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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.

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    $\begingroup$ The above map is only defined at $x$ if $c^Tx+d\neq 0$. So there should be some condition on $(A,b,c,d)$ to avoid the image to be contained in a hyperplane (since otherwise the composition would be ill-defined). $\endgroup$
    – YCor
    Jan 4, 2018 at 21:34
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    $\begingroup$ They're called (real) projective transformations, or homographies. $\endgroup$
    – Gro-Tsen
    Jan 4, 2018 at 21:44
  • $\begingroup$ @YCor: regardless from conditions on the denominators, they could be seen as elements of the group of birational automorphisms of real projective space. $\endgroup$
    – Qfwfq
    Jan 4, 2018 at 22:09
  • $\begingroup$ @Qfwfq regardless from conditions on denominators? which birational transformation do you recognize when $(A,b,c,d)=(0,0,0,0)$? $\endgroup$
    – YCor
    Jan 4, 2018 at 22:11
  • $\begingroup$ If these encode homographies, the condition is probably that the determinant of $\begin{pmatrix}A & b\\c^T & d\end{pmatrix}$ is nonzero. $\endgroup$
    – YCor
    Jan 4, 2018 at 22:13

1 Answer 1

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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.

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  • $\begingroup$ I see now that Gro-Tsen has posted a comment to the same effect and that the question has picked up a vote to close. I'm happy to delete this answer if others feel it's not necessary. $\endgroup$
    – j.c.
    Jan 4, 2018 at 21:47
  • $\begingroup$ ... the question could be moved to MathSE too and the answer would be useful too. $\endgroup$
    – YCor
    Jan 4, 2018 at 22:19
  • $\begingroup$ @YCor that sounds fine to me. I will vote for migration. $\endgroup$
    – j.c.
    Jan 4, 2018 at 22:20

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