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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 3 (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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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.
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.
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.
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