Do maps of the form $$ x \in \mathbb{R}^n \mapsto \frac{Ax+b}{c^Tx+d} \in \mathbb{R}^n, $$ where $A \in \mathbb{R}^{n\times n}, b, c \in \mathbb{R}^n, d\in \mathbb{R}$ have a name? Have they been studied anywhere?
It looks somehow familiar to Möbius-transformation but it is different as $A, b, c, d$ are not complex numbers.
It is easy to see that the above maps form a group.
I am interested in this because of an application in optics where I found that for a thin lense the map which maps image to object points is of the above form. I am especially interested in the $n=2$ and $n=3$ cases.
These seem to be projective transformations / homographies / collineations. See particularly the formulas given when projective spaces are defined by adding points at infinity to affine spaces.
This is no surprise since there is a long history of projective geometry in optics, going back to the study of perspective. I think you are probably already aware of this, but these maps provide a good description of image transformations by lenses only in the paraxial approximation.
Here's a chapter by Douglas S. Goodman from the Optical Society of America's Handbook of Optics which contains a discussion of these transformations in Section 1.15 (page 59 of the PDF, page 1.60 in the internal numbering of the book). It seems the preferred terminology in optics is "collineation"; note however that Wikipedia distinguishes collineations from homographies, though they agree for real projective spaces.