# Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Journal Article (Journal Article)

Let P be a set of n points and Q a convex k-gon in R . 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 nλ (n)) on the number of topological changes experienced by the diagrams throughout the motion; here λ (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. 2 4 r r

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Kaplan, H; Rubin, N; Sharir, M

### Published Date

- September 8, 2015

### Published In

### Volume / Issue

- 54 / 4

### Start / End Page

- 871 - 904

### Electronic International Standard Serial Number (EISSN)

- 1432-0444

### International Standard Serial Number (ISSN)

- 0179-5376

### Digital Object Identifier (DOI)

- 10.1007/s00454-015-9729-3

### Citation Source

- Scopus