Voronoi pole

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In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram.

Definition[edit]

Let V_p be the Voronoi cell of the site p\in P. If V_p is bounded then its positive pole is the Voronoi vertex in V_p with maximal distance to the sample point p. Furthermore, let \bar{u} be the vector from p to the positive pole. If the cell is unbounded, then a positive pole is not defined, and \bar{u} is defined to be a vector in the average direction of all unbounded Voronoi edges of the cell.

The negative pole is the Voronoi vertex v in V_p with the largest distance to p such that the vector \bar{u} and the vector from p to v make an angle larger than \frac{\pi}{2}.

Example[edit]

Example of poles in a Voronoi diagram

Here x is the positive pole of V_p and y its negative. As the cell corresponding to q is unbounded only the negative pole z exists.

References[edit]