Talk:Poincaré half-plane model

message: 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)
replies: text: 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)
thread_title: Key to hyperbolic geometry

message: 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)
thread_title: New reference

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)

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