# Difference between revisions of "Voronoi pole"

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In [[geometry]], the positive and negative '''Voronoi poles''' of a [[cell (geometry)|cell]] in a [[Voronoi diagram]] are certain vertices of the diagram. | |||

==Definition== | ==Definition== | ||

Let <math>V_p</math> be the | Let <math>V_p</math> be the Voronoi cell of the site <math>p\in P</math>. If <math>V_p</math> is bounded then its ''positive pole'' is the Voronoi vertex in <math>V_p</math> with maximal distance to the sample point <math>p</math>. Furthermore, let <math>\bar{u}</math> be the vector from <math>p</math> to the positive pole. If the cell is unbounded, then a positive pole is not defined, and <math>\bar{u}</math> is defined to be a vector in the average direction of all unbounded Voronoi edges of the cell. | ||

The ''negative pole'' is the Voronoi vertex <math>v</math> in <math>V_p</math> with the largest distance to <math>p</math> such that the vector <math>\bar{u}</math> and the vector from <math>p</math> to <math>v</math> make an angle larger than <math>\frac{\pi}{2}</math>. | The ''negative pole'' is the Voronoi vertex <math>v</math> in <math>V_p</math> with the largest distance to <math>p</math> such that the vector <math>\bar{u}</math> and the vector from <math>p</math> to <math>v</math> make an angle larger than <math>\frac{\pi}{2}</math>. |

## Latest revision as of 19:25, 18 September 2012

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

## Definition

Let be the Voronoi cell of the site . If is bounded then its *positive pole* is the Voronoi vertex in with maximal distance to the sample point . Furthermore, let be the vector from to the positive pole. If the cell is unbounded, then a positive pole is not defined, and is defined to be a vector in the average direction of all unbounded Voronoi edges of the cell.

The *negative pole* is the Voronoi vertex in with the largest distance to such that the vector and the vector from to make an angle larger than .

## Example

Example of poles in a Voronoi diagram

Here is the positive pole of and its negative. As the cell corresponding to is unbounded only the negative pole exists.

## References

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