Complex Descartes Theorem
To determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (x, y) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem.
Given four circles with curvatures ki and centers zi (for i = 1...4), the following equality holds in addition to equation (1):
Once k4 has been found using equation (2), one may proceed to calculate z4 by rewriting equation (4) to a form similar to equation (2):
Again, in general, there are two solutions for z4, corresponding to the two solutions for k4.
Read more about this topic: Descartes' Theorem
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