TY - JOUR

T1 - On Topological Changes in the Delaunay Triangulation of Moving Points

AU - Rubin, Natan

N1 - Funding Information:
I would like to thank my former Ph.D. advisor Micha Sharir whose help made this work possible. In particular, I would like to thank him for the insightful discussions, and, especially, for his invaluable help in the preparation of this paper. In addition, I would like to thank the anonymous DCG referees for valuable suggestions that helped to improve the presentation. Work on this paper was partly supported by the Minerva Fellowship Program of Max Planck Society, and by Grant 338/09 from the Israel Science Fund.

PY - 2013/6/1

Y1 - 2013/6/1

N2 - Let P be a collection of n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants s,c>0 such that any four points are co-circular at most s times, and any three points are collinear at most c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during the motion of the points of P. In this paper, we obtain an upper bound of O(n2+ε), for any ε>0, under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.

AB - Let P be a collection of n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants s,c>0 such that any four points are co-circular at most s times, and any three points are collinear at most c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during the motion of the points of P. In this paper, we obtain an upper bound of O(n2+ε), for any ε>0, under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.

KW - Combinatorial complexity

KW - Delaunay triangulation

KW - Discrete changes

KW - Moving points

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=84879322684&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9512-2

DO - 10.1007/s00454-013-9512-2

M3 - Article

AN - SCOPUS:84879322684

SN - 0179-5376

VL - 49

SP - 710

EP - 746

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -