Talk:Poincaré half-plane model

Key to hyperbolic geometry

This article is an important portal to the general reader who might be interested in getting started in hyperbolic geometry.As such I have attempted to show how the half-plane model needs only elementary geometry for its foundation.The introduction I composed for this page has been deleted by User:Linas for reasons he considers adequate.I am requesting Wikipedians to look at the introduction I wrote (click on history) and consider his reasons and the content of my logical introduction.Hyperbolic geometry will remain an obscure and arcane topic if an adequate inroad is not prepared.The complete reversion of my introduction is contrary to the dispute-avoidance policy posted on this encyclopedia.I hope other editors will come forward to take my side in this incipient conflict.Rgdboer 18:28, 17 September 2005 (UTC)

Hi Rgdboer; please let me extend my apologies. Can you repost that intro here, and we can perhaps copy-edit a bit? When I read your intro, I saw some problems, which I then attempted to copy-edit. Before long, I realized I was re-writing the whole thing from scratch, and, as it was late and I was tired, I did not want to do. I decided is was easier to block-delete, and wait for you to challenge. :) Which I guess you did. Again, copy it here, and we can discuss and edit. linas 22:02, 17 September 2005 (UTC)

Introduction in vet

Prelim remarks:

• Please don't use HTML markup for line breaks, paragraphs. Its not needed on wikipedia, and makes things look ugly.
• An "abcissa" is not the same thing as the "x-axis", and the text below confuses the one for the other, that makes it hard to read and understand.
• Not every word has to be hyperlinked. Its acceptable to assume the reader is familiar with the most basic concepts.

After picking apart the intro, I started to understand what you were trying to do, and, in general, I agree with you. This topic does need a "ruler-and-compass" description, using just basic trig and no appeal to complex numbers, which would be accessible to a broader audience (I'm thinking of high school students and the lay public). The current article, as written, assumes a college math degree, and so is fairly horrid in that respect. However, I also suspect that such a non-calculus type presentation might be best done in a distinct article, as mixing it up with this one just may lead to trouble.

On to the intro, which I put in italic here:

>>>Slightly modified version of introduction posted and deleted by Linas >>>Rgdboer 16:41, 22 September 2005 (UTC)

Double-indented comments by me. linas 18:37, 23 September 2005 (UTC)

The half-plane is the key to hyperbolic geometry because it is accessible with elementary Euclidean geometry and hence provides a model needed to prove consistency of the theory of the hyperbolic plane.

I had trouble imagining what might have been "inconsistent", so this sentence confused me. The link to model points at model theory which is something totally and completely different.

Some of the "lines" of the upper half-plane model are semicircles with diameter on the abscissa.

They're not lines, they're geodesics. Its not "some of them", its "all of them". The abcissa is not the same as the x-axis. Thus, reword as:

The geodesics of the the upper half plane model are either semi-circles whose origin lies on the x-axis, or are vertical lines (which can be thought of as semi-circles whose origin lies at infinity).

Given two points in the upper half-plane, the perpendicular bisector of the segment between them may intersect the abscissa.

Is the perpendicular bisector another geodesic or a Euclidean line? If the former, then *all* such bisectors intersect twice (once at infinity for vertical lines). From the next sentence, I'm guessing "line" means "Euclidean line".

If so, the point of intersection is a center for a semicircular arc connecting them.

This is a curious theorem from basic Euclidean geometry, but I am not sure why this needs to be stated in the introduction (or why it needs to be stated at all). Other than that semi-circles are involved, I don't see any particular connection between this theorem and hyperbolic geometry.

If the perpendicular bisector is parallel to the abscissa,

There are no geodesics that are parallel to the x-axis, so this is wrong if "line" is a geodesic. So I guess you mean "Euclidean line", but this is contradicted by the next sentence.

then the points lie vertically in the plane and are connected by the vertical ray to the abscissa.Thus there is a ruler-and-compass construction of the "line" between any pair of points.

Using "line" in quotes implies its a geodesic. Also, per above, I find this ruler-and-compass construction marginally curious, but don't see why it is important.

The metric geometry of the upper half-plane is also accessible by elementary methods when its congruence mappings are developed with hyperbolic motions.

By congruence mapping, do you mean an isometry? Hmm. I see, the article congruence (geometry) is the one you want; however, it promptly assumes Euclidean geoemtry, which makes it rather useles to link to. linas 18:51, 23 September 2005 (UTC)

This modern geometrical approach is consistent with the Erlangen program initiated by Felix Klein.

What "modern geometrical approach" does this sentence refer to? Surely not the ruler-and-compass construction given earlier? After skimming the article hyperbolic motion, it seems to be distinctly old-fashioned in its approach.

The half-plane model may also be developed using complex arithmetic and Mobius transformations. However complex arithmetic is a logically independent subject necessitating its own foundations including admissibility of alternative complex number systems.

I'm not sure what this is getting at. One can certainly write everything in terms of coordinates "x" and "y", but many of the formulas are easier when written with complex-valued variables. Although "complex numbers" are "logically independent" of "2D geometry", I'm not sure why this is an important distinction to make here. Indeed, what is remarkable about hyperbolic geometry is the interplay between complex analysis, and Riemann surfaces (and number theory and ergodic theory and so on). I don't know what an "alternative complex number system" is, or why the complex numbers might not be "admissible".

Err, I guess a whole section could be added at the begining (after the intro) that presented ruler-and-compass type constructions, and that reviewed the various properties using basic trigonometry, as, indeed, much of the subject is accessible without the need for calculus (or complex numbers; point now taken; it took me a while to "get it".). Such a treatment is lacking in WP. It could be done either here, as a section in this aricle, or in a distinct article. If you care to undertake this, do so; my only advice would be to carefully distinguish "Euclidean line" and "geodesic", since terms like "perpendicular bisector" are well defined and meaningful for both. My other suggestion would be to use a lot of diagrams illustrating things. If written well, I think this could be a marvelous portal for e.g. high school students looking into the matter. Again, illustration and good graphics would make it even better. linas 19:04, 23 September 2005 (UTC)

New reference

One author who has advanced the cause of popularization of the concept behind this encyclopedia entry is John Stillwell in his book Numbers and Geometry (1998).In the preface he says the aim of his book is "to give a broad view of arithmetic, geometry, and algebra at the level of calculus, without being a calculus book (or a precalculus book)."

The Poincare model at hand ought to be known to students before they begin differential geometry and metric spaces.It should be one of the particulars that are encircled by the general concepts of the abstract theories.Rgdboer 01:30, 15 March 2006 (UTC)

Additional material needed

At present this article seems to focus on the symmetry group of isometries too much. We need to add some other stuff, such as:

• What is the formula for the metric tensor?
• What is the formula for the distance between two given points?
• Are circles modeled by circles? And how does one adjust the center?
• How one can use Compass and straightedge constructions#The basic constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points.
• Give a projection or formula which translates from this model to the Hyperboloid model and its inverse function.

JRSpriggs (talk) 23:56, 7 August 2010 (UTC)

I started to work out a conversion to/from the hyperboloid model, but because I need two intermediate steps (hyperboloid ↔ conformal disc ↔ hemisphere ↔ half-plane) it got messy enough that the probability of my mis-copying something approached 1. —Tamfang (talk) 19:01, 29 April 2018 (UTC)

It was later pointed out to me that inversion would save a step, but I have not yet tried it. —Tamfang (talk) 05:11, 30 June 2023 (UTC)

The y coordinate in the half-plane model represents some plane (not through the origin) in the 3-dimensional Minkowski space (of signature +, +, −) in which the 2-sheeted hyperboloid is embedded which contains a light-like line. Or if you think of the conical asymptote of the hyperboloid, the section our plane cuts is a parabola (the part of the hyperboloid it cuts is a horocycle). –jacobolus (t) 05:41, 30 June 2023 (UTC)

If you want, you can think of the half-plane model as a stereographic projection where the center of the projection is an ideal point (the disk model is likewise a stereographic projection where the center of projection is any ordinary point of the hyperbolic plane). –jacobolus (t) 06:15, 30 June 2023 (UTC)

As it stands, this article presumes knowledge from differential geometry for definition of a metric. Since the topic is key to learning hyperbolic geometry, an elementary approach to the metric was developed some time ago at hyperbolic motion. Above discussions show interest in making this model accessible without differential geometry or complex numbers, though those tools are generally used to enter this realm. — Rgdboer (talk) 02:42, 30 April 2018 (UTC)

I added some figures illustrating the distance formulas. –jacobolus (t) 02:22, 10 July 2022 (UTC)

Other isometries between the half-plane model and the Poincaré disk model?

The article mentiones the Cayley transform as an isometry between the half-plane model and the Poincaré disk model. I'm wondering if not the model resulting from the mapping $f(\langle x,y\rangle )=k\langle x,y+1\rangle$ (where $z$ is a complex number with magnitude < 1), in the limit when $k\to \infty$, is also an isometry? That is, basically an infinite zoom on the lowest point of the Poincaré disk? —Kri (talk) 18:39, 18 January 2023 (UTC)

That’s pretty well the same idea: the result of a far enough zoom is approximately the same as the composition of a Möbius transformation (which fixes the disk and moves a point very close to the edge to the center; a 'translation' of the hyperbolic plane) followed by a Cayley transform from disk -> plane. But when you make the zoom infinite the result is not well defined as far as I can tell.

Or perhaps it would be better to say, if you can come up with a may to make this explicit / well defined, what you will end up with is the Cayley transform (possibly composed with a hyperbolic-plane translation). Given any 3 initial points in the projectively extended complex plane which you want to map to 3 final points in the plane, the Möbius transformation matching that description is unique. –jacobolus (t) 20:59, 18 January 2023 (UTC)

To correct my oversimplified language, the zoom is never actually infinite. What we do is we take the space that is the limit of the space we get when we zoom in on the boundary of the Poincaré disk with magnification $k\to \infty$. I also don't care that much about which point in the disk model is mapped to which point in the upper half-plane model, as this is undefined when $k\to \infty$. What is interesting is the resulting model, which is described solely by the shape and the metric we obtain.

I understand that the Möbius transformation results in the upper half-plane model. I was just wondering if this is an alternative way to obtain the same model in a perhaps intuitively simpler way (I consider zooming more easy to grasp than a Möbius transform). —Kri (talk) 03:58, 29 January 2023 (UTC)