Minkowski products of unit quaternion sets

Rida T. Farouki; Graziano Gentili; Hwan Pyo Moon; Caterina Stoppato

doi:10.1007/s10444-019-09687-9

Minkowski products of unit quaternion sets

Rida T. Farouki, Graziano Gentili, Hwan Pyo Moon, Caterina Stoppato

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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 R⁴, 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 R³ 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.

Original language	English
Pages (from-to)	1607-1629
Number of pages	23
Journal	Advances in Computational Mathematics
Volume	45
Issue number	3
DOIs	https://doi.org/10.1007/s10444-019-09687-9
State	Published - Jun 2019

Keywords

3-sphere
Boundary evaluation
Lie algebra
Minkowski products
Spatial rotations
Stereographic projection
Unit quaternions

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10.1007/s10444-019-09687-9

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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 R4, 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 R3 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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AU - Gentili, Graziano

AU - Moon, Hwan Pyo

AU - Stoppato, Caterina

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KW - Lie algebra

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Minkowski products of unit quaternion sets

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