Many geometric constraint solvers use a combinatorial or graph algorithm to generate a decomposition-recombination (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 algebraic-numeric 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 DR-plan. 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 DR-plan 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.

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 non-constraint based representations. We trace the challenge to a requirement on constraint decomposition algorithms or decomposition-recombination (DR) planners used by most variational constraint solvers, formalize the feature hierarchy incorporation problem for DR-planners, 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.

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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

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.

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.

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 self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly 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.

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 self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly 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

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 1-degree-of-freedom(dof) Henneberg-I 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 1-dof Henneberg-I 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.