Let *C* be a circle in the complex plane with center *c* and radiusÂ *r*. Assume *C* does not pass through the origin.

Let *f*(*z*) = 1/*z*. Then *f*(*C*) is also a circle. We will derive the center and radius of *f*(*C*) in this post.

***

Our circle *C* is the set of points *z* satisfying

Define *w* = 1/*z* and substitute 1/*w* for *z* above.

A little algebra shows

and a little more shows

This is the equation of a circle with center

and radius

As a way to check the derivation above, here’s some Python code for making circles and taking their reciprocal.

import numpy as np import matplotlib.pyplot as plt theta = np.linspace(0, 2*np.pi, 1000) # plot image of circle of radius r centered at c def plot(c, r): cc = np.conj(c) d = r**2 - c*cc print("Expected center: ", -cc/d) print("Expected radius: ", r/abs(d)) u = np.exp(1j * theta) # unit circle w = 1/(r*u + c) print("Actual radius:", (max(w.real) - min(w.real))/2) plt.plot(w.real, w.imag) plt.gca().set_aspect("equal") plt.show() plot(1 + 2j, 3) plot(0.5, 0.2) plot(1j, 0.5)