Voronoi pole

In computational geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram, chosen in pairs in each cell of the diagram to be far from the site generating that pair. They have applications in surface reconstruction.

Definition

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.

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.^[1]

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 ${\displaystyle {\tfrac {\pi }{2}}}$.^[1]

History and application

The poles were introduced in 1998 in two papers by Nina Amenta, Marshall Bern, and Manolis Kellis, for the problem of surface reconstruction. As they showed, any smooth surface that is sampled with sampling density inversely proportional to its curvature can be accurately reconstructed, by constructing the Delaunay triangulation of the combined set of sample points and their poles, and then removing certain triangles that are nearly parallel to the line segments between pairs of nearby poles.^[2][3]