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.
Let
V
P
Vp
V
p\inP
Vp
Vp
p
Furthermore, let
\bar{u}
p
\bar{u}
v
Vp
p
\bar{u}
p
v
\tfrac{\pi}{2}
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.