constraint ◽

New Method ◽

Geometric Representation ◽

Constraint Systems ◽

Fundamental Difficulty ◽

Direct Modeling ◽

Rigid Body Motions ◽

Constrained Model

Abstract The very recent CAD paradigm of direct modeling gives rise to the need of processing 3D geometric constraint systems defined on boundary representation (B-rep) models. The major issue of processing such variational B-rep models (in the STEP format) is that free motions of a well-constrained model involve more than just rigid-body motions. The fundamental difficulty lies in having a systematic description of what pattern these free motions follow. This paper proposes a geometric perturbation method to study these free motions. This method is a generalization of the witness method, allowing it to directly deal with variational B-rep models represented by the STEP format. This generalization is essentially achieved by using a direct, geometric representation of the free motions, and then expressing the free motions in terms of composites of several basis motions. To demonstrate the effectiveness of the proposed method, a series of comparisons and case studies are presented. Highlights A new method to analyze geometric constraint systems for direct modeling. A generalization of the witness configuration method. A new method to characterize the constraint states of variational B-rep models.

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