Time evolution of integrals

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 $\overline{u}$ be the vector from $p$ to the positive pole. If the cell is unbounded, then a positive pole is not defined, and $\overline{u}$ 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 $\overline{u}$ and the vector from $p$ to $v$ make an angle larger than $\frac{\pi }{2}$.

Example

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

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