Abstract
We present constructions inspired by the Ma–Schlenker example of “Non-rigidity of spherical inversive distance circle packings” (Discrete Comput Geom 47(3):610–617, 2012). In contrast to the use in Ma and Schlenker (2012) of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inversive geometry of the 2-sphere.
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Notes
In both Luo [4] and Ma–Schlenker [5] there is a typo in the expression for the spherical formula for inversive distance. They report the negative of this formula. A quick 2nd order Taylor approximation shows that this formula reduces to Expression (1.1) in the limit as the arguments of the sines and cosines approach zero.
Any circle in \({\mathbb {S}}^{2}\) bounds two distinct disks. Without explicitly stating so, we always assume that one of these has been chosen as a companion disk. The center and radius of a circle in \({\mathbb {S}}^{2}\) are the center and radius of its companion disk. The ambiguity should cause no confusion.
It is in no way obvious that formulæ (1.1) and (1.2) are Möbius invariants of circle pairs. There is a not so well-known development of inversive distance, which applies equally in spherical, Euclidean, and hyperbolic geometry, that uses the cross-ratio of the four points of intersection of \(C_{1}\) and \(C_{2}\) with a common orthogonal circle. It is computationally less friendly than (1.1) and (1.2), but has the theoretical advantage of being manifestly Möbius-invariant. See [1].
By an application of (3.14).
References
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Bowers, P.L., Stephenson, K.: Uniformizing dessins and Belyĭ maps via circle packing. Mem. Am. Math. Soc. 170(805), 1–97 (2004)
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Luo, F.: Rigidity of polyhedral surfaces, III. Geom. Topol. 15(4), 2299–2319 (2011)
Ma, J., Schlenker, J.-M.: Non-rigidity of spherical inversive distance circle packings. Discrete Comput. Geom. 47(3), 610–617 (2012)
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Bowers, J.C., Bowers, P.L. Ma–Schlenker c-Octahedra in the 2-Sphere. Discrete Comput Geom 60, 9–26 (2018). https://doi.org/10.1007/s00454-017-9928-1
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DOI: https://doi.org/10.1007/s00454-017-9928-1