# How could you extend hyperbolic geometry beyond the edge of the Poincare disk?

How can I extend hyperbolic geometry beyond the edge of the Poincare disk?

The reason I'm asking has to do with the ideal shape for a mirror to focus light from different distances, according to the following idea:

• A circle focuses light from its origin.
• An ellipse focuses light emitted a finite distance from its first focus.
• A parabola focuses light from infinity.
• And so a hyperbola focuses light from beyond infinity? I say this last bit, partly joking.

When I said this to a great optics guy, he replied:

Personally I would formulate it a bit differently: the hyperbola can correctly focus a convergent beam (generally containing a spherical aberration). This in contrast to elliptical shape, where the beam coming from the object is divergent.

This made me wonder if there's an equivalence between light emitted from beyond infinity, and spherical aberation. I think spherical aberation makes point sources appear as if they were emitted from a finite area, rather than an infinitesimal point.

This makes me think that in this extension of the Poincare disk, light emitted from a point outside the disk would spread out, while bending in as it approaches the edge of the disk from the outside, then converge along a line segment within the disk, as would light focused by a spherical lens.

I thought it might be possible to to observe light being focused from beyond infinity, so to speak, in a simulation that extended the Poincare disk, by placing a point source beyond the edge of the disk, and then placing a hyperbola within the Poincare disk to focus the light emitted from the other side of the disk's edge.

I've seen an extension of Penrose diagrams, which seems to deal with crossing over infinities, but other than that, I'm not sure how to approach it. Maybe there are reasons the question doesn't make sense, or is impossible, and I'd like to hear those reasons.

Sorry for the long explanation, but the question is short: How can I extend hyperbolic geometry beyond the edge of the Poincare disk? Any ideas are welcome. Cheers.

Edit: Reading comments and answers, I see that focusing by conics in hyperbolic space is more complex than I imagined.

I now realize that I only want this hypothetical extension of the Poincare disk so that I can follow the path of light rays from a point beyond the finite region of the disk, back into the finite region of the disk.

The further propagation of those rays within the finite region and their focusing, could be done after applying the inverse transform to take the Poincare disk and light rays back into Euclidean space, where the focusing properties of conics would be less difficult.

I guess here I'm assuming that you can transform back and forth between the Poincare disk and Euclidean space. That might be a whole question on its own.

I really appreciate all the input. It's fascinating. Many thanks.

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– Pedro
Jun 17, 2021 at 13:00

How can I extend hyperbolic geometry beyond the edge of the Poincare disk?

I'm coming from a background in projective geometry. From that angle I'd say the part of the plane outside the unit circle in the Poincaré model is essentially a reflection of the part within. The two are related by an inversion in the unit circle. Personally I'm inclined to simply identify the points outside with those within. Examples:

• A geodesic is usually described by a circular arc orthogonal to the unit circle, but adding that mirror image makes it a full circle.
• Intersecting two geodesics you need to intersect two circles then usually pick the intersection inside the unit circle. But if you include that mirror image, both intersections together represent the same point, and it doesn't matter which one you pick to describe that point.
• Hyperbolic isometries can be described as either Möbius or anti Möbius transformations that leave the unit disk fixed. An anti Möbius transformation adds a complex conjugation to express orientation-reversing transformation. But with the mirror image included, you can instead use Möbius transformations that exchange the inside and the outside of the unit circle, allowing you to avoid the complex conjugation which makes some things easier.

That said, the Beltrami-Klein model does allow for points outside the unit circle, and those are indeed separate points. Their distance to points in the inside would be a complex number, the logarithm of a negative number. The fact that this model has a somewhat unintuitive angle metric makes the behaviour of mirrors harder to grasp, though.

... placing a hyperbola within the Poincare disk ...

It is unclear what exactly a hyperbola in hyperbolic geometry would be. Your reference to the Poincaré disk suggests you might be referring to a hyperbola in the Euclidean plane where the Poincaré disk is embedded. Such a reinterpretation of geometries from one world in the other may seem fairly unmotivated, and the placement within the model would likely impact it's geometric properties.

The concept of a conic section can be expressed as pure incidence geometry, e.g. using Pascal's theorem. Since the Beltrami-Klein model models geodesics using (segments of) Euclidean lines, the conic sections defined in this fashion would correspond to conic sections in the Euclidean plane into which the Beltrami-Klein disk model is embedded. I feel this would be a more reasonable generalisation.

The classification into ellipses, parabolas and hyperbolas however is inherently Euclidean. Taking a conic section, you essentially distinguish whether it has zero, one or two intersections with the line at infinity. In the Beltrami-Klein model, the unit circle corresponds in a way to the line at infinity. More specifically, the line at infinity seen as a degenerate conic with multiplicity two plays that role for Euclidean geometry. So you could say that an ellipse has two complex conjugate points of intersection of multiplicity 2 each. A parabola has one real point of multiplicity 4, and the hyperbola has 2 real points of multiplicity 2 each. Intersecting the unit circle with conics, a lot of additional cases would enter the picture.

In the following section I'll just assume some of the typical properties of an Euclidean parabola and hyperbola, but I'm not sure they can actually be achieved by anything we would consider a conic section in hyperbolic geometry.

I thought it might be possible to to observe light being focused from beyond infinity...

I wonder whether you are looking at the wrong curvature sign here. I'll explain that in a moment.

First off, let's reverse the direction of the rays of light. Start with an omnidirectional point source in the intended focus, then watch where the rays go. In the Euclidean plane, the ellipse will have rays from one focus meet again at the other focus. The parabola will have rays from the focus become parallel, but for the hyperbola those rays would be divergent.

So now so the same in a curved surface, taking the parabola as an example. If that parabola is small and close to the focus, compared to the matrix of the plane, then the behaviour will be almost Euclidean. So without having done a rigorous examination, is assume that the rays coming from that would be close to rays that have a common orthogonal line in that area. For the Euclidean case, rays having a common orthogonal are parallel, so this is one of several generalisations is that concept.

Now in hyperbolic geometry, rays that have a common orthogonal will still diverge from one another, with distances between points increasing the further you move away from the common orthogonal. So even for something where rays stay the same distance apart in Euclidean, they now move father apart. This is the opposite of what you want.

Take spherical geometry on the other hand. Lines with a common orthogonal will still intersect in a single point, a finite distance away. The geodesics orthogonal to the equator of the earth meet in the poles. So even a parabola-like reflection should stand a fair chance to focus light from a source a finite distance away. A small team from there on the direction of a hyperbola should be able to preserve that property.

All of this hinges on my assumption that a small enough mirror wouy still essentially work like its Euclidean counterpart at small scales. I wouldn't be surprised to learn that this is only an approximation, and therefore you only get your rays to meet in approximately the same point. Our conversely, if it's actually the same point on both ends you might need a mirror that's only approximately a hyperbola. There might be more bath to be done here to actually find out.

... spherical aberration ...

To be honest, I don't know enough optics to know how this figures in this whole idea. I might be missing something important here.

And so a hyperbola focuses light from beyond infinity? I say this last bit, partly joking.

Not a joke, it is well known in optics.. a single nappe of a hyperboloid of two sheets that collects all radially converging rays into one virtual focus of hyperboloid $$F_v$$ are re-directed to the other real converging focus $$F_r$$ but after a reflection. ( A sketch is given for confirmation).

( Tangents gray, normals in red, rays before and after reflection brown). Geogebra rough sketch.

Btw, a spherical aberration does not give an accurate convergence, the points do not arrive in phase and in general are not aplanatic. Only paraxial rays near the axis of symmetry focus well. Parabola reflection removes the caustic fully for all rays parallel to axis. For rays not parallel to axis even these parabolic reflections leave behind a caustic.

Earlier I had stated Hyperbolic geodesics can exist beyond the Poincare boundary

Next, with reference the following optics related ellipse foci idealization of this question it can be stated that an incoming ray ( geodesic) redirects to the other focus endlessly , so that all rays coming in ( converge ) to a single point on the Poincare disc boundary analogously can go out ( diverge ) from the same point as a pole or point at infinity on the boundary along chosen (shown four) hyperbolic geodesic rays shown here in the "out of bounds" region: