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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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inversion: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inversion: 1.