Minkowski products of unit quaternion sets

• Published in: Advances in computational mathematics Vol. 45; no. 3; pp. 1607 - 1629
• Main Authors: Farouki, Rida T., Gentili, Graziano, Moon, Hwan Pyo, Stoppato, Caterina
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2019; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Lie groups; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Quaternions; Rotation; Spherical caps; Visualization; Unit quaternions; 65G30; 30G35; Minkowski products; 3-sphere; Stereographic projection; Lie algebra; Boundary evaluation; Spatial rotations
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09687-9

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 S 3 in 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 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.