Semi Voronoi Diagrams - Cheng, Yongxi; Li, Bo; Xu, Yinfeng (Computational Geometry, Graphs and Applications, 2011)
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 pi in P. For each pi, the direction $\vec{d_i}$ defines a visible half plane of pi. A point p in the plane is said to be controlled by pi if: (1) p is visible to pi; (2) among all the points in P that p is visible to, pi 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.