On Kinetic Line Voronoi Operations and Finite Fields

In this paper, we show an isomorphism between the set of complex operations on the kinetic Voronoi diagram of points and open oriented line segments and the sets of numbers of new / deleted Voronoi regions induced by these operations, and its explanation using the finite field of residual classes of integers modulo 5: F₅ = Z/5Z. We show also an isomorphism between the set of complex operations on the kinetic Voronoi diagram of points and open oriented line segments and the set of differences of new and deleted quad-edge edges induced by these operations, and its explanation using the finite field F₁₅ = Z/15Z. We show finally the application of these theoretical results to the logging of a kinetic line Voronoi data structure.