Minkowski products of unit quaternion sets

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.Comment: 29 pages, 1 figur

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed

Computation of Minkowski values of polynomials over complex sets

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X . Given any polynomial P (z) with prescribed coefficients in the complex variable z, the Minkowski value P (X ) is de ned to be the set of all complex values generated by evaluating P , through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X . The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield dierent Minkowski value sets P (X ). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial value Pm (X ), for which the monomial terms are evaluated separately (incurring n(n+1) independent values of z) and summed; the Minkowski factor value P f (X ), where P is represented as the product (z r 1 ) (z r n ) of n linear factors | each incurring an independent choice z 2 X | and r 1 ; : : : ; r n are the roots of P (z); and the Minkowski Horner value P h (X ), where the evaluation is performed by \nested multiplication" and incurs n independent values z 2 X . A new algorithm for the evaluation of P h (X ), when 0 62 X , is presented