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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.7 1.5 2 2 Figure 1: ( left ) Wellconstrained System; middle left: underconstrained system; middle right: overconstrained system; ( right ) consistently overconstrained system 1 Rigidity Geometric constraint systems have been studied in the context of variational constraint solving in CAD for nearly 2 decades. [65, 40, 64, 50, 34, 36, 39, 48, 13, 60, 31, 5, 49, 61, 17, 62, 63] [6, 11, 28, 29, 20, 21, 1] [23, 26, 27, 25] [30, 19, 25, 14, 15, 43, 3, 4] [44, 7, 42, 47, 2, 45, 46] [18, 37, 41, 52]. For recent reviews of the extensive literature on geometric constraint solving more elaborate descriptions and examples for the definitions below, see, e.g, [23, 33, 12, 53]. 1.1 Definitions A geometric constraint system consists of a finite set of primitive geometric objects such as points, lines, planes, conics etc. and a finite set of geometric constraints between them such as distance, angle, incidence etc. The constraints can usually be written as algebraic equations and inequalities whose variables are the coordinates of the participating geometric objects. For example, a distance constraint of d between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) in 2D is written as ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 = d 2 . In this case the distance d is the parameter associated with the constraint. Most of the constraint solvers so far deal with 2D constraint systems. With the exception of work [22, 24, 26, 27], [25], [18, 38, 37, 41, 51, 58, 56, 57], related to the FRONTIER geometric constraint solver [52], to the best of our knowledge, work on stand-alone 3D geometric constraint solvers is relatively sparse [7, 42]. A solution or realization of a geometric constraint system is the (set of) real zero(es) of the corresponding algebraic system. In other words, the solution is a class of valid instantiations of (the position, orientation and any other parameters of) the geometric elements such that all constraints are satisfied. Here, it is understood that such a solution is in a particular geometry, for example the Euclidean plane, the sphere, or Euclidean 3 di- mensional space. A constraint system can be classified as overconstrained , wellconstrained , or underconstrained . Well-constrained systems have a finite, albeit potentially very large number of rigid solutions; i.e., solutions that cannot be infinitesimally flexed to give another nearby solution: the solution space (modulo rigid body transfor- mations such as rotations and translations) consists of isolated points - it is zero-dimensional. Underconstrained systems have infinitely many solutions; their solution space is not zero-dimensional. Overconstrained systems do not have a solution unless they are consistently overconstrained . In that case, they could be embedded within overall underconstrained systems, see Figure 1. Systems that are not underconstrained are called rigid systems. A geometric constraint graph G = ( V, E, w ) corresponding to geometric constraint system is a weighted graph with vertex set (representing geometric objects) V and edge set (representing constraints) E ; w ( v ) is the weight of vertex v and w ( e ) is the weight of edge e , corresponding to the number of degrees of freedom available

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