geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the positive and negative Voronoi poles of a

cell Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

in a

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed th ...

are certain vertices of the diagram.

Definition

Let

V

be the Voronoi diagram for a set of sites

P

, and let

V_p

be the Voronoi cell of

V

corresponding to a site

p\in P

. If

V_p

is bounded, then its positive pole is the vertex of the boundary of

V_p

that has maximal distance to the point

p

. If the cell is unbounded, then a positive pole is not defined. Furthermore, let

\bar

be the vector from

p

to the positive pole, or, if the cell is unbounded, let

\bar

be a vector in the average direction of all unbounded Voronoi edges of the cell. The negative pole is then the Voronoi vertex

v

V_p

with the largest distance to

p

such that the vector

\bar

and the vector from

p

v

make an angle larger than

\tfrac

References

* {{Cite book , last = Boissonnat , first = Jean-Daniel , title = Effective Computational Geometry for Curves and Surfaces , publisher =

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

, location = Berlin , year = 2007 , isbn = 978-3-540-33258-9 Computational geometry