Abstract
Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.
For the group \(\mathcal{C}_{3}\) which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46:762–773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with \(\mathcal{C}_{3}\) symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints.
In addition, we establish symmetric versions of Henneberg’s theorem and Crapo’s theorem for \(\mathcal{C}_{3}\) which provide alternate characterizations of “generically” isostatic graphs with \(\mathcal{C}_{3}\) symmetry.
As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups \(\mathcal{C}_{2}\) and \(\mathcal{C}_{s}\) , in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups \(\mathcal{C}_{2}\) and \(\mathcal{C}_{s}\) which are generated by a half-turn and a reflection in the plane, respectively.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Altmann, S.L., Herzig, P.: Point-Group Theory Tables. Clarendon, Oxford (1994)
Asimov, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978)
Bishop, D.M.: Group Theory and Chemistry. Clarendon, Oxford (1973)
Connelly, R., Fowler, P.W., Guest, S.D., Schulze, B., Whiteley, W.J.: When is a symmetric pin-jointed framework isostatic? Int. J. Solids Struct. 46, 762–773 (2009)
Crapo, H.: On the generic rigidity of plane frameworks. Inst. Nat. Rech. en Informatique et Automatique (INRIA), No. 1278 (1990)
Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37, 1793–1804 (2000)
Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Geometric Topology, Proc. Conf., Park City, Utah, 1974. Lecture Notes in Mathematics, vol. 438, pp. 225–239. Springer, Berlin (1975)
Graver, J.: Counting on Frameworks. Mathematical Association of America, Washington (2001)
Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics. AMS, Providence (1993)
Guest, S.D., Schulze, B., Whiteley, W.: When is a symmetric body-bar structure isostatic? Int. J. Solids Struct. (2009, submitted)
Hall, L.H.: Group Theory and Symmetry in Chemistry. McGraw-Hill, New York (1969)
Hendrickson, B., Jacobs, D.: An algorithm for two-dimensional rigidity percolation: the pebble game. J. Comput. Phys. 137, 346–365 (1997)
Henneberg, L.: Die Graphische Statik der Starren Systeme. Leipzig (1911). Johnson reprint (1968)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
Lomeli, L.C., Moshe, L., Whiteley, W.: Bases and circuits for 2-rigidity: constructions via tree partitions. Tech. report, http://www.math.yorku.ca/Who/Faculty/Whiteley/menu.html
Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods 3, 91–98 (1982)
Owen, J.C., Power, S.C.: Frameworks, symmetry and rigidity. Preprint (2009)
Schulze, B.: Block-diagonalized rigidity matrices of symmetric frameworks and applications. Contributions to Algebra and Geometry (2009, submitted). arXiv:0906.3377
Schulze, B.: Combinatorial and geometric rigidity with symmetry constraints. Ph.D. thesis, York University, Toronto, Canada (2009)
Schulze, B.: Injective and non-injective realizations with symmetry. Contributions to Discrete Math. (2009, to appear). arXiv:0808.1761
Schulze, B.: Symmetrized Laman theorems for the groups \(\mathcal{C}_{2}\) and \(\mathcal {C}_{s}\) (2009, in preparation)
Schulze, B., Watson, A., Whiteley, W.: Symmetry, flatness, and necessary conditions for independence (2009, in preparation)
Servatius, B., Shai, O., Whiteley, W.: Combinatorial characterization of the assur graphs from engineering. Eur. J. Comb. (2008, to appear). arXiv:0801.2525
Servatius, B., Shai, O., Whiteley, W.: Geometric properties of assur graphs. Eur. J. Comb. (2008, to appear). arXiv:0801.4113v1
Tay, T.-S.: Rigidity of multigraphs I: linking rigid bodies in n-space. J. Comb. Theory, Ser. B 26, 95–112 (1984)
Tay, T.-S.: A new proof of Laman’s theorem. Graphs Comb. 9, 365–370 (1993)
Tay, T.-S., Whiteley, W.: Generating isostatic frameworks. Topol. Struct. 11, 21–69 (1985)
Tay, T.-S., Whiteley, W.: Recent advances in generic rigidity of structures. Topol. Struct. 9, 31–38 (1985)
White, N., Whiteley, W.: The algebraic geometry of bar and body frameworks. SIAM J. Algebr. Discrete Methods 8, 1–32 (1987)
Whiteley, W.: A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4, 75–95 (1989)
Whiteley, W.: Some matroids from discrete applied geometry. In: Matroid Theory. Contemporary Mathematics, vol. 197, pp. 171–311. AMS, Providence (1996)
Whiteley, W.: Counting out to the flexibility of molecules. Phys. Biol. 2, 1–11 (2005)
Whiteley, W.: Handbook of discrete and computational geometry. In: Goodman, J.E., O’Rourke, J. (eds.) Rigidity and Scene Analysis, pp. 1327–1354. Chapman & Hall, London (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research for this article was supported, in part, under a grant from NSERC (Canada), and final preparation occurred at the TU Berlin with support of the DFG Research Unit 565 “Polyhedral Surfaces.”
Rights and permissions
About this article
Cite this article
Schulze, B. Symmetric Versions of Laman’s Theorem. Discrete Comput Geom 44, 946–972 (2010). https://doi.org/10.1007/s00454-009-9231-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9231-x