Talk:Ideal triangle

paraphrase please

Any point in the triangle is within constant distance of some two sides of the triangle.

Does this mean there is a constant d such that every point is within d of two sides (which seems trivial); or that for every point there is a constant r such that that point is within r of two sides (which is even more trivial); or something else? —Tamfang (talk) 20:45, 19 July 2009 (UTC)[reply]

The former. Note that d is an absolute constant, independent of the specific triangle. It's not so trivial, since it's not true for triangles in the Euclidean plane: for any d, one can find a Euclidean triangle that contains a circle of radius 2d, and the center of the circle is more than d away from any side of the triangle. —David Eppstein (talk) 21:18, 19 July 2009 (UTC)[reply]

It should be made clear that the same constant works not just for ideal triangles, but for all triangles. It would be the radius of the circle which could be inscribed in an ideal triangle. Can you calculate what its value is and provide it? JRSpriggs (talk) 08:59, 21 July 2009 (UTC)[reply]

I don't think it's the circumradius, exactly. It's the maximum distance that any point on one side of the triangle can be from the other sides, which is the distance between the three points of tangency of the inscribed circle. Here, it's calculated to be 4 ln φ, approximately 1.925, where φ is the golden ratio. —David Eppstein (talk) 15:14, 21 July 2009 (UTC)[reply]

I was referring to the inradius, not the circumradius.

However, you are correct that d is not the inradius; rather d is the side of that inscribed equilateral triangle. JRSpriggs (talk) 17:58, 21 July 2009 (UTC)[reply]

Uh, hang on. If my understanding of hyperbolic geometry is correct, ideal triangles don't exist. Right? Professor M. Fiendish, Esq. 05:40, 13 September 2009 (UTC)[reply]

What do you mean by "exist"? They are well defined geometric objects. If you object to the fact that they are unbounded in size, do you think a Euclidean line exists? —David Eppstein (talk) 06:25, 13 September 2009 (UTC)[reply]

In the hyperbolic plane it isn't possible to get to the "edge" of the disk, right? Professor M. Fiendish, Esq. 11:51, 14 September 2009 (UTC)[reply]

The points at the edge of the disk are not part of the hyperbolic plane, true. So an ideal triangle doesn't have vertices within the hyperbolic plane. That doesn't mean it's not a proper geometric figure, though — a Euclidean line doesn't have vertices within the Euclidean plane, either, but that doesn't make it improper. —David Eppstein (talk) 14:46, 14 September 2009 (UTC)[reply]

Generality of description in models

The models section makes some statements that aren't fully general:

"In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles" excludes the case where it is bounded by two semicircles and a line through the center of the disk.

"In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles" excludes the case where it is bounded by two lines and a semicircle.

Both of these are non-issues if we consider lines to be circles of infinite radius, but I think a clarifying sentence or two would be in order. Sarahtheawesome (talk) 02:00, 20 March 2012 (UTC)[reply]

That sounds like a good idea — please go ahead. (Also, you mean diameter in the disk case and ray in the halfplane case, I guess, rather than lines.) —David Eppstein (talk) 02:32, 20 March 2012 (UTC)[reply]

Ideal triangle tiling

I'd like someone to explain the difference between these tilings. The first is generated as a reflective kaleidoscope of triangle group (∞ ∞ ∞), and can be seen as a limiting regular triangular tiling {3,∞}, but I don't know what the second tiling represents, and concerned because I have no tools of analysis to identify its symmetry. All I can tell for sure is that it has order-3 reflective symmetry at the center. Tom Ruen (talk) 03:32, 4 February 2013 (UTC)[reply]

The Poincaré disk model tiled with ideal triangles

Hm, I (who made the first of these) think I see a way to generate an image of the latter under a translation that brings the incenter of one of the other triangles to the center of the projection; that might tell us something. (One of these years.) —Tamfang (talk) 19:29, 29 July 2017 (UTC)[reply]

The second one often appears (incorrectly) as an image of the Farey tiling (which is the first tiling). It appears to have been constructed by, each time you want to add a new ideal vertex, adding a new ideal vertex that subdivides the circular arc in half. The shear coordinates on the edges beyond the initial three edges are non-zero, so it's definitely a different tiling. I don't think it has any symmetries besides the obvious symmetries mentioned above. I'm also not aware of a name, despite its ubiquity in depictions. Maybe the "fake Farey tesellation"? Dylan Thurston (talk) 16:24, 23 September 2018 (UTC)[reply]

What is triangle called with only 1 or two ideal vertices?

It seems like a triangle with 1 or more ideal vertex might also be considered ideal or something? Tom Ruen (talk) 22:46, 27 August 2013 (UTC)[reply]

Like these:

Here's one answer, I found this paper, [1], which divides an ideal quadrilateral into four semi-ideal triangles (each with 2 ideal vertices). Tom Ruen (talk) 22:59, 27 August 2013 (UTC)[reply]

Another paper [2] defines, A hyperbolic triangle S is called semi-ideal if two of its three vertices lie on the line at infinity. Tom Ruen (talk) 23:01, 27 August 2013 (UTC)[reply]