University⏱ 4 min

Complex Inversion

1/z - inversion in complex plane

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inputoutputz1/z

z = 1.5+0.5i

Input circle

\ rac1a+bi=\ raca−bia2+b2\ rac{1}{a+bi} = \ rac{a-bi}{a^2+b^2}
w=1z=zˉ∣z∣2=0.600−0.200iw = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = 0.600 -0.200i

w=1/z is a Möbius transformation: it maps circles to circles (or lines). Circles through the origin map to lines!