Generalised Circle - Equation in The Extended Complex Plane

Equation in The Extended Complex Plane

The extended plane of inversive geometry can be identified with the extended complex plane, so that equations of complex numbers can be used to describe lines, circles and inversions.

A circle Γ is the set of points z in a plane that lie at radius r from a center point γ.

Using the complex plane, we can treat γ as a complex number and circle Γ as a set of complex numbers.

Using the property that a complex number multiplied by its conjugate gives us the square of the modulus of the number, and that its modulus is its Euclidean distance from the origin, we can express the equation for Γ as follows:

We can multiply this by a real constant A to get an equation of the form

$A z \bar z + B z + C \bar z + D = 0$

where A and D are real, and B and C are complex conjugates. Reversing the steps, we see that in order for this to be a circle, the radius squared must be equal to BC/A^2 - D/A > 0. So the above equation defines a generalized circle whenever AD < BC. Note that when A is zero, this equation defines a straight line.

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