In the Poincaré disk model geodesics are defined as all co-axal circles of unity tangent length orthogonal to unit boundary circle with $K=-1$. ( They can be intersecting, parallel through a single boundary point or be also ultra parallel).
When they are not geodesics, one is naturally curious to know how the curves of constant hyperbolic geodesic curvature $ k_{gH}$ are defined in the Poincaré model.
Iirc, numerical integration results in polar curves pass through the origin. But at first can we get to see the ode and sketches from standard references? How are the second fundamental form and the Liouville's formulation modified?
Thanks in advance.
EDIT 1:
The following is not a Poincaré or other hyperbolic model describing constant geodesic deviation.
There are four models in hyperbolic geometry. I have introduced a fifth one described in what follows.
I had authored to define hyperbolic geodesics as inversions with respect to a circle having center at origin in a 2d polar form. Inversion circle not shown here.
An inversion process in my model changes nature of geodesics and from zero elliptic to zero hyperbolic curvature. It reverses the sense of angle rotation from clockwise to anticlockwise and vice-versa depending on the chosen angle convention.
The following image depicts elliptic and hyperbolic geodesics (green, black lines).
$$ \psi'\pm \frac{\sin \psi}{r} =\kappa_{g ~Elliptic,~Hyperbolic} $$
Let $\kappa_{g(Elliptic,Hyperbolic)}=0$ and integration leads to the two geodesic types with the two minimum and maximum distance invariants as integration constants from origin as:
$$ r \cdot \sin \psi~~ \text {and }\frac{r}{\sin \psi}$$
These definitions of either curvature result in geodesics viewed as all straight lines and co-origin passing circles respectively. The latter Hyperbolic geodesic passes through the origin. The first one is the Clairaut elliptic constant and the second hyperbolic constant was introduced here. In 3d the elliptic geodesics are skew to cartesian axes and hyperbolic geodesics are asymptotic lines in all negative surfaces.
The above sketch shows both Curvature lines $$ k_{g~Elliptic, Hyperbolic} =0.1 ~ $$
passing through a common tangent point $(2,0)$ (blue and magenta) curvature lines respectively and the hyperbolic geodesic (green).
When applied to 3d we obtain the Chebyshev Net lines satisfying Sine Gordon pde asymptotic lines that can be seen as hyperbolic geodesics in my model.