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
Famous quotes containing the words complex, descartes and/or theorem:
“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)
“I think, therefore I am.
[Cogito, ergo sum.]”
—René Descartes (1596–1650)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)