The distance trisect or curve

Tetsuo Asano, Jiří Matoušek, Takeshi Tokuyama

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Citations (Scopus)

Abstract

Given points p and q in the plane, we are interested in separating them by two curves C1 and C2 such that every point of C 1 has equal distance to p and to C2, and every point of C2 has equal distance to C1 and to q. We show by elementary geometric means that such C1 and C2 exist and are unique. Moreover, for p = (0,1) and q = (0,-1), C1 is the graph of a function f: ℝ → ℝ, C2 is the graph of - f, and f is convex and analytic (i.e., given by a convergent power series at a neighborhood of every point). We conjecture that f is not expressible by elementary functions and, in particular, not algebraic. We provide an algorithm that, given x ∈ ℝ and ε > 0, computes an approximation to f(x) with error at most ε in time polynomial in log 1+|x|/ε. The separation of two points by two "trisector" curves considered here is a special (two-point) case of a new kind of Voronoi diagram, which we call the Voronoi diagram with neutral zone and which we investigate in a companion paper.

Original languageEnglish
Title of host publicationSTOC'06
Subtitle of host publicationProceedings of the 38th Annual ACM Symposium on Theory of Computing
Pages336-343
Number of pages8
Publication statusPublished - 2006
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: 2006 May 212006 May 23

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume2006
ISSN (Print)0737-8017

Other

Other38th Annual ACM Symposium on Theory of Computing, STOC'06
Country/TerritoryUnited States
CitySeattle, WA
Period06/5/2106/5/23

Keywords

  • Algorithms
  • Theory

ASJC Scopus subject areas

  • Software

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