Generalised Circle - The Transformation w = 1/z

The Transformation w = 1/z

It is now easy to see that the transformation w = 1/z maps generalized circles to generalized circles:

\begin{align} A z \bar z + B z + C \bar z + D & = 0 \\ A \frac{1}{w} \frac{1}{\bar w} + B \frac{1}{w} + C \frac{1}{\bar w} + D & = 0 \\ A + B \bar w + C w + D w \bar w & = 0 \\ D \bar w w + C w + B \bar w + A & = 0. \end{align}

We see that straight lines through the origin (A = D = 0) are mapped to straight lines through the origin, straight lines not containing the origin (A = 0; D ≠ 0) to circles containing the origin, circles containing the origin (A ≠ 0; D = 0) to straight lines not containing the origin, and circles not containing the origin (A ≠ 0; D ≠ 0) to circles not containing the origin.

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