In Euclidean geometry Thales' Theorem says that if you view a diameter of a circle from any point on the perimeter it occupies exactly $90$ degrees in your field of view.
More generally for any segment $s$ and any angle $\theta$ the set of points from which $s$ occupies exactly an angle of $\theta$ in the field of view is the union of two circle arcs. This follows from the Inscribed Angle Theorem.
In hyperbolic geometry Thales' theorem is false. In fact, there is a distance $d > 0$ such that if you are at distance $d$ from a geodesic then the whole thing occupies exactly $90$ degrees in your field of view. Therefore on a circle of radius $d$ or more there are points from which a diameter occupies less than $90$ degrees.
At least in the case of a geodesic the set of points from which it occupies $90$ degrees is a nice and well known curve (i.e. an equidistant: a curve at a fixed distance from a geodesic). In fact the curve is even an Euclidean circle arc in either the Poincaré disk or the upper half-plane model.
My question is the following: Given a finite segment $s$ in the hyperbolic plane, what does the set of points from which the segment occupies exactly $90$ degrees look like? Is it a well known curve? Has it appeared or been used in relation to other questions?
Also one can ask the same questions for a general angle $\theta$.
Edit: Just realized this is has been asked before on MO. Sorry.