Inverse of Curve under Inversive Transformation

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Theorem

Let $\CC$ be a circle embedded in a Cartesian plane $\EE$ whose center $O$ is at the origin $\tuple {0, 0}$ and whose radius is $r$.

Let $f$ be the inversive transformation of $\EE$ with respect to $\CC$.

Let $P = \tuple {x, y}$ be an arbitrary point of $\CC$.


The inverse point $P' = \tuple {x', y'}$ of $P$ under $f$ is given by:

\(\ds x'\) \(=\) \(\ds \dfrac {r^2 x} {x^2 + y^2}\)
\(\ds y'\) \(=\) \(\ds \dfrac {r^2 y} {x^2 + y^2}\)


Proof


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