Power diagram
In computational geometry, a power diagram is a partition of the Euclidean plane into polygonal cells defined from a set of circles, where the cell for a given circle C consists of all the points for which the power distance to C is smaller than the power distance to the other circles. It is a form of generalized Voronoi diagram, and coincides with the Voronoi diagram of the circle centers in the case that all the circles have equal radii.
If C is a circle and P is a point outside C, then the power of P with respect to C is the square of the length of a line segment from P to a point T of tangency with C. Equivalently, if P has distance d from the center of the circle, and the circle has radius r, then (by the Pythagorean theorem) the power is d2 − r2. The same formula d2 − r2 may be extended to all points in the plane, regardless of whether they are inside or outside of C: points on C have zero power, and points inside C have negative power.
The power diagram of a set of n circles Ci is a partition of the plane into n regions Ri (called cells), such that a point P belongs to Ri whenever circle Ci is the circle minimizing the power of P.
In the case n = 2, the power diagram consists of two halfplanes, separated by a line called the radical axis or chordale of the two circles. Along the radical axis, both circles have equal power. More generally, in any power diagram, each cell Ri is a convex polygon, the intersection of the halfspaces bounded by the radical axes of circle Ci with each other circle. Triples of cells meet at vertices of the diagram, which are the radical centers of the three circles whose cells meet at the vertex.
The power diagram may be seen as a weighted form of the Voronoi diagram of a set of point sites, a partition of the plane into cells within which one of the sites is closer than all the other sites. Other forms of weighted Voronoi diagram include the additively weighted Voronoi diagram, in which each site has a weight that is added to its distance before comparing it to the distances to the other sites, and the multiplicatively weighted Voronoi diagram, in which the weight of a site is multiplied by its distance before comparing it to the distances to the other sites. In contrast, in the power diagram, we may view each circle center as a site, and each circle's squared radius as a weight that is subtracted from the squared distance before comparing it to other squared distances. In the case that all the circle radii are equal, this subtraction makes no difference to the comparison, and the power diagram coincides with the Voronoi diagram.
...
Wikipedia