## Abstract

Let P be a set of n points and Q a convex k-gon in R^{2}. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of P, under the convex distance function defined by Q, as the points of P move along prespecified continuous trajectories. Assuming that each point of P moves along an algebraic trajectory of bounded degree, we establish an upper bound of O(k^{4}nλ_{r}(n)) on the number of topological changes experienced by the diagrams throughout the motion; here λ_{r}(n) is the maximum length of an (n, r)-Davenport–Schinzel sequence, and r is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

Original language | English |
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Pages (from-to) | 871-904 |

Number of pages | 34 |

Journal | Discrete and Computational Geometry |

Volume | 54 |

Issue number | 4 |

DOIs | |

State | Published - 8 Sep 2015 |

## Keywords

- Convex distance function
- Delaunay triangulation
- Discrete changes
- Kinetic data structure
- Moving points
- Voronoi diagram

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics