TY - JOUR

T1 - The distance trisector curve

AU - Asano, Tetsuo

AU - Matoušek, Jiří

AU - Tokuyama, Takeshi

N1 - Funding Information:
* Corresponding author. E-mail address: matousek@kam.mff.cuni.cz (J. Matoušek). 1 The part of this research by T.A. was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research on Priority Areas and Scientific Research (B). 2 Parts of this research by J.M. were done during visits to the Japanese Advanced Institute for Science and Technology (JAIST) and to the ETH Zürich; the support of these institutions is gratefully acknowledged. 3 The part of this research by T.T. was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research on Priority Areas.

PY - 2007/6/20

Y1 - 2007/6/20

N2 - 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 C1 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 frac(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 zone diagram and which we investigate in a companion paper.

AB - 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 C1 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 frac(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 zone diagram and which we investigate in a companion paper.

KW - Analytic function

KW - Generalized Voronoi diagram

KW - Planar curve

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U2 - 10.1016/j.aim.2006.10.006

DO - 10.1016/j.aim.2006.10.006

M3 - Article

AN - SCOPUS:34047263277

VL - 212

SP - 338

EP - 360

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -