Definition:Inversive Transformation/Inverse Point
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Definition
Let $\EE$ denote the Euclidean plane.
Let $f: \EE \to \EE$ be the inversive transformation on $\EE$ with respect to the circle $\CC$ whose center is $O$ and whose radius is $r$.
The image $P' := \map f P$ of a point $P$ under the inversive transformation $f$ is referred to as the inverse point of $P$ under $f$.
From Inverse Transformation is Involution it also follows that also $P$ is the inverse point of $P'$ under $f$.
Also known as
An inverse point is also referred to as just an inverse, but this is a word which applies to a number of contexts.
Similarly, the images of other geometric objects under an inversive transformation are likewise referred to as just the inverse of those objects.
Also see
- Results about inverse points can be found here.
Sources
- 1996: Richard Courant, Herbert Robbins and Ian Stewart: What is Mathematics? (2nd ed.): Chapter $\text{III}$ / $\text{II}$ Section $4$: "Geometrical Transformations. Inversion."
- 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
- Weisstein, Eric W. "Inversion Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InversionCircle.html