The action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane
We have seen that the fixed points and qualitative behavior of the action of $\sigma\in \mathrm{SL}_2(\mathbb{R})$ on the upper half plane $H^2$ (via linear fractional transformation) depends on the sign of $\mathrm{tr}^2(\sigma)-4$. The animations below were created via this sage worksheet.
Elliptic transformations
If $\mathrm{tr}^2\sigma < 4$, we call $\sigma$ an elliptic transformation. In this case, $\sigma$ has a unique fixed point in $H^2$ and acts as a "hyperbolic rotation" about its fixed point. In particular, all the hyperbolic circles centered at the fixed point are stable under $\sigma$, and the geodesics passing through the fixed point are taken to each other. The following animation illustrates how $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ acts on some geodesics passing through $i$ as $\theta$ varies from $0$ to $2\pi$.
Parabolic transformations
If $\mathrm{tr}^2\sigma = 4$, we call $\sigma$ a parabolic transformation. This is a sort of limiting case of elliptic transformations where the fixed point is actually on $\mathbb{R} = \partial H^2$. The circles parallel to $\mathbb{R}$ at the fixed point are stable under the action, and geodesics with one endpoint at the fixed point are taken to each other. The following animation illustrates how $\begin{pmatrix} 1 & 0 \\ t & 1\end{pmatrix}$ acts on some geodesics with endpoint $0$ as $t$ varies from $0$ to $12$.
Hyperbolic transformations
When $\mathrm{tr}^2\sigma > 4$, we call $\sigma$ a hyperbolic transformation. Here $\sigma$ has two distinct fixed points on $\mathbb{R} = \partial H^2$ and we think of $\sigma$ as a "hyperbolic translation." Arcs of circles passing through both fixed points are stable under $\sigma$, and geodesics with one of the fixed points as center are taken to each other. The following animation illustrates how $\begin{pmatrix} e^t & 0 \\ e^t-e^{-t} & e^{-t} \end{pmatrix}$ acts on some geodesics with center either $0$ or $1$ as $t$ varies from $0$ to $3$.
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Kyle M. Ormsby