Definition:Inversive Transformation
Definition
Let $\CC$ be a circle in the Euclidean plane $\EE$ whose center is $O$ and whose radius is $r$.
For a point $P$ such that $P \ne O$, let Euclid's First Postulate be used to construct a ray $\LL$ starting from $O$ and passing through $P$.
Let $f: \EE \to \EE$ be the mapping defined as:
- $\forall P \in \EE: \map f P = P'$
such that:
- $P'$ is also on $OP$
- $OP \times OP' = r^2$
Then $f$ is known as the inversive transformation of $\EE$ with respect to $\CC$.
Inversion Circle
The circle $\CC$ is known as the inversion circle of $f$.
Inversion Radius
The radius $r$ of the inversion circle $\CC$ is known as the inversion radius of $f$.
Inversion Center
The center $O$ of the inversion circle $\CC$ is known as the inversion center of $f$.
Inverse Point
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 $f$.
Inversive Transformation with respect to Sphere
The same inversive transformation can be performed in $3$-dimensional Euclidean space:
Let $\SS$ be a sphere in the Euclidean space $\EE$ whose center is $O$ and whose radius is $r$.
For a point $P$ such that $P \ne O$, let Euclid's First Postulate be used to construct a ray $\LL$ starting from $O$ and passing through $P$.
Let $f: \EE \to \EE$ be the mapping defined as:
- $\forall P \in \EE: \map f P = P'$
such that:
- $P'$ is also on $OP$
- $OP \times OP' = r^2$
Then $f$ is known as the inversive transformation of $\EE$ with respect to $\CC$.
Also known as
An inversive transformation is also known as:
- a circular reflection
- an inversion.
Also see
- Results about inversive transformations can be found here.
Historical Note
The study of inversive transformations was first performed systematically by Jakob Steiner.
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