Exact Minkowski Products of N Complex Disks

Exact Minkowski Products of N Complex Disks An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N > 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable ϕ over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic." http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

Exact Minkowski Products of N Complex Disks

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Publisher
Springer Journals
Copyright
Copyright © 2002 by Kluwer Academic Publishers
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1014737602641
Publisher site
See Article on Publisher Site

Abstract

An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N > 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable ϕ over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic."

Journal

Reliable ComputingSpringer Journals

Published: Oct 13, 2004

References

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