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38
Geometric constraint solving
 Computing in Euclidean Geometry
, 1995
"... We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized. ..."
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Cited by 31 (3 self)
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We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized.
Decomposition plans for geometric constraint systems
 J. Symbolic Computation
, 2001
"... A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past ..."
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Cited by 24 (0 self)
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A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)planning problem as well as several performance measures by which DRplanning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best and worstchoice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DRplanners that use two wellknown types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DRplanning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.
Using Graph Decomposition for Solving Continuous CSPs
 in "Principles and Practice of Constraint Programming, CP’98", LNCS
, 1998
"... In practice, constraint satisfaction problems are often structured. By exploiting this structure, solving algorithms can make important gains in performance. In this paper, we focus on structured continuous CSPs defined by systems of equations. We use graph decomposition techniques to decompose the ..."
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Cited by 18 (7 self)
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In practice, constraint satisfaction problems are often structured. By exploiting this structure, solving algorithms can make important gains in performance. In this paper, we focus on structured continuous CSPs defined by systems of equations. We use graph decomposition techniques to decompose the constraint graph into a directed acyclic graph of small blocks. We present new algorithms to solve decomposed problems which solve the blocks in partial order and perform intelligent backtracking when a block has no solution. For underconstrained problems, the solution space can be explored by choosing some variables as input parameters. However, in this case, the decomposition is no longer unique and some choices lead to decompositions with smaller blocks than others. We present an algorithm for selecting the input parameters that lead to good decompositions. First experimental results indicate that, even on small problems, significant speedups can be obtained using these algorithms.
Choosing Consistent Constraints for Beautification of Reverse Engineered Geometric Models
 ComputerAided Design
, 2004
"... Boundary representation models reconstructed from 3D range data suffer from various inaccuracies caused by noise in the data and the model building software. Such models can be improved in a beautification step, which finds geometric regularities approximately present in the model and imposes a cons ..."
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Cited by 12 (6 self)
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Boundary representation models reconstructed from 3D range data suffer from various inaccuracies caused by noise in the data and the model building software. Such models can be improved in a beautification step, which finds geometric regularities approximately present in the model and imposes a consistent subset of them on the model. Methods to select regularities consistently such that they are likely to represent the original, ideal design intent are presented. Efficiency during selection is achieved by considering degrees of freedom to analyse the solvability of constraint systems representing the regularities (without actually solving them). Priorities are used to select regularities in case of inconsistencies. The selected set of constraints is solved numerically and an improved model is rebuild from the solution. Experiments show that the presented methods can beautify models by selecting consistent regularities and enforcing major intended regularities.
A branchandprune solver for distance constraints
 IEEE Trans. on Robotics
, 2005
"... Abstract—Given some geometric elements such as points and lines in Q, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position an ..."
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Cited by 11 (9 self)
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Abstract—Given some geometric elements such as points and lines in Q, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branchandprune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one. Index Terms—Branchandprune, Cayley–Menger determinant, direct and inverse kinematics, distance constraint, interval method, kinematic and geometric constraint solving, octahedral manipulator. I.
A Constraint Programming Approach for Solving Rigid Geometric Systems
, 2000
"... . This paper introduces a new rigidification methodusing interval constraint programming techniquesto solve geometric constraint systems. Standard rigidification techniques are graphconstructive methods exploiting the degrees of freedom of geometric objects. They work in two steps: a planning p ..."
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Cited by 11 (5 self)
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. This paper introduces a new rigidification methodusing interval constraint programming techniquesto solve geometric constraint systems. Standard rigidification techniques are graphconstructive methods exploiting the degrees of freedom of geometric objects. They work in two steps: a planning phase which identifies rigid clusters, and a solving phase which computes the coordinates of the geometric objects in every cluster. We propose here a new heuristic for the planning algorithm that yields in general small systems of equations. We also show that interval constraint techniques can be used not only to efficiently implement the solving phase, but also generalize former adhoc solving techniques. First experimental results show that this approach is more efficient than systems based on equational decomposition techniques.
Efficient Triangle Counting in Large Graphs via Degreebased Vertex Partitioning
"... The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering t ..."
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Cited by 9 (2 self)
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The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering the hidden thematic structures in the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting approximation algorithm which can be adapted to the semistreaming model [23]. The key idea of our algorithm is to combine the sampling algorithm of [51,52] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [5], treating each set appropriately. From a mathematical perspective, we show a simplified proof of [52] which uses the powerful KimVu concentration inequality [31] based on the HajnalSzemerédi theorem [25]. Furthermore, we improve bounds of existing triple sampling ( techniques based on a theorem of Ahlswede and Katona [3]. We obtain a running time O m + m3/2 log n tɛ2) and an (1 ± ɛ)
Multiloop position analysis via iterative linear programming
 in Proc. of Robotics, Science, and Systems
, 2006
"... Abstract — This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology, ..."
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Cited by 8 (5 self)
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Abstract — This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology, and complete, in the sense that all possible solutions get accurately bounded, irrespectively of whether the analyzed linkage is rigid or mobile. The problem is tackled by formulating a system of linear, parabolic, and hyperbolic equations, which is here solved by a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions at the desired accuracy in short computation times. I.
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 Constraint Solving via Ctree Decomposition
 Proc.ACM SM03
, 2003
"... Abstract. This paper has two parts. First, we propose a method which can be used to decompose a geometric constraint graph into a ctree. With this decomposition, solving for a wellconstrained problem is reduced to the solving for smaller rigids if possible. Second, we give the analytical solutions ..."
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Cited by 7 (3 self)
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Abstract. This paper has two parts. First, we propose a method which can be used to decompose a geometric constraint graph into a ctree. With this decomposition, solving for a wellconstrained problem is reduced to the solving for smaller rigids if possible. Second, we give the analytical solutions to one of the basic merge patterns used to solve a ctree: the 3A3D general Stewart platform, which is to determine the position of a rigid relative to another rigid when we know three angular and three distance constraints between the two rigids.