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.
Given a [[point (geometry)|point]] [[set (mathematics)|set]] and the corresponding [[Voronoi diagram]], then for each [[cell (geometry)|cell]] at most two poles are defined, namely the ''positive'' and ''negative'' poles.


==Definition==
==Definition==
Let <math>V_p</math> be the corresponding cell of the point <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.
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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