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.

  • $\begingroup$ I guess you may need to be more precise about what you mean by hyperbolic conic. I would assume that you could define them e.g. as conics in the Beltrami-Klein model, or using counterparts to the Euclidean distance definitions with foci and perhaps a directrix. I would be surprised if these definitions were to coincide nicely. I would assume the distance definitions will lead to a curve of degree higher than 2, making them harder to handle. But that's just guessing. If you go for conics in the Beltrami-Klein model, then the existence of foci is far from clear. $\endgroup$
    – MvG
    Jun 3, 2018 at 21:19
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    $\begingroup$ Articles I found were basically based in Beltrami-Klein model and they classify conics by number and type of their intersections with the horizon. A Russian article I saw (mathnet.ru/links/0116e85b4ef8e4fdf19cbe340a1eb771/ivm1975.pdf) showed foci and axes and all that, as far as I could understand it. Though right now I found this: arxiv.org/pdf/1603.09285.pdf. So I should probably look through that first. $\endgroup$
    – Marek14
    Jun 4, 2018 at 17:30
  • $\begingroup$ Can't read Russian. That Chao and Rosenberg article does look interesting. One thing worth noting: if you are looking at Beltrami-Klein model, then one way to translate the definition of a focus into this world would be defining it as the intersection between common tangents of your conic and the fundamental conic (i.e. horizon). In general this will yield 4 complex tangents, resulting in 6 points of intersection. 2 of them appear to be real in general, so you might restrict investigation to those but the algebra should hold for any tangent-disjoint pair. Reflective property seems to hold. $\endgroup$
    – MvG
    Jun 5, 2018 at 7:45
  • $\begingroup$ MvG: Yes, that's basically what the Russian article says :) $\endgroup$
    – Marek14
    Jun 5, 2018 at 7:58


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