Semi Voronoi Diagrams

Abstract

We consider a problem that is a variant of the Voronoi diagram problem on the Euclidean plane, with the association of a given direction $\vec{d_i}$ to each point p _i in P. For each p _i , the direction $\vec{d_i}$ defines a visible half plane of p _i . A point p in the plane is said to be controlled by p _i if: (1) p is visible to p _i ; (2) among all the points in P that p is visible to, p _i is the closest one to p. The members in P partition the plane into different connected regions, each region is controlled by a member in P or is not controlled by any member in P. We give some preliminary results on this partition and propose some problems for future studies.