by **Marek14** » Mon Jun 15, 2015 3:24 pm

Well, the foci of the conics look pretty fun.

So, each hyperbolic conic has four "absolute tangents"; these are Euclidean lines tangent to both the horizon (or absolute) and to the conic itself. These absolute tangents can be real or imaginary.

The four absolute tangents intersect in six points (three pairs) which are the foci of the conic. If a focus lies outside the horizon, then we can consider its polar line instead, the directrix.

I think that the foci have special reflective properties. In Euclidean ellipse, a ray sent from one focus will reflect into the other. In hyperbola, it will reflect, as if it was coming from the other focus. In both cases, a tangent to the curve at a point is an axis of some angle between straight lines containing that point and first, resp. second focus.

If this holds in hyperbolic plane, then a conic would reflect rays in various ways.

Double real focus: circle

Any ray from the center will be reflected back to the center.

Two real foci: ellipse or convex hyperbola

These would work just like the Euclidean case, ellipse sending all rays from one focus to the other and convex hyperbola sending rays from one focus as if they were coming from the other.

One real and one ideal focus: elliptic parabola and convex hyperbolic parabola.

In elliptic parabola, rays from the real focus will be reflected towards the ideal focus, along one of convergent parallels.

In convex hyperbolic parabola, rays from the real focus will be reflected as if they were coming from the ideal focus.

One real focus and one directrix: semihyperbola

Rays from the focus will be reflected perpendicular to the directrix. The directrix will be perpendicular to the axis of semihyperbola, but it might have different positions. If it passes through the semihyperbola, some rays reflected from the focus will converge before passing the directrix and starting to diverge. But it might also lie entirely outside the semihyperbola, in which case the rays from focus will always diverge after reflection.

One double ideal focus: horocycle

Any ray from the focus will rebound back into it.

Two ideal foci: equidistant

Any ray from one ideal focus will be reflected to the other one. This suggest that an equidistant-shaped mirror would be very useful for distance optical communication as it would guide any ray towards its destination with only one rebound! So an equidistant conic has 2 independent focus systems.

Ideal focus and directrix: 2-branched concave hyperbolic parabola

Rays from the ideal focus will reflect perpendicular to the directrix.

One double directrix: equidistant

Apart from "two ideal foci", equidistant can be also considered to have a double directrix (the line it's equidistant from). Any line perpendicular to it will be also perpendicular to the equidistant, so it will reflect back.

Two directrices: here, ray perpendicular to one directrix will be reflected in a direction perpendicular to the other.

The directrices can intersect (concave hyperbola), be convergent parallels (1-branched concave hyperbolic parabola), or be ultraparallels (concave hyperbola). In case of concave hyperbola, there seem to be three focus pairs, one with intersecting directrices and two with ultraparallel ones.

The only case not explored yet is the osculation parabola. We can imagine this as a limit case of 2-branched concave hyperbolic parabola, where we keep position of one branch and move the other branch to infinity. This conic has an ideal focus and a directrix like 2-branched concave hyperbolic parabola, but it's special: the directrix actually passes through the ideal focus.