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Minkowski products of unit quaternion sets. (English) Zbl 1458.53058

Summary: 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 \(\mathbb {R}^{4}\), 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.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53A20 Projective differential geometry
30G35 Functions of hypercomplex variables and generalized variables
65G30 Interval and finite arithmetic
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

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