Talk:Beltrami–Klein model

message: I am trying to rewrite this to a bit simpler text:
- more about the two dimensional case 
- and more of that ilk 
please help WillemienH (talk) 07:09, 25 April 2015 (UTC)
replies: text: I'm unconvinced that bringing the comparison with the Poincaré disk and the detail of the distance formula into the lead is a simplification. What is it about the prior version that you wanted to simplify? I can see that a section on the two-dimensional case may be beneficial. —Quondum 21:20, 25 April 2015 (UTC)

text: We need to be clear about whether this article is about the general model of the hyperbolic geometry of any number of dimensions, or whether it is primarily about the plane.  I see it as being the former, with the "Beltrami–Klein disk model" being the special case in two dimensions. This would argue for keeping mention of two dimensions to a section (which could be quite extensive and could even become its own page), and for keeping it from becoming prominent in the lead. —Quondum 14:27, 26 April 2015 (UTC)

text: I tweaked your edits a little, not changing much.  I did remove the mention of the line at infinity; searching about, I noticed that its name is the absolute. It is incorrect to call it a line: it is really a conic. Hyperbolic geometry is a projective geometry, and so cohesiveness of terminology is important.  I suspect that some authors may have adopted some terms in an incompatible way, but not many seem to refer to the absolute as "the line at infinity" in this context. —Quondum 02:42, 22 May 2015 (UTC)

text: Simply constructing a list as a series of paragraphs is a bit awkward.  It comes across as a staccato series of unrelated statements.  What you may want to try is to tie points in a list together by introducing the list followed by bullets. Yes, the ideal points can be stated as being "at infinite distance", rather than "at infinity". Horocycles and hypercycles are circles in the Poincaré disk model (AFAICT), but are funny shapes in the Klein disk model.  The ultraideal points are naturally mapped to points outside the Klein disk model (these points must be expanded on), but the points outside the Poincaré disk model naturally map to points inside the disk (it is just an inversive map of the interior).  Also, in the Poincaré disk model, horocycles would be circles tangent to the border, and hypercycles would be circles intersecting the border at two distinct points (excluding those that are lines).  The metric tensor should not be difficult to determine, though I'm a little rusty at the moment.  I've not looked much at the metric in non-Euclidean geometries, though these can be deduced from the hyperboloid model.  What do you mean by "making lines from chords"?  There is a one-to-one correspondence between lines of the geometry and chords of the model.  Extending the chords beyond the boundary adds the ideal and ultraideal points, which is useful for reasoning about properties of points inside the geometry.  (The ultraideal points form another type of geometry, which is nicely modelled by the exterior of the Klein disk.  But this is diverting.)  —Quondum 14:08, 22 May 2015 (UTC)
thread_title:  simplification rewrite