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I recently started reading The Princeton Companion to Mathematics. I am currently stuck in the introduction to hyperbolic geometry and have some doubts about its models.

Isn't the hyperbolic space produced by rotating a hyperbola? That is, isn't hyperbolic geometry carried out on a hyperboloid?

Also, how is the half plane model generated? Is it generated by taking the crossection of the hyperboloid on the upper half of the complex plane or by taking a projection?

Also, can someone explain to me how "hyperbolic distances become larger and larger, relative to Euclidean ones, the closer you get to the real axis". Is the axis referred to here the real axis of the complex plane or some other axis. And why do distances become larger the closer we get to the real axis? The part of the line close to the real axis simply looks like a part of a circle whose distance does not increase abnormally.

Any visuals will be appreciated.

Akshit
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  • What is your question. I think some people actually have the question "How is it possible that distance doesn't satisfy some of the properties I thought were so obvious?" meaning that it doesn't have some of the properties of the Euclidean metric. I can understand that question. It doesn't appear that that is the question you have. What actually is your question? I know that you already got an answer that solved your problem. I'm still curious what your question actually was. – Timothy Oct 11 '19 at 02:12

2 Answers2

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There are at least four "common" models of the hyperbolic plane:

  1. The "upper" sheet of the hyperboloid of two sheets in Minkowski space (a.k.a., the set of future-pointing unit timelike vectors): $$ x_{1}^{2} + x_{2}^{2} - x_{3}^{2} = -1,\quad x_{3} > 0. $$ Hyperbolic lines turn out to be the intersections of planes through the origin with the hyperboloid.

  2. The Klein disk model, viewed as the unit disk $$ x_{1}^{2} + x_{2}^{2} < 1,\qquad x_{3} = 1, $$ identified with the hyperboloid model by radial projection from the origin. Hyperbolic lines are Euclidean chords. (Patrick Ryan's Euclidean and Non-Euclidean Geometry is a good reference for these two models.)

  3. The Poincaré disk model, viewed as the unit disk $$ x_{1}^{2} + x_{2}^{2} < 1,\qquad x_{3} = 0, $$ identified with the hyperboloid model by radial projection from the point $(0, 0, -1)$ (diagram below). Hyperbolic lines are arcs of Euclidean circles meeting the boundary of the disk orthogonally. Unlike the Klein model, the Poincaré model is conformal; hyperbolic angles coincide with Euclidean angles.

  4. The upper half-plane model, also conformal, obtained from the Poincaré disk model via the fractional linear transformation $$ z \mapsto -i \frac{z + i}{z - i} = \frac{-iz + 1}{z - i}. $$ Hyperbolic lines are Euclidean semicircles (meeting the real axis orthogonally).

Projection from the hyperboloid to the Poincare disk

The Riemannian metrics in the Poincaré and upper half-plane models have well-known formulas in Euclidean coordinates $z = x + iy$: $$ ds^{2} = \frac{4(dx^{2} + dy^{2})}{\bigl(1 - (x^{2} + y^{2})\bigr)^{2}},\qquad ds^{2} = \frac{dx^{2} + dy^{2}}{y^{2}}. $$

Particularly, in the upper half-plane model, a Euclidean distance $ds = \sqrt{dx^{2} + dy^{2}}$ corresponds to a hyperbolic distance $ds/y$; as $y \to 0^{+}$, the hyperbolic length of a Euclidean segment of fixed length grows without bound.

Don Hatch has created an extensive gallery of hyperbolic tessellations (in the Poincaré model) that make this "length distortion" vivid. The "tiles" have fixed hyperbolic shape (and size), and their Euclidean representations shrink toward the boundary of the disk.

Another famous example is the Circle Limit series of prints by M. C. Escher. A web search for "Poincare disk" or "Poincare metric" should turn up many more diagrams.

Andrew D. Hwang
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  • "fractional linear transformation" please explain. I am almost certain of 2 things: that this type of conformal transform is called a Möbius transform, that this specific "fractional linear transform" is called the "Cayley Transform," and that lines become circles 100% of the time and that it does not have a representation via a 2x2 matrix (rather some other kind of matrix). Like literally let me just edit it – cmarangu Apr 10 '20 at 02:40
  • EDIT: please excuse my harsh tone. I see this is a formal term. Not sure why they would call a class of transforms which often transforms lines to circles and requires larger-than 2x2 matrix a "fractional linear transform," oh well I suppose many naming in math is misleading anyways "real numbers" "improper fraction" – cmarangu Apr 10 '20 at 02:46
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It looks like you are talking about the Poincaré half-plane model of hyperbolic geometry (see https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model )

Be aware this is only a model of hyperbolic geometry.

For doing "real-life" hyperbolic geometry you need a "surface with a constant negative curvature " or pseudo spherical surface see https://en.wikipedia.org/wiki/Pseudosphere

Also the hyperboloid is only a model of hyperbolic geometry (the curvature of an hyperboloid is positive and not constant, so wrong on both counts of the curvature )

Having said all this you seem to investigate the Poincaré half-plane model of hyperbolic geometry and indeed the scale of this model becomes 0 when you get near the x=0 line

also see

https://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane

for more information

Hopes this helps

Willemien
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