Exact Minkowski products of N complex disks

Rida T. Farouki*, Helmut Pottmann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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.".

Original languageEnglish (US)
Pages (from-to)43-66
Number of pages24
JournalReliable Computing
Issue number1
StatePublished - Feb 1 2002

ASJC Scopus subject areas

  • Software
  • Safety, Risk, Reliability and Quality


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