Abstract
The structural rigidity property, a generalisation of Laman’s theorem which characterises generically rigid bar frameworks in 2D, is generally considered a good heuristic to detect rigidities in geometric constraint satisfaction problems (GCSPs). In fact, the gap between rigidity and structural rigidity is significant and essentially resides in the fact that structural rigidity does not take geometric properties into account. In this article, we propose a thorough analysis of this gap. This results in a new characterisation of rigidity, the extended structural rigidity, based on a new geometric concept: the degree of rigidity (DOR). We present an algorithm for computing the DOR of a GCSP, and we prove some properties linked to this geometric concept. We also show that the extended structural rigidity is strictly superior to the structural rigidity and can thus be used advantageously in the algorithms designed to tackle the major issues related to rigidity.
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References
Angeles, J.: Spatial Kinematic Chains. Springer, Berlin (1982)
Bouma, W., Fudos, I., Hoffmann, C.M., Cai, J., Paige, R.: Geometric constraint solver. Computer Aided Design 27(6), 487–501 (1995)
Chou, S.C.: Mechanical Theorem Proving. Reidel Publishing Co., Dordrechtz (1988)
Dufourd, J.-F., Mathis, P., Schreck, P.: Formal resolution of geometrical constraint systems by assembling. In: Hoffmann, C., Bronswort, W. (eds.) Proc. Fourth Symposium on Solid Modeling and Applications, pp. 271–284 (1997)
Fudos, I., Hoffmann, C.M.: Correctness proof of a geometric constraint solver. Technical Report TR-CSD-93-076, Purdue University, West Lafayette, Indiana (1993)
Grbler, M.: Getriebelehre. Springer, Berlin (1917)
Graver, J.: Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures. Dolciani Mathematical Expositions, vol. 25. Mathematical Association of America (2002)
Hendrickson, B.: Conditions for unique realizations. SIAM j Computing 21(1), 65–84 (1992)
Hoffmann, C.M., Lomonosov, A., Sitharam, M.: Finding solvable subsets of constraint graphs. In: Smolka, G. (ed.) CP 1997. C.M. Hoffmann, A. Lomonosov, and M. Sitharam, vol. 1330, pp. 463–477. Springer, Heidelberg (1997)
Hoffmann, C.M., Lomonosov, A., Sitharam, M.: Decomposition plans for geometric constraint systems. In: Proc. J. Symbolic Computation 2000 (2000)
Jermann, C.: Rsolution de contraintes gomtriques par rigidification rcursive et propagation d’intervalles. Thse de doctorat en informatique, Universit de Nice Sophia-Antipolis (2002)
Jermann, C., Neveu, B., Trombettoni, G.: Algorithms for identifying rigid subsystems in geometric constraint systems. In: 18th International Joint Conference in Artificial Intelligence, IJCAI 2003 (2003)
Jermann, C., Trombettoni, G., Neveu, B., Rueher, M.: A constraint programming approach for solving rigid geometric systems. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 233–248. Springer, Heidelberg (2000)
Kramer, G.: Solving Geometric Constraint Systems. MIT Press, Cambridge (1992)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
Latham, R.S., Middleditch, A.E.: Connectivity analysis: A tool for processing geometric constraints. Computer Aided Design 28(11), 917–928 (1996)
Sitharam, M.: Personal communication on the minimal dense algorithm. University of Florida at Gainesville (2000)
Wang, D.: Geother 1.1: Handling and proving geometric theorems automatically. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 194–215. Springer, Heidelberg (2004)
Whiteley, W.: Applications of the geometry of rigid structures. In: Crapo, H. (ed.) Computer Aided Geometric Reasoning, pp. 219–254. INRIA (1987)
Wu, W.: Basic principles of mechanical theorem proving in elementary geometries. J. Automated Reasoning 2, 221–254 (1986)
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Jermann, C., Neveu, B., Trombettoni, G. (2004). A New Structural Rigidity for Geometric Constraint Systems. In: Winkler, F. (eds) Automated Deduction in Geometry. ADG 2002. Lecture Notes in Computer Science(), vol 2930. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24616-9_6
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DOI: https://doi.org/10.1007/978-3-540-24616-9_6
Publisher Name: Springer, Berlin, Heidelberg
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