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12
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.
COMBINATORIAL DECOMPOSITION, GENERIC INDEPENDENCE AND ALGEBRAIC COMPLEXITY OF GEOMETRIC CONSTRAINTS SYSTEMS: APPLICATIONS IN BIOLOGY AND ENGINEERING
, 2006
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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.
Combinatorial Classification of 2D Underconstrained Systems (Abstract)
, 2005
"... Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D [18, 4]. Significant progress has also been made in understanding ge ..."
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Cited by 2 (2 self)
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Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be important and crucial for
Characterizing 1Dof HennebergI graphs with efficient configuration spaces
, 810
"... We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance Constraint Systems(EDCS, Linkages, Frameworks) based on 1degreeoffreedom(dof) HennebergI graphs. Each representation corresponds to a choice of parameters and y ..."
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Cited by 1 (1 self)
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We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance Constraint Systems(EDCS, Linkages, Frameworks) based on 1degreeoffreedom(dof) HennebergI graphs. Each representation corresponds to a choice of parameters and yields a different parametrized configuration space. Our notion of efficiency is based on the algebraic complexities of sampling the configuration space and of obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely combinatorial characterizations that capture (i) the class of graphs that have efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. Our results automatically yield an efficient algorithm for sampling realizations, without missing extreme or boundary realizations. In addition, our results formally show that our definition of efficient configuration space is robust and that our characterizations are tight. We choose the class of 1dof HennebergI graphs in order to take the next step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. In particular, the results presented here are the first characterizations that go beyond graphs that have connected and convex configuration spaces.
Characterizing graphs with convex and connected configuration spaces. arXiv:0809.3935 [cs.CG
"... spaces ..."
–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.
Solution space navigation for geometric constraint systems
"... Abstract. We study the welldocumented problem of systematically navigating the potentially exponentially many roots or realizations of wellconstrained, variational geometric constraint systems. We give a scalable method called the ESM or Equation and Solution Manager that can be used both for auto ..."
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Abstract. We study the welldocumented problem of systematically navigating the potentially exponentially many roots or realizations of wellconstrained, variational geometric constraint systems. We give a scalable method called the ESM or Equation and Solution Manager that can be used both for automatic searches and visual, userdriven searches for desired realizations. The method incrementally assembles the desired solution of the entire system and avoids combinatorial explosion, by offering the user a visual walkthrough of the solutions to recursively constructed subsystems and by permitting the user to make gradual, adaptive solution choices. We isolate requirements on companion methods that are essential and desirable for efficient, meaningful solution space navigation. Specifically, they permit (a) incorporation of many existing approaches to solution space steering or navigation into the ESM; and (b) integration of the ESM into a standard geometric constraint solver architecture. We address the latter challenge and explain how the integration is achieved. Additionally, we sketch the ESM implementation as part of an opensource, 2D and 3D geometric constraint solver FRONTIER developed at the University of Florida. Keywords. Root selection for geometric constraint systems. Wellconstrained systems. Underconstrained and Overconstrained systems. Constraint graphs. Cyclical and 3D geometric constraint systems. Variational geometric constraint solving. Decomposition of geometric constraint systems. Degree of Freedom analysis. Conceptual design. Featurebased and assembly modeling.
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 ..."
Abstract
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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
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.