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Decomposition of Geometric Constraint Systems: a Survey
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2006
"... Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition tech ..."
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Cited by 18 (6 self)
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Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones. In this paper, we propose a survey of the decomposition techniques for geometric constraint problems a. We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the different approaches, and point out key issues for meeting industrial requirements such as generality and reliability.
Solving minimal, wellconstrained, 3d geometric constraint systems: combinatorial optimization of algebraic complexity
, 2004
"... Many geometric constraint solvers use a combinatorial or graph algorithm to generate a decompositionrecombination (DR) plan. A DR plan recursively decomposes the system of polynomial equations into small, generically rigid subsystems that are more likely to be successfully solved by algebraicnumer ..."
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Cited by 7 (7 self)
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Many geometric constraint solvers use a combinatorial or graph algorithm to generate a decompositionrecombination (DR) plan. A DR plan recursively decomposes the system of polynomial equations into small, generically rigid subsystems that are more likely to be successfully solved by algebraicnumeric solvers. In this paper we show that, especially for 3D geometric constraint systems, a further optimization of the algebraic complexity of these subsystems is both possible, and often necessary to successfully solve the DRplan. To attack this apparently undocumented challenge, we use principles of rigid body manipulation and quaternion forms and combinatorially optimize a function over the minimum spanning trees of a graph generated from DRplan information. This approach follows an interesting connection between the algebraic complexity of the system and the topology of the corresponding constraint graph. The optimization has two secondary advantages: in navigating the solution space of the constraint system and in mapping solution paths in the configuration spaces of the subsystems. We formally compare the reduction in algebraic complexity of the subsystem after optimization with that of the unoptimized subsystem and illustrate the practical benefit with a natural example that could only be solved after optimization.
Geometric constraints within feature hierarchies
 COMPUTERAIDED DESIGN
, 2006
"... We study the problem of enabling general 2D and 3D variational constraint representation to be used in conjunction with a feature hierarchy representation, where some of the features may use procedural or other nonconstraint based representations. We trace the challenge to a requirement on constrai ..."
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Cited by 7 (0 self)
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We study the problem of enabling general 2D and 3D variational constraint representation to be used in conjunction with a feature hierarchy representation, where some of the features may use procedural or other nonconstraint based representations. We trace the challenge to a requirement on constraint decomposition algorithms or decompositionrecombination (DR) planners used by most variational constraint solvers, formalize the feature hierarchy incorporation problem for DRplanners, clarify its relationship to other problems, and provide an efficient algorithmic solution. The new algorithms have been implemented in the general, 2D and 3D opensource geometric constraint solver FRONTIER developed at the University of Florida.
Wellformed Systems of Point Incidences for Resolving Collections of Rigid Bodies
"... For tractability, many modern geometric constraint solvers recursively decompose an input geometric constraint system into standard collections of smaller, generically rigid subsystems or clusters. These are recursively solved and their solutions or realizations are recombined to give the solution ..."
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Cited by 3 (1 self)
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For tractability, many modern geometric constraint solvers recursively decompose an input geometric constraint system into standard collections of smaller, generically rigid subsystems or clusters. These are recursively solved and their solutions or realizations are recombined to give the solution or realization of the input constraint system. The recombination of a standard collection of solved clusters typically reduces to positioning and orienting the rigid realizations of the clusters with respect to each other, subject to incidence constraints representing primitive, shared objects between the clusters and other external constraints relating objects in different clusters. Even for generically wellconstrained systems in 3D, and even when the shared objects are restricted to be points, finding a system of incidence constraints that extends to a wellconstrained system for recombining a cluster decomposition is a significant hurdle faced by geometric constraint solvers. In general, we would like a wellformed system of incidences that generically preserves the classification of the original, undecomposed system as a well, under or overconstrained system. Here we motivate, formally state and give an efficient, greedy algorithm to find such a wellformed system for a general constraint system, when the shared objects in the cluster decomposition are restricted to be points. Our solution relies on isolating an interesting new matroid structure underlying collections of rigid clusters with shared point objects.
Nucleationfree 3D rigidity
"... All known examples of generic 3D barandjoint frameworks where the distance between a nonedge pair is implied by the edges in the graph contain a rigid vertexinduced subgraph. In this paper we present a class of arbitrarily large graphs with no nontrivial vertexinduced rigid subgraphs, which hav ..."
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Cited by 1 (1 self)
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All known examples of generic 3D barandjoint frameworks where the distance between a nonedge pair is implied by the edges in the graph contain a rigid vertexinduced subgraph. In this paper we present a class of arbitrarily large graphs with no nontrivial vertexinduced rigid subgraphs, which have implied distances between pairs of vertices not joined by edges. As a consequence, we obtain (a) the first class of counterexamples to a potential combinatorial characterization of 3D generic independence and rigidity proposed by Sitharam and Zhou [5] and (b) the first example of a 3D rigidity circuit which has no nontrivial rigid induced subgraphs. 1
Modeling Virus SelfAssembly Pathways Using Computational Algebra and Geometry
 APPLICATIONS OF COMPUTER ALGEBRA (ACA2004)
, 2004
"... We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are d ..."
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We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. The model uses static geometric constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their
COUNTING AND ENUMERATION OF SELFASSEMBLY PATHWAYS FOR SYMMETRIC MACROMOLECULAR STRUCTURES
"... We consider the problem of explicitly enumerating and counting the assembly pathways by which an icosahedral viral shell forms from identical constituent protein monomers. This poorly understood assembly process is a remarkable example of symmetric macromolecular selfassembly occuring in nature and ..."
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We consider the problem of explicitly enumerating and counting the assembly pathways by which an icosahedral viral shell forms from identical constituent protein monomers. This poorly understood assembly process is a remarkable example of symmetric macromolecular selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. We use the new model of���that employs a static geometric constraint graph to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The model was developed to answer focused questions about the structural properties of the most probable types of successful assembly pathways. Specifically, the model reduces the study of pathway types and their probabilities to the study of the orbits of the automorphism group of the underlying geometric constraint graph, acting on the set of pathways. Since these are highly symmetric polyhedral graphs, it seems a viable approach to explicitly enumerate these orbits and count their sizes. The contribution of this paper is to isolate and simplify the core combinatorial questions, list related work and indicate the advantages of an explicit enumerative approach. 1.
THE INFLUENCE OF SYMMETRY ON THE PROBABILITY OF ASSEMBLY PATHWAYS FOR ICOSAHEDRAL VIRAL SHELLS
"... This paper motivates and sets up the mathematical framework for a new program of investigation: to isolate and clarify the precise influence of symmetry on the probability space of assembly pathways that successfully lead to icosahedral viral shells. Several tractable open questions are posed. Besid ..."
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This paper motivates and sets up the mathematical framework for a new program of investigation: to isolate and clarify the precise influence of symmetry on the probability space of assembly pathways that successfully lead to icosahedral viral shells. Several tractable open questions are posed. Besides its virology motivation, the topic is of independent mathematical interest for studying constructions of symmetric polyhedra. Preliminary results are presented: a natural, structural classification of subsets of facets of T = 1 polyhedra, based on their stabilizing subgroups of the icosahedral group; and a theorem that uses symmetry to formalize why increasing depth increases the numeracy (and hence probability) of an assembly pathway type (or symmetry class) for a T = 1 viral shell. 1.
–The Virus Assembly Model: Pathways and Effort
"... We develop a model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are desirable whi ..."
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We develop a model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. The model uses static geometric and tensegrity constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their properties are carefully defined and computed using computational algebra and geometry, specifically stateofart concepts in geometric constraint decomposition. The model is analyzable and refinable and avoids expensive dynamics. We show that it has a provably tractable and accurate computational simulation and that its predictions are roughly consistent with known information about viral shell assembly. Justifications for mathematical and biochemical assumptions are provided, and comparisons are drawn with other virus assembly models. A method for more conclusive experimental validation involving specific viruses is sketched. Overall the paper indicates a strong and direct, mutually beneficial interplay between (a) the concepts underlying macromolecular assembly; and (b) a wide variety of established as well as novel concepts from combinatorial and computational algebra, geometry and algebraic complexity.
c © World Scientific Publishing Company Decomposition of Geometric Constraint Systems: a Survey
, 2006
"... Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition tech ..."
Abstract
 Add to MetaCart
Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones. In this paper, we propose a survey of the decomposition techniques for geometric constraint problemsa. We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the different approaches, and point out key issues for meeting industrial requirements such as generality and reliability.