Wikipedia:Reference desk/Archives/Mathematics/2023 July 13

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July 13

sources about antipodal points in hyperbolic geometry

Lexell's theorem: Orange triangles △ABC share a base AB and area. The locus of their apex C is a small circle (dashed green) passing through the points antipodal to A and B.

In the half-plane model, antipodal points are reflections into the opposite half-plane (shaded gray). Hyperbolic triangles △ABC (orange) share a base AB and area. The locus of apex C is a hypercycle (dashed green) passing through points antipodal to A and B.

Just like spherical geometry, hyperbolic geometry has a perfectly useful and reasonable concept of antipodal points, but in my (not very exhaustive) skimming through hyperbolic geometry sources, basically nobody ever describes or uses it. To be precise, in the hyperboloid of two sheets embedded in a pseudo-Euclidean space of signature $(-,+,+)$ (the "hyperboloid model"), the antipodal transformation is a point reflection across the origin, so that two antipodal points are on opposite sheets of the hyperboloid. Just as on the sphere, a geodesic ("straight line") is the intersection between the hyperboloid and a plane through the origin. Any generalized circle (which for either the sphere or the hyperboloid is the intersection with an arbitrary plane) that passes through a pair of antipodal points must be a geodesic. In the conformal disk model (stereographic projection of the hyperboloid centered at an ordinary point), the antipodal transformation is an inversion across the boundary circle of the disk into the other paired hyperbolic plane found in the complement of the disk. In the conformal half-plane model (stereographic projection of the hyperboloid centered at an ideal point), the antipodal transformation is a reflection across the linear boundary of the half-plane into the other hyperbolic plane found in the lower half-plane. (In the Beltrami–Klein model (gnomonic projection of the hyperboloid), both antipodal points project to the same point, just as in spherical geometry; the exterior of the disk in gnomonic projection instead represents points on the hyperboloid of one sheet, a model for the De Sitter plane.

Anyway, the only source I can find discussing hyperbolic antipodal points is:

Johnson, Norman W. (1981), "Absolute Polarities and Central Inversion", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift, Springer, pp. 443–464, doi:10.1007/978-1-4612-5648-9_28 (see especially p. 452ff.)

I can find a few other sources which discuss points in the second hyperbolic plane, but not which explicitly talk about antipodal points.

What I am wondering here is whether anyone here knows of other sources about this. Are there historical (or modern) sources about hyperbolic geometry which mention antipodal points at all? I really can't figure out the oversight; antipodal points seem like a basic and important tool, very useful when e.g. trying to implement code or use interactive geometry software to draw pictures of hyperbolic geometry. I was expecting to easily find dozens of sources. So I'm kind of puzzled. –jacobolus (t) 08:35, 13 July 2023 (UTC)

A definition of antipodal pair for arbitrary metric spaces – which appears to imply that only spaces of finite diameter can have such pairs – is given in: P. G. Hjorth, S. L. Kokkendorff, and S. Markvorsen (2002). "Hyperbolic spaces are of strictly negative type". Proceedings of the American Mathematical Society. 130 (1): 175–181.{{cite journal}}: CS1 maint: multiple names: authors list (link)

For a different concept, creating a potential ambiguity, see: Károly Bezdek, Márton Naszódi and Déborah Oliveros (2006). "Antipodality in hyperbolic space". Journal of Geometry. 85 (1–2): 22–31. doi:10.1007/s00022-006-0038-0.  --Lambiam 09:36, 13 July 2023 (UTC)

That second one is an entirely unrelated concept which should never been called "antipodal" in my opinion. People need to stop just picking whatever vaguely evocative name first comes to mind when making up new concepts. :-) –jacobolus (t) 10:57, 13 July 2023 (UTC)

Not entirely unrelated; it was conceived by Victor Klee as a generalization of the spherical sense to arbitrary convex bodies. See here, section K11.  --Lambiam 13:46, 13 July 2023 (UTC)

Oh I was misunderstanding the idea. I see, this is a generalization of the spherical concept to include arbitrary blobs, as in rotating calipers. I don't think there's much risk of confusion between these distinct concepts.

We can also more specifically look for antipodal points on circles/spheres in hyperbolic space, which work more or less the same as in Euclidean space. We can also look for antipodal points on a hypercycle (a "diameter" being a a geodesic orthogonal to the hypercycle) with the caveat that there's no well-defined "center"; or even on a horocycle (every point of which is is antipodal to the ideal point where the horocycle is tangent to the boundary). –jacobolus (t) 16:00, 13 July 2023 (UTC)

Your "antipodal points" appear to be in different connected components of your space. For the embedded hyperboloid, all of the things you say seem to be connected to your embedding, not to any intrinsic property of the space, unlike the antipodal points in a sphere (which can be recovered by various intrinsic concepts like points of maximal geodesic distance or as Conjugate points or as cut locus). —Kusma (talk) 09:52, 13 July 2023 (UTC)

They are indeed in different connected components, but they are absolutely an "intrinsic property" of the (double) hyperbolic plane. The two separate components always transform together, despite being separated; a hypercycle or geodesic can cross the boundary and include points in both components. The situation is almost exactly analogous to the sphere, except for the difference that the double hyperbolic plane is separated while the sphere is not. (In just the same way that the hyperbola has 2 branches whereas the circle does not.) You can measure distance between points on the two branches, but you can't use hyperbolic angle measure for it. You instead have to use a more appropriate canonical representation like a unit-magnitude hyperbolic number (analogous to using a unit complex number to represent a circular angle) or the hyperbolic half-tangent. (If you start from a hyperbolic angle measure $\psi$ between points on the same branch, you can get the hyperbolic half-tangent as $\tanh {\tfrac {1}{2}}\psi ,$ but for points on separate components the hyperbolic half-tangent is >1, so the hyperbolic angle measure is not well defined per se.) –jacobolus (t) 10:53, 13 July 2023 (UTC)

Can you think of a direct intrinsic definition?  --Lambiam 13:27, 13 July 2023 (UTC)

I guess this depends on what you consider to be "intrinsic". One definition could be: take two lines through the given point and find their second intersection: that is the antipodal point.

(As for the circle, this definition breaks down in the 1-dimensional hyperbola case; in that case you need to use some notion based on maximal distance; for both hyperbola and circle you could say e.g. the antipodal point is the unique point with half-tangent distance of $\infty$) –jacobolus (t) 15:45, 13 July 2023 (UTC)

By "intrinsic definition" I mean a definition given in terms of the metric tensor of the manifold, without reference to any models given by embedding it in other spaces.  --Lambiam 17:17, 13 July 2023 (UTC)

By "intrinsic" you seem to mean "local" or "continuously measurable along a path" or something like that. But that's too limiting a concept for a disconnected structure like this. After all, to get to the ideal points at the boundary, you need to go "infinitely far" if hyperbolic angle measure is your concept of distance. So if you want to get to the antipodal point you need to go "infinitely far" two times over. But we can still make meaningful sense of the "distance" or "separation" between points on opposite branches of the hyperboloid (hyperbolic double plane), and this concept is not "extrinsic" in the sense of relying on some external structure; it is inherent in the geometry of the double plane, irrespective of representation. But this may explain why such a concept hasn't really showed up too much in the literature: if people try to apply concepts of "local" Euclidean geometry to the inherently separated hyperbolic geometry, they end up missing obvious structures, because the mismatched conceptual frame serves as a kind of blinkers. –jacobolus (t) 17:33, 13 July 2023 (UTC)

I thought hyperbolic space is a simply connected manifold.  --Lambiam 23:15, 13 July 2023 (UTC)

Well again, it's a matter of definition. I think by far the most natural hyperbolic analog of the sphere is what we might call a "double plane" with two disconnected parts. Obviously that version hasn't been studied as much. That is why I am asking for references. –jacobolus (t) 23:41, 13 July 2023 (UTC)

For context, I've been working on a draft about Lexell's theorem. I added a couple pictures in this section which might give folks a clearer idea. –jacobolus (t) 11:03, 13 July 2023 (UTC)

Some quick mentions without too much systematic study:

Coxeter (1979) "Angles and Arcs in the Hyperbolic Plane", "The investigation of these cycles is facilitated by using one of Poincare's Euclidean models for the hyperbolic plane. In this model the ends appear as the points of a horizontal line Ω, and the hyperbolic lines are represented by the circles and lines orthogonal to Ω, that is, by the circles with their centres on Ω, and the vertical lines. [...] The point of intersection of the two lines thus appears as a pair of points which are images of each other by reflection in Ω, or as a single point in the upper half-plane (if we choose to disregard the lower half-plane and represent lines by semicircles and vertical rays)."

Eriksson & Lagarias (2005) "Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings" "However we will allow hyperbolic Apollonian circle packings which lie in the entire plane. We have a second copy of the hyperbolic plane which is the exterior of the closed unit disk, and we use the convention that (positively oriented) circles entirely contained in the exterior region have the sign of their curvature reversed, in which case coth r ranges from −1 to −∞. Euclidean circles that intersect the ideal boundary (unit circle) are treated as hyperbolic circles of pure imaginary radius, and these cover the remaining range −1 ≤ coth r ≤ 1. A hyperbolic circle of curvature 0 is a hyperbolic geodesic, which corresponds to a Euclidean circle which intersects the ideal boundary (unit circle) at right angles, a "

A source that gets slightly more into it is:

Akopyan (2009) "On some classical constructions extended to hyperbolic geometry" "Theorem 4 (analogue in hyperbolic geometry of Lexell’s theorem). Let points A* and B* be the inverses of points A and B with respect to the absolute circle of the Poincaré disk model. Let ω be any Euclidean circle passing through A* and B*. Then for any point X on ω, the area of triangle XAB (in the hyperbolic sense) is constant, i.e., will not depend on the choice of X" [...] "The second question that occurs is what is the “analogue” of the points A* and B*, which are the inverses of A and B in the Poincaré disk model? Note that the point A has the property that any line through A (which is a circle in the Poincaré disk) also passes through the point A*. On the sphere, this property attaches to the antipode of A, the point symmetric to A with respect to the center of the sphere."

Akopyan also includes a suitable distance measurement (which I would call hyperbolic half-tangent or stereographic distance): "Let the pseudolength d_E(A,B) of a hyperbolic line segment AB be defined as follows. After a suitable motion that places A at the center of a Poincaré disk of radius 1, the pseudolength d_E(A, B) is defined as the Euclidean length of segment AB."

Note that Eriksson & Lagarias's concept of hyperbolic curvature is the reciprocal of Akopyan's "pseudolength" of the diameter of a hyperbolic circle, horocycle, hypercycle, or geodesic. Caratheodory (1954) Theory of Functions of a Complex Variable, §§1.3.86–88, calls it the "pseudo-chordal distance" in the context of the complex unit disk as a model for the the hyperbolic plane. –jacobolus (t) 20:42, 13 July 2023 (UTC)

In "On hyperbolic analogues of some classical theorems in spherical geometry" (doi:10.2969/aspm/07310225) a proof is given of the hyperbolic analogue of Lexell's theorem. Curiously, the article states, "In the spherical case the solution is different. The locus of the vertices of the triangles having a given basis and a given area is a horocycle passing through the points antipodal to the extremities of the bases; this fact has no immediate analogue in the hyperbolic case."  --Lambiam 09:57, 14 July 2023 (UTC)

Yes, I don't think Papadopoulos and Su had by the point of writing that known about Akopyan's paper. They construct the same hypercycle in a slightly roundabout way by finding the "midpoint geodesic" which passes through the same ideal points (it is parallel to the hypercycle in the sense that the two curves remain equidistant from each-other at every point along their length, like parallel circles on a sphere or concentric circles in the plane), but they did not notice that it passes through the analog of antipodal points in the paired opposite hyperbolic plane. –jacobolus (t) 13:00, 14 July 2023 (UTC)