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

Definition

Let ${\displaystyle V}$ be the Voronoi diagram for a set of sites ${\displaystyle P}$, and let ${\displaystyle V_{p))$ be the Voronoi cell of ${\displaystyle V}$ corresponding to a site ${\displaystyle p\in P}$. If ${\displaystyle V_{p))$ is bounded, then its positive pole is the vertex of the boundary of ${\displaystyle V_{p))$ that has maximal distance to the point ${\displaystyle p}$. If the cell is unbounded, then a positive pole is not defined.

Furthermore, let ${\displaystyle {\bar {u))}$ be the vector from ${\displaystyle p}$ to the positive pole, or, if the cell is unbounded, let ${\displaystyle {\bar {u))}$ be a vector in the average direction of all unbounded Voronoi edges of the cell. The negative pole is then the Voronoi vertex ${\displaystyle v}$ in ${\displaystyle V_{p))$ with the largest distance to ${\displaystyle p}$ such that the vector ${\displaystyle {\bar {u))}$ and the vector from ${\displaystyle p}$ to ${\displaystyle v}$ make an angle larger than ${\displaystyle {\tfrac {\pi }{2))}$.