Abstract
We extend to arbitrary finite n the notion of immobilization of a convex body O in \({\mathbb {R}}^n\) by a finite set of points \({\mathcal {P}}\) in the boundary of O. Because of its importance for this problem, necessary and sufficient conditions are found for the immobilization of an n-simplex. A fairly complete geometric description of these conditions is given: as n increases from \(n = 2\), some qualitative difference in the nature of the sets \({\mathcal {P}}\) emerges.
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Bracho, J., Fetter, H., Mayer, D., Montejano, L.: Immobilization of solids and mondriga quadratic forms. J. Lond. Math. Soc. 51(1), 189–200 (1995)
Bracho, J., Montejano, L., Urrutia, J.: Immobilization of smooth convex figures. Geom. Dedic. 53(2), 119–131 (1994)
Bracho, J., Montejano, L.: Rotors in triangles and tetrahedra. J. Geom. 108(3), 851–859 (2017)
Czyzowicz, J., Stojmenovic, I., Urrutia, J.: Immobilizing a shape. Int. J. Comput. Geom. Appl. 9(2), 181–206 (1999)
Gilbert, A.D., Nsubuga, S.H.: Immobilization of Convex Bodies in \({\mathbb{R}}^{n}\). arXiv:1810.11381 (2018)
Kuperberg, W.: Dimacs Workshop on Polytopes. Rutgers University, Camden (1990)
Markenscoff, X., Ni, L., Papadimitriou, C.H.H.: The geometry of grasping. Int. J. Robot. Res. 9(1), 61–74 (1990)
Markenscoff, X., Papadimitriou, C.H.: Optimum grip of a polygon. Int. J. Robot. Res. 8(2), 17–29 (1989)
O’Neill, B.: Elementary Differential Geometry. Academic Press, London (1997)
Rimon, E., Burdick, J.W.: New bounds on the number of frictionless fingers requied to immobilize. J. Robot. Syst. 12(6), 433–451 (1995)
Rimon, E., Burdick, J.W.: Mobility of bodies in contact. I. A 2nd-order mobility index for multiple-finger grasps. IEEE Trans. Robot. Autom. 14(5), 696–708 (1998)
Rimon, E., Burdick, J.W.: Mobility of bodies in contact. II. How forces are generated by curvature effects. IEEE Trans. Robot. Autom. 14(5), 709–717 (1998)
van der Stappen, A.F.: Immobilization: analysis, existence, and output-sensitive synthesis a. frank van der stappen. In: Dimacs Workshop Computer Aided Design and Manufacturing, vol. 67, p. 165 (2005)
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The authors wish to thank Elmer Rees for his generous advice towards this paper.
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Gilbert, A.D., Nsubuga, S.H. Immobilization of convex bodies in \({\mathbb {R}}^n\). J. Geom. 110, 3 (2019). https://doi.org/10.1007/s00022-018-0458-7
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DOI: https://doi.org/10.1007/s00022-018-0458-7