Voronoi pole

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{{ safesubst:#invoke:Unsubst||$N=Context |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Given a point set and the corresponding Voronoi diagram, then for each cell at most two poles are defined, namely the positive and negative poles.

Definition

Let be the corresponding cell of the point . If is bounded then its positive pole is the Voronoi vertex in with maximal distance to the sample point . Furthermore, let be the vector from to the positive pole. If the cell is unbounded, then a positive pole is not defined, and 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 in with the largest distance to such that the vector and the vector from to make an angle larger than .

Example

Example of poles in a Voronoi diagram

Here is the positive pole of and its negative. As the cell corresponding to is unbounded only the negative pole exists.

References

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