ELSEVIER Computer Aided GeometricDesign 16 (1999) 89-106 COMPUTER AIDED GEOMETRIC DESIGN Swap conditions for dynamic Voronoi diagrams for circles and line segments Marina Gavrilova *, Jon Rokne Department of Computer Science, University of Calga~, Calga~, Alberta, Canada T2N 1N4 Received March 1997; revised May 1998 Abstract Dynamic maintenance of Voronoi diagrams for a set of disks moving independently in the plane along given trajectories is considered in this paper. The domain is limited by boundary represented by straight-line segments. Maintenance of a Voronoi diagram for moving objects (disks and/or line segments) over time requires calculation of topological events which occur when four objects arrive at positions where they are tangent to a common circle. Criteria for determination of such topological events for circles and line segments in the Euclidean metric have been derived using a standard fractional transformation in complex plane. These criteria are represented in the form of polynomial algebraic equations, based on the coordinates and trajectories of the moving objects. These equations can normally only be solved using numerical methods. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Dynamic Voronoi diagrams; Computational geometry 1. Introduction Problems involving geometric objects that move with time arise in many applications, such as motion planning, geometric modeling, computer simulation of physical system and virtual environments, robotics, computer graphics and computer animation. In such problems, a collection of geometric objects such as points, circles or line segments, is given along with analytic functions describing their motion, often specified as a polynomial of time. * Correspondingauthor. 0167-8396/99/$ - see front matter © 1999 ElsevierScience B.V. All rights reserved. PII: $0167-8396(98 )00039-9