Geometric constraints are at the heart of CAD/CAM applications and also arise in many geometric m... more Geometric constraints are at the heart of CAD/CAM applications and also arise in many geometric modeling contexts such as virtual reality, robotics, molecular modeling, teaching geometry, etc. Informally, a geometric constraint problem consists of a finite set of geometric objects and a finite set of constraints between them. The geometric objects are drawn from a fixed set of types such as points, lines, circles and conics in the plane, or points, lines, planes, cylinders and spheres in 3 dimensions. The constraints are spatial and include logical constraints such as incidence, tangency, perpendicularity and metric constraints such as distance, angle, radius. The spatial constraints can usually be written as algebraic equations whose variables are the coordinates of the participating geometric objects. A solution of a geometric constraint problem is a real zero of the corresponding algebraic system. Currently there is a lack of effective spatial variational constraint solvers and assembly constraint solvers that scale to large problem sizes and can be used interactively by the designer as conceptual tools throughout the design process. The requirement is a constraint solver that uses geometric domain knowledge to develop a plan for decomposing the constraint system into small subsystems, whose solutions can be recombined by solving other small subsystems. The primary aim of this decomposition plan is to restrict the use of direct algebraic/numeric solvers to subsystems that are as small as possible. Hence the optimal or most efficient decomposition plan would minimize the size of the largest such subsystem. Any geometric constraint solver should first solve the problem of efficiently finding a close-to-optimal decomposition-recombination (DR) plan, because that dictates the usability of the solver. In this thesis we state this problem of finding a close-to-optimal solution as a problem that deals with weighted graphs and also identify several important subproblems. One class of such subproblem involves finding dense subgraphs—graphs such that sum of weights of its edges is greater than sum of weights of its vertices. Dense graphs that present interest for finding a DR-plan are (a) minimum (smallest possible dense graphs), (b) minimal (not containing any other dense subgraphs), (c) maximum (largest dense ones), (d) maximal (not contained in any other dense subgraph). This thesis presents polynomial time algorithms for problems (b), (c) and (d). Problem (a) is shown to be NP-complete, and various approximation algorithms are suggested, as well as explicit solutions for special cases that arise from CAD/CAM applications.
We define and study exact, efficient representations of realization spaces Euclidean Distance Con... more We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS). These are graphs with distance assignments on the edges (frameworks) or graphs with distance interval assignments on the edges. Each representation corresponds to a choice of non-edges or Cayley parameters. The set of realizable distance assignments to the chosen parameters yields a parametrized configuration space. We initialize a systematic and graded program of combinatorially characterizing graphs with configuration spaces of different geometry and algebraic complexity. Our notion of efficiency is based on the convexity and connectedness of the configuration space, as well as algebraic complexity of sampling realizations, i.e., sampling the configuration space and obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture the class of graphs that always admit efficient configuration spaces and the possible choices of representation parameters that yield efficient configuration spaces for a given graph. We completely characterize EDCS that have connected, convex and efficient configuration spaces, based on precise and formal measures of efficiency. It should be noted that our results do not rely on genericity of the EDCS. Some of our proofs employ an unusual interplay of classical analytic and algebraic results related to positive semi-definiteness of Euclidean distance matrices, and Cayley-Menger conditions, with recent forbidden minor characterizations and algorithms related to realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a strategy that we anticipate will be useful in other situations. We study 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. We prove an algebraic theorem that makes combinatorial classification meaningful. We give the graph characterization according to the classification. We prove our results are tight and our definitions are robust. Our results have immediate CAD applications. We give preliminary results and conjectures for two natural extensions: which 2D EDCS have configuration space with at most two connected components and which 3D EDCS have connected configuration space. Finally, we discuss two application problems: characterizing configuration space of packing Zeolite and Helix. We give a surprising configuration space description theorem for Zeolite problem. We show that our novel simulation of Helix packing via geometric constraint solving provides quality and efficiency guarantees that other methods do not.
Geometric constraints are at the heart of CAD/CAM applications and also arise in many geometric m... more Geometric constraints are at the heart of CAD/CAM applications and also arise in many geometric modeling contexts such as virtual reality, robotics, molecular modeling, teaching geometry, etc. Informally, a geometric constraint problem consists of a finite set of geometric objects and a finite set of constraints between them. The geometric objects are drawn from a fixed set of types such as points, lines, circles and conics in the plane, or points, lines, planes, cylinders and spheres in 3 dimensions. The constraints are spatial and include logical constraints such as incidence, tangency, perpendicularity and metric constraints such as distance, angle, radius. The spatial constraints can usually be written as algebraic equations whose variables are the coordinates of the participating geometric objects. A solution of a geometric constraint problem is a real zero of the corresponding algebraic system. Currently there is a lack of effective spatial variational constraint solvers and assembly constraint solvers that scale to large problem sizes and can be used interactively by the designer as conceptual tools throughout the design process. The requirement is a constraint solver that uses geometric domain knowledge to develop a plan for decomposing the constraint system into small subsystems, whose solutions can be recombined by solving other small subsystems. The primary aim of this decomposition plan is to restrict the use of direct algebraic/numeric solvers to subsystems that are as small as possible. Hence the optimal or most efficient decomposition plan would minimize the size of the largest such subsystem. Any geometric constraint solver should first solve the problem of efficiently finding a close-to-optimal decomposition-recombination (DR) plan, because that dictates the usability of the solver. In this thesis we state this problem of finding a close-to-optimal solution as a problem that deals with weighted graphs and also identify several important subproblems. One class of such subproblem involves finding dense subgraphs—graphs such that sum of weights of its edges is greater than sum of weights of its vertices. Dense graphs that present interest for finding a DR-plan are (a) minimum (smallest possible dense graphs), (b) minimal (not containing any other dense subgraphs), (c) maximum (largest dense ones), (d) maximal (not contained in any other dense subgraph). This thesis presents polynomial time algorithms for problems (b), (c) and (d). Problem (a) is shown to be NP-complete, and various approximation algorithms are suggested, as well as explicit solutions for special cases that arise from CAD/CAM applications.
We define and study exact, efficient representations of realization spaces Euclidean Distance Con... more We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS). These are graphs with distance assignments on the edges (frameworks) or graphs with distance interval assignments on the edges. Each representation corresponds to a choice of non-edges or Cayley parameters. The set of realizable distance assignments to the chosen parameters yields a parametrized configuration space. We initialize a systematic and graded program of combinatorially characterizing graphs with configuration spaces of different geometry and algebraic complexity. Our notion of efficiency is based on the convexity and connectedness of the configuration space, as well as algebraic complexity of sampling realizations, i.e., sampling the configuration space and obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture the class of graphs that always admit efficient configuration spaces and the possible choices of representation parameters that yield efficient configuration spaces for a given graph. We completely characterize EDCS that have connected, convex and efficient configuration spaces, based on precise and formal measures of efficiency. It should be noted that our results do not rely on genericity of the EDCS. Some of our proofs employ an unusual interplay of classical analytic and algebraic results related to positive semi-definiteness of Euclidean distance matrices, and Cayley-Menger conditions, with recent forbidden minor characterizations and algorithms related to realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a strategy that we anticipate will be useful in other situations. We study 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. We prove an algebraic theorem that makes combinatorial classification meaningful. We give the graph characterization according to the classification. We prove our results are tight and our definitions are robust. Our results have immediate CAD applications. We give preliminary results and conjectures for two natural extensions: which 2D EDCS have configuration space with at most two connected components and which 3D EDCS have connected configuration space. Finally, we discuss two application problems: characterizing configuration space of packing Zeolite and Helix. We give a surprising configuration space description theorem for Zeolite problem. We show that our novel simulation of Helix packing via geometric constraint solving provides quality and efficiency guarantees that other methods do not.
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Papers by Meera Sitharam