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Minkowski products of unit quaternion sets

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Abstract

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere S3 in \(\mathbb {R}^{4}\), closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in \(\mathbb {R}^{3}\) are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.

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Funding

The second and fourth authors are partly supported by the Italian Ministry of Education (MIUR) through Finanziamento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environments” and by Istituto Nazionale di Alta Matematica (INdAM) through Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA).

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California, Davis, CA, 95616, USA

    Rida T. Farouki

  2. Dipartimento di Matematica e Informatica “U. Dini,”, Università di Firenze, Viale Morgagni 67/A, I–50134, Firenze, Italy

    Graziano Gentili & Caterina Stoppato

  3. Department of Mathematics, Dongguk University–Seoul, Seoul, 04620, Republic of Korea

    Hwan Pyo Moon

Authors
  1. Rida T. Farouki
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  2. Graziano Gentili
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  3. Hwan Pyo Moon
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  4. Caterina Stoppato
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Correspondence to Rida T. Farouki.

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Communicated by: Axel Voigt

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Farouki, R.T., Gentili, G., Moon, H.P. et al. Minkowski products of unit quaternion sets. Adv Comput Math 45, 1607–1629 (2019). https://doi.org/10.1007/s10444-019-09687-9

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  • DOI: https://doi.org/10.1007/s10444-019-09687-9

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