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I have a nice description of the geodesics of hyperbolic space in the hyperboloid model as intersections of $2$-planes with the hyperboloid, as given in this answer. As discussed there, if such a geodesic goes through $p$ with unit velocity $v$, it is parametrized by

$$\gamma_v^p(t) = (\cosh t)p + (\sinh t)v.$$

Moreover, I know how to map the hyperboloid model to the Poincaré ball and half-space models as in this answer:

\begin{align*} \alpha:\begin{pmatrix} y_0 \\ y_1 \\ y_2 \\ \vdots \\ y_{n-1} \\ y_n \end{pmatrix} &\mapsto \frac1{y_0+1} \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1} \\ y_n \end{pmatrix} =: \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1} \\ x_n \end{pmatrix} \\ \beta:\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1} \\ x_n \end{pmatrix} &\mapsto \frac{2}{1-2x_n+\lVert x\rVert^2} \begin{pmatrix}x_1\\ x_2\\ \vdots\\ x_{n-1}\\ 1-x_n\end{pmatrix} - \begin{pmatrix}0\\ 0\\ \vdots\\ 0\\ 1\end{pmatrix} =: \begin{pmatrix}z_1\\ z_2\\ \vdots\\ z_{n-1}\\ z_n\end{pmatrix} \end{align*}

I’d like to check that the images of the geodesics in the hyperboloid model under these maps are what I expect (circles intersecting the boundary of the ball perpendicularly in the ball model and vertical lines and half-circles in the half-space model). How could I do this?

Attempt: To map a geodesic $\gamma_v^p$ from the hyperboloid model to the Poincaré ball model, I first write $\gamma_v^p$ in coordinates. To simplify the task, I first perform a rotation of $\mathbb R^{1,n}$ in the last $n$ coordinates to get $p$ in the form $p=(p_0,p_1,0,0,…,0)$, where $p_1 = \sqrt{p_0^2-1}$ and $p_0\ge 1$. Using $\langle p, v\rangle = 0$ and $\langle v, v\rangle = 1$ and performing a second rotation in the last $n-1$ coordinates, I also get $v=(v_0,v_1,v_2,0,0,…,0)$ where $v_1=\frac{v_0p_0}{\sqrt{p_0^2-1}}$ and $v_2=\sqrt{\frac{p_0^2+v_0^2-1}{p_0^2-1}}$. Then, $$\alpha(\gamma_v^p(t)) = \left(\frac{(\cosh t)\sqrt{p_0^2-1}+(\sinh t)\frac{v_0p_0}{\sqrt{p_0^2-1}}}{(\cosh t)p_0+(\sinh t)v_0+1},\frac{(\sinh t)\sqrt{\frac{p_0^2+v_0^2-1}{p_0^2-1}}}{(\cosh t)p_0+(\sinh t)v_0+1},0,0,…,0\right).$$ How can I see from this equation that $\alpha(\gamma_v^p(t))$ parametrizes the intersection of a circle with the ball?

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Here's how I would approach this:

First, using the half-space and ball models only:

  1. Verify that in the half-space model vertical lines are geodesics and in the ball model lines through the origin are geodesics
  2. Verify that given any two geodesics, there exists an isometry that maps one geodesic to the other
  3. Use the explicit formulas for hyperbolic isometries to verify that geodesics in either model are circular segments that are orthogonal to the boundary

Now check that:

  1. The map from the hyperboloid model to the half-space model maps any intersection of the hyperboloid with a vertical 2-plane to a vertical line

  2. The map from the hyperboloid model to the ball model maps any intersection of the hyperboloid with a vertical 2-plane to a line through the origin.

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