Discrete & Computational Geometry

, Volume 51, Issue 3, pp 702–721

On Spaces of Infinitesimal Motions and Three Dimensional Henneberg Extensions



We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.


Frameworks Rigidity Henneberg moves 

Mathematics Subject Classification



  1. 1.
    Bartlett, M.S.: An inverse matrix adjustment arising in discriminant analysis. Ann. Math. Stat. 22, 107–111 (1951)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33(4), 549–563 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Gortler, S.J., Gotsman, C., Liu, L., Thurston, D.: On affine rigidity (2010). http://arxiv.org/pdf/1011.5553v1
  4. 4.
    Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2. American Mathematical Society, Providence, RI (1993)Google Scholar
  5. 5.
    Jackson, B., Jordán, T.: On the rank function of the 3-dimensional rigidity matroid. Int. J. Comput. Geom. Appl. 16(5–6), 415–429 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Jackson, B., Owen, J.: Notes on Henneberg Moves. Unpublished ManuscriptGoogle Scholar
  7. 7.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat. 21, 124–127 (1950)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Tay, T.-S., Whiteley, W.: Generating isostatic frameworks. Struct. Topol. 11, 21–69 (1985)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland GalwayGalwayIreland

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