Voronoi pole

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

Definition

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

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

Boissonnat, Jean-Daniel (2007). Effective Computational Geometry for Curves and Surfaces. Berlin: Springer. ISBN 978-3-540-33258-9.

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