Projective transformation that fixes the unit circle and sends a point to the origin

Question

I'd like to find a projective transformation that fixes the unit circle and sends some point on the $x$-axis within the unit circle to the origin (or I guess a random point in the unit circle, however, as I know that I can rotate, I thought this might be an easier case). Now, a fellow mathematician gave me a matrix that worked (under the assumption that we've already rotated our plane such that the remaining transformation works in the $XZ$-plane), but I had no idea how he got that matrix. He found $$ \begin{pmatrix} -\sqrt{1+c^2}&0&c\\ 0&1&0\\ c&0&-\sqrt{1+c^2} \end{pmatrix} $$ I know how to find some contraints; our conic is given by $x^2+y^2=z^2$, which corresponds to the diagonal matrix $D$ with $1,1,-1$ on the diagonal (in that order). So we're looking for a transformation $A$ which satisfies $A^TDA=D$. Furthermore, we'd like $A(x,0,z)=(0,0,z')$ for some $\vert x\vert<1$. But I'm kind of stuck how to derive the matrix above, or something similar. Any ideas?

Answer 1

A projective transformation that fixes the unit circle is an automorphic collineation in the Beltrami-Klein model of the hyperbolic plane. So this is the projective version of the better known automorphism of a unit disk as taught in complex analysis courses.

This is neither here nor there, but there is a handy practical guide Hyperbolic Geometry in Klein’s Model (by Franz Rothe) that can help with your question. In particular, the proof of Proposition 3.6 (pg 813) gives four source/target point pairs that define the projective transformation you want.

The URL (on the wayback machine) should be fairly stable, but here is the crux diagram:

Points $(x_1,x_2,x_3,x_4)$ map to $(x'_1,x'_2,x'_3,x'_4)$. Presumably you can derive a matrix from there, using math software or methods like this.

Addendum: OP asked what's meant by "counting with multiplicity, five points are mapped to other five points" in Rothe's proof. This is a shorthand for the scenario where two points on a curve converge to one and the chord they define becomes a tangent. You see it for example when the hexagon in Pascal's theorem degenerates to a polygon with less than 6 sides(e.g. page 4-5 here). In general, two tangents and three points will define 4 conics. But it will be unique if two of the points are on the tangents. A fuller discussion of the two tangents and three points case can be found in Pamfilos' A Gallery of Conics by Five Elements, Section 4.