Definition:Inversive Transformation/Spherical
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Definition
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.
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.