Talk:Horocycle

approximation by hypercycles

From the convex side the horocycle is approximated by hypercycles whose distances go towards infinity.

I'm not sure I'm picturing this properly. (The following sentence quoted from Coxeter doesn't illuminate much.) Here's what I'm thinking. Start with a line L which penetrates the intended horocycle, and an equidistant to that line which is tangent to the horocycle at P. Consider the line M which contains P and is perpendicular to L. Now move L away from P, keeping it perpendicular to M, and increase the distance of the equidistant so that it continues to pass through P.

If this is indeed what's meant, can it be made a bit more explicit? Perhaps with an animation? —Tamfang (talk) 03:30, 8 July 2010 (UTC)

See the math site from stackexchange http://math.stackexchange.com/q/867317/88985 i did ask the question there as well and got a quite nice answer 09:46, 7 August 2014 (UTC)

horoball / horosphere

This page has been redirected to horoball/horosphere (same thing, different dimension)Selfstudier (talk) 14:37, 12 February 2012 (UTC)

Clearly this should be a separate article. Would you argue for redirecting the euclidean plane to euclidean n-space or, better, to abstract vector space? Tkuvho (talk) 13:55, 13 February 2012 (UTC)

Nonsense, I already left a comment and a reference about this on the horosphere page; following your logic, you now need to produce separate pages for the treatment of horocycles in each MODEL of hyperbolic space. I am content to leave that task (as well as fixing the horosphere page) to you.Selfstudier (talk) 18:09, 13 February 2012 (UTC)

Perhaps you are right. However, the consensus here seems to be against a merge. Your contribution is welcome here. It seems to be a minor issue whether there are two pages or one page. Tkuvho (talk) 13:38, 14 February 2012 (UTC)

Convergent and divergent apeirogon curves

I see an apeirogon in the hyperbolic plane can converge at infinity with a single ideal point (with all points on a horocycle), or diverge (converge outside the Poincare disk model radius). Is there a name for this divergent curve? Tom Ruen (talk) 06:17, 1 March 2014 (UTC)

Hypercycle? —Tamfang (talk) 09:27, 1 March 2014 (UTC)

I see! So the H3 analogy of the hypercycle is a 2-hypercyle, for example the points of a {3,7} in H3. Tom Ruen (talk) 20:01, 1 March 2014 (UTC)

How should articles about hyperbolic 'cycles' be organized?

Right now we have articles entitled Horocycle and Hypercycle (geometry), but as far as I can tell no dedicated articles about geodesics ("hyperbolic lines") or circles (points equidistant from a common center) in hyperbolic space. I could imagine either making 4 separate articles about these, or else one unified article about them since quite a lot of the material is going to be duplicative. (Other possibilities include 1 unified article and also 4 separate sub-articles, or one unified article and also a dedicated sub-article about hyperbolic lines.)

The current situation seems like a serious problem, since hyperbolic lines (geodesics) are the most fundamental kind of 1-dimensional object in hyperbolic space, and there is plenty to say about them, but all of our various pages about hyperbolic geometry just link to either straight line, line segment, or geodesic when discussing them, but none of these linked articles really discuss the subject. There's also plenty we can say about circles which currently goes unmentioned. I think what happened was that since only "horocycle" and "hypercycle" have unusual names, when people tried to look them up they found nothing and then were motivated to create a stub about it. But since "line"/"geodesic"/"circle" are names repurposed from a Euclidean context there was no similar demand for a new stub, and these just never got created. –jacobolus (t) 20:50, 7 November 2023 (UTC)