More of the Stretched Model
More of the Stretched Model
This continues my first post about a stretched model of the hyperbolic plane: plus.google.com/+RoiceNelson/posts/ESxR6WRvS9z
I've added pictures of various hyperbolic tilings (like the {4,∞} below) to the album: goo.gl/photos/QZvK9FyJQp1jGTbd9
I've also been able to reason out some properties of this model.
- Geodesics are half-hyperbolas with a center at the origin.
- Lines through the origin are degenerate hyperbolas and are geodesic as well.
- Hyperbolic circles with finite radius appear as circles if origin-centered, or ellipses otherwise.
- Horocycles are parabolas.
But perhaps the most useful feature of this model is that equidistant curves are also equidistant in the euclidean sense! For example, take a geodesic and simply offset it to or away from the origin and you have an equidistant curve of the hyperbolic plane. This also explains my observation in the first post that tile widths appeared to remain constant even though stretched in the radial direction.
This property makes sense if you think about how the {4,4} tiling looks in the fisheye model of euclidean geometry, like here: goo.gl/photos/WzSJhLfVb1egZBPX7
Interpreted as a hyperbolic pattern in the Poincaré disk rather than a euclidean tiling in the fisheye model, those are a bunch of equidistant curves. When exploded out to the entire complex plane, they become equidistant in the euclidean sense as well.
We need a better name for this model!
By the way, if you apply circle inversion to the images in this model, there are some lovely results. The hyperbolas turn into lemniscates. But that's another album for another day.
You mean half-hyperbolas, not hyperbolas (you only get one branch per line), right?
ReplyDeleteYep, thanks, "half-hyperbola" is more descriptive, so I'll update the post with that wording.
ReplyDeleteThis also makes me think... I should do an {∞,3} render to get a better sense of a parabola showing up (even though it will only the tile vertices on a parabola). I'll go mess with that now :)
ok, {∞,3} is pretty cool. It's here:
ReplyDeletehttps://goo.gl/photos/ejwquMoFsdEmSBpw9
This model looks very much like Weeks' version of the hyperboloid model, in his software http://geometrygames.org/KaleidoTile/index.html -- ah, well, in the old version of KaleidoTile. The new version uses Poincare/Klein. Perhaps contact him?
ReplyDeleteI guess I'd call this the gnomonic model of hyperbolic geometry?
Similar to how I think of the Poincaré disk as analogous to sterographic projection (both conformal, both with arc geodesics), I think of the Klein model as the analogue of the gnomonic model (geodesics being straight lines in both).
ReplyDeleteThanks for the tip on this model having been in Jeff's software. I'll get to meet him at your ICERM conference in late June, so that will be a good chance to ask him about his version of the hyperboloid model in person.
Your comment does make me think this model is probably a vertically projected version of the hyperboloid model, similar to how the Klein model projects from the hemisphere model, as you and Henry show so well: plus.google.com/+HenrySegerman/posts/Yxrs851YHJa
This is now causing me confusion as to which models are really analogous to which :)
The projection here is the analog of the orthographic projection: the one that preserve circumferences at the centre.
ReplyDeleteIt's how i visualise the hyperbolic plane. It's also the basis of the methods that I use in my calculations. In essence, the radial projections are Euclidean lengths.