Talk:Ultraparallel theorem
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Diagrams
[edit]The proofs in this article would be enhanced if diagrams were available. If you can help, please consider this project.Rgdboer (talk) 20:14, 14 December 2007 (UTC)
- I try an image of demostration. Rocchini (talk) 13:57, 7 May 2008 (UTC)
- Thank you very much. The image contains key features of the proof. The use of color for correspondences is impressive. Confidence in this theorem is now more sure.Rgdboer (talk) 20:23, 8 May 2008 (UTC)
Klein model proof
[edit]This proof depends on the notion of "pole" of a line with respect to a conic. Though this concept is very elementary in Projective Geometry, our articles do not yet establish it. The related concept of Harmonic Cojugates is not in place either. So far WP understands the two real functions in an analytic function as harmonic conjugates. When the tools are in place then the concept of "perpendicular lines" in the Klein model will be available. The proof given is quite satisfactory given such foundation. Due to the delay in foundation, I have put the Klein model proof into second position.
The reference (Borsuk) uses a proof based on metric properties of the curvature -1 plane.Rgdboer (talk) 23:11, 20 February 2008 (UTC)
On the German WP I have found "Pol und Polare" which is what we need. Trusting the translation will come easy, then the Projective Geometry in English will be improved soon. In the meantime, the article Klein model has a section on Angle and Perpendicularity.Rgdboer (talk) 23:48, 3 March 2008 (UTC)
After looking over the German article, and thinking about the relation to Projective Harmonic Conjugates, I posted this brief synopsis with an English reference.Rgdboer (talk) 22:15, 4 March 2008 (UTC)
Definition?
[edit]This article does not actually have a definition of "ultraparallel" nor is there a wikilink to a separate definition page. So it's a little difficult to see what the theorem is actually saying. Michael Kinyon (talk) 14:59, 26 October 2021 (UTC)
- Fixed on 5 January 2023. Michael Kinyon (talk) 03:47, 2 February 2023 (UTC)