In Euclidean geometry, conics have many interesting properties. We can define them as a geometric places (points on an ellipse all have constant sum of distances from two foci, points on a hyperbola all have constant difference of distances, points on a parabola are all equally distant from a given point and a given line). We can also study their reflective properties -- ellipse reflects rays from one focus to converge in the other focus, parabola reflects rays from the focus to be parallel to the axis, hyperbola reflects rays from one focus into rays coming from the other focus).
Seeing that hyperbolic plane has greater variety of possible conics, are properties along this line known for them?
Edit: A bit of reflexion (pun intended) made me realize that the three reflection cases in Euclidean system are actually the same thing if we add the line at infinity. Parabola has this line as a tangent while hyperbola has it as a secant. So all three have the same property of reflecting lines from a pencil passing through one focus into lines from a pencil passing through the others. Parabola has one focus in infinity and a pencil passing through a point at infinity is a bunch of parallel lines.
If this works in hyperbolic plane as well, then a hyperbolic conic can have three types of foci: real (pencil is a set of straight lines passing through a given point), ideal (pencil is a set of mutually convergent lines) and ultraideal (pencil is a set of lines that are all perpendicular to a given line). So, let's say, a semihyperbola would have one real and one ultraideal focus, so rays coming from the real focus would be reflected into rays perpendicular to a line dual to the ultraideal focus (and semihyperbolas would thus naturally separate into three sub-cases based on the mutual position of semihyperbola and that line).
Still no idea if hyperbolic conics can be defined as geometrical places of points with specific properties, though.