How to define a sphere inversion transformation in 3D?
I would like to define a circle inversion like transformation for a sphere?
I saw inversion in higher dimensions on wiki, but it doesn't elaborate much on how to do it.
Let $S$ be a set of $N$ 3D points s.t. $S=\{ \vec{x}_i \}_{i=0} ^ N$ where $\vec{x}_i \in \mathbb{R}^3$.
We assume the origin isn't in $S$.
How to define a 3D inversion with respect to a sphere with radius $R= \max \{ \|\vec{x}_i \|_2 \}$. We assume the origin to be $(0,0,0)$.
Do I need to define a complex mapping using Quaternions? Or is there a simple way to find the "circle" inversion inf 3D?
From Wikipedia visualization of circle (2D) inversion:
1 Answer
$\begingroup$It’s the same regardless of the number of dimensions. Points that are inverses of each other lie on the same ray that emanates from the center of the sphere, and the product of their distances from the center is equal to the square of the radius. That is, if the radius of the sphere (centered on the origin) is $r$ then the inversion of a point $\mathbf x$ is ${r^2\over\|\mathbf x\|^2}\mathbf x$.
$\endgroup$ 3
