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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.〔.〕〔.〕〔.〕〔.〕
==Definition==
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 ''d''2 − ''r''2. The same formula ''d''2 − ''r''2 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 ''C''''i'' is a partition of the plane into ''n'' regions ''R''''i'' (called cells), such that a point ''P'' belongs to ''R''''i'' whenever circle ''C''''i'' 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 ''R''''i'' is a convex polygon, the intersection of the halfspaces bounded by the radical axes of circle ''C''''i'' 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.〔〔〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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