Minkowski roots of complex sets

R. T. Farouki, Weiqung Gu, Hwan Pyo Moon

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

10 Scopus citations

Abstract

An n-th Minkowski root ⊗1/n script A of a given complex set script A is defined by the property {z1z2⋯zn|zi∈⊗1/n script A}≡script A, i.e., the set of all products of n independently chosen values from ⊗1/nscript A is identical to script A. Hence, the n-th Minkowski power of ⊗1/nscript A yields the original set script A. Minkowski root extractions are fundamental operations in the Minkowski geometric algebra of complex sets: depending on the nature of script A, subtle issues concerning the existence, uniqueness and minimality or maximality of ⊗1/nscript A may arise. For a domain script A with a smooth boundary that is strictly logarithmically convex, we show that each connected component of the "ordinary" root script A1/n={z|zn∈script A} is a Minkowski n-th root. ⊗1/nscript A has a more intricate structure, however when ∂script A has logarithmic inflections. For example, if script A is a circular disk, ⊗1/nscript A is (a single loop of) the "n-th order" ovals of Cassini or lemniscate of Bernoulli when 0∉script A or 0∈∂script A, respectively. But when 0 is in the interior of script A, a composite curve (portions of the Cassini ovals and a higher-order curve) is required to describe ⊗1/nscript A.

Original languageEnglish
Title of host publicationProceedings - Geometric Modeling and Processing 2000
Subtitle of host publicationTheory and Applications
EditorsRalph Martin
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages287-300
Number of pages14
ISBN (Electronic)0769505627, 9780769505626
DOIs
StatePublished - 2000
EventGeometric Modeling and Processing 2000, GMP 2000 - Hong Kong, China
Duration: 11 Apr 200012 Apr 2000

Publication series

NameProceedings - Geometric Modeling and Processing 2000: Theory and Applications

Conference

ConferenceGeometric Modeling and Processing 2000, GMP 2000
Country/TerritoryChina
CityHong Kong
Period11/04/0012/04/00

Keywords

  • lemniscate of Bernoulli
  • logarithmic convexity
  • minimal and maximal roots
  • Minkowski geometric algebra
  • Minkowski roots
  • ordinary roots
  • ovals of Cassini

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