TY - GEN

T1 - The distance trisect or 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 - 2006

Y1 - 2006

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 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.

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 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.

KW - Algorithms

KW - Theory

UR - http://www.scopus.com/inward/record.url?scp=33748095498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748095498&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33748095498

SN - 1595931341

SN - 9781595931344

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 336

EP - 343

BT - STOC'06

T2 - 38th Annual ACM Symposium on Theory of Computing, STOC'06

Y2 - 21 May 2006 through 23 May 2006

ER -