Mapping an element of the euclidean group to an isometry of a horosphere in the hyperboloid model

Question

Horospheres have the geometry of euclidean space. I was wondering how to take an element of the euclidean group and map it to the isometry group of the horosphere, as represented in the hyperboloid model. For the purpose of this question, we will consider the horosphere which contains the origin and with the center in the direction of the positive x-axis.

Answer 1

Suppose that the element of the euclidean group is $\vec x \mapsto A \vec x + \vec b$, where $A$ is an orthogonal matrix. Then, in the hyperboloid model, the isometry is represented by the following block matrix:

$$\begin{pmatrix} 1 + \frac {\|\vec b\|^2} 2 & -\frac {\|\vec b\|^2} 2 & \vec b ^ \top A \\ \frac {\|\vec b\|^2} 2 & 1 - \frac {\|\vec b\|^2} 2 & \vec b ^ \top A \\ \vec b & - \vec b & A \end{pmatrix}$$

This transformation preserves any horosphere whose ideal center is in the direction of the positive x-axis.

You can use conjugation to find isometries around other ideal points, of course.