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.
be the Voronoi diagram for a set of sites
is bounded, then its positive pole is the vertex of the boundary of
that has maximal distance to the point
If the cell is unbounded, then a positive pole is not defined.
to the positive pole, or, if the cell is unbounded, let
be a vector in the average direction of all unbounded Voronoi edges of the cell.
The negative pole is then the Voronoi vertex
make an angle larger than
[1] 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.
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.