Closed Rotation Sequences

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A finite sequence of rotations is closed if a sequential application of all the rotations from the sequence results in no net orientation change. A complete characterization of closed rotation sequences involving a given set of rotation axes is presented, and the set of such sequences is shown to be a smooth manifold under a nondegeneracy condition on the rotation axes. The characterization is used to derive several examples of closed rotation sequences, some of which are then shown to specialize to classical examples of such sequences provided by the Rodrigues–Hamilton theorem and the Donkin’s theorem. Discrete versions of the Goodman–Robinson and Ishlinskii theorems are also derived and illustrated using the so-called Codman’s paradox.

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    Both the coplanarity conditions imply that the angles between \((i-2)\)th and \((i-1)\)th faces of \(\fancyscript{M}_\mathrm{P}\) is the same as that between the corresponding faces of \(\fancyscript{F}_\mathrm{P}\). In other words, while the cones \(\fancyscript{M}\) and \(\fancyscript{F}\) have the same corresponding sides, the polar cones have the same corresponding angles.


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The second author is thankful to Professor Christoph Ament and Dr. Thomas Glotzbach, Technische Universität Ilmenau, for providing financial support through the project MORPH (EU FP7 under Grant agreement No. 288704).

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Correspondence to Sanjay P. Bhat.

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Bhat, S.P., Crasta, N. Closed Rotation Sequences. Discrete Comput Geom 53, 366–396 (2015) doi:10.1007/s00454-014-9653-y

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  • Rotation sequences
  • Spherical polygons
  • Rodrigues–Hamilton theorem
  • Donkin’s theorem
  • Goodman–Robinson theorem
  • Ishlinkskii’s theorem
  • Codman’s paradox

Mathematics Subject Classification

  • 51F25
  • 70B10