Inverse of Curve under Inversive Transformation/Mistake

Source Work

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.)

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.):

inversion 1.

Mistake

A curve $\map f {x, y} = 0$ has an inverse $\map f {x', y'} = 0$, where

$x' = \dfrac {r^2 x} {x^2 + y^2} \qquad y' = \dfrac {r^2 y} {x^2 + y^2}$

Correction

The above statement is true only when the inversion center is at the origin.

For a general inversion circle of radius $r$ and inversion center $\tuple {x_0, y_0}$, the inverse point $P' = \tuple {x', y'}$ of a general point $P$ under $f$ is given by:

\(\ds x'\)

\(=\)

\(\ds \dfrac {r^2 \paren {x - x_0} } {\paren {x - x_0}^2 + \paren {y - y_0}^2}\)

\(\ds y'\)

\(=\)

\(\ds \dfrac {r^2 \paren {y - y_0} } {\paren {x - x_0}^2 + \paren {y - y_0}^2}\)

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.

• Weisstein, Eric W. "Inversion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Inversion.html