This problem cropped up in the context of scale-insensitive methods for generating random variables.

Let $X=R^n \cup \{\infty\}$. Suppose we consider a set of transforms $\cal{T}$ from $X\rightarrow X$. We construct them by concatenating functions chosen from the following set:

- Invertible linear transform ($x \mapsto Ax$, for $A\in GL_n(R)$)
- Translation ($x \mapsto x+b$)
- Circle inversion ($x \mapsto x/|x|^2$; 0 and $\infty$ swap)

The set $\mathcal{T}$ is very similar to the Mobius transformations, which are built from:

- Rotation and scaling ($x \mapsto sAx$, for $A\in SO_n(R)$ and $s$ a positive scalar)
- Translation ($x \mapsto x+b$)
- Circle inversion and reflection ($x \mapsto Mx/|x|^2$, where $M$ reflects through the first coordinate; 0 and $\infty$ swap)

I would like to know if $\cal{T}$ has a standard name, and if any of the properties of the Mobius transformations generalize to $\cal{T}$. For instance, Mobius transformations in $R^2$ preserve generalized circles; are generalized ellipsoids in $R^n$ preserved by $\mathcal{T}$? Is there a property analogous to the cross-ratio? Any references would be greatly appreciated.