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.


Visible Angle Half Plane Voronoi Diagram Visible Area Euclidean Plane 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Yongxi Cheng
    • 1
  • Bo Li
    • 2
  • Yinfeng Xu
    • 1
  1. 1.School of ManagementXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Mathematics, School of ScienceXi’an Jiaotong UniversityXi’anChina

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