This is very cool. It makes me realize we can make disk models of spherical geometry too, by inserting an additional projection at the beginning (stereographic, aka polar projection). Once we have an image on the plane, we just do the same steps you laid out for the euclidean case.
I think I need to experiment with some (animated) images of euclidean and spherical tilings in these models!
Neat. The historical euclidean models of non-euclidean geometries together with duality in projective geometry form an obviously incomplete perspective to which the likes of non-euclidean models of euclidean geometry -- or like here non-trivial euclidean models of euclidean geometry itself, are valuable additions.
+Roice Nelson Or go directly from the sphere to the hemisphere via a Möbius transformation, then project that to the disk.
There are also many ways to do the projection from the hemisphere to the disk (perpendicular, perspective, azimuthal equidistant, azimuthal equal-area). I think there's no conformal way to do it; that would just explode the map back out to the whole plane.
+Matt McIrvin, +Roice Nelson Correct - there is no conformal (hence holomorphic) map from the plane to the disk. This is due to Liouville's theorem. This result starts us moving in the direction of uniformization: there are exactly three "simply connected" types of conformal surface: the sphere, the plane, and the disk.
After staring at this for a bit, I guessed that this is too nice to not already be known. Since this is a model of projective geometry, I searched around that for a bit and found "Recycling circle planes" by Burkard Polster (see Figure 4.2). He calls this model the "classical flat Mobius plane" but does not give earlier references. These must exist... Hmm...
+Saul Schleimer I don't think they are models of projective geometry but rather (as headlined and detailed in the post) models of euclidean geometry itself. And I concur with you that they've likely already been explored long ago, yet my point with my top-posted comment, is that they very rarely get the limelight. I had myself thought of it as an abstract want but had never seen any advertised.
+Boris Borcic Ah, yes - it is euclidean if you remove the boundary circle, and projective if you add the boundary circle (and mod it out by 180 degree rotation).
Yes, but the mod is an unintuitive thing to do with a disk. You might as well use the spherical model of projective geometry where you mod diameter pairs of points on the whole sphere — that's much more symmetric.
+Saul Schleimer You could do a stereographic projection from the hemisphere to a disk... that would be a conformal mapping just from the surface of the hemisphere, but not from the original plane.
Roice Nelson - 2016-05-18 13:52:18-0700 - Updated: 2016-05-18 13:54:38-0700
(aside: I already had functions for transforming between stereographic and gnomonic models of spherical geometry on the plane. Transforming from stereographic to gnomonic tends to stretch everything way away from the origin. To get these images, it turns out all I had to do was apply the gnomonic to stereographic function. This pulls in all the material far out from the unit disk.)
In any case, I've seen the complete Euclidean plane. It's what is commonly known as Möbius geometry, the plane where every circle represents a straight line. It's really the epitome of 0/0, actually.
It came to me by a rather unusual route: the horizon or infinity of the Euclidean and of the Hyperbolic planes must be a horocycle, and the apparent differences is that in the former, we see eighty digits, while in the latter, just eighty.
Even more extraordinary, is you set the point U on the plane, and straight lines are the intersections of E2 and the S2 sphere, where the E2 contains U, give the conformal mapping of inversion.
We can move U inwards or outwards slightly to get H2 or S2 conformal mappings, which aligns with the notion the universe might be curved one way or the other, without shifting from one geometry to another.
Likewise, we see if we take a plane M, and project the E2 that contain the point U, give straight lines on M, represent a projective model of the geometries on the plane M, the 'line at infinity' is a tangent to the sphere at U.
I think the usefulness of such perspective for visualization is much lower than the Poincaré model of hyperbolic plane, though.
The main reason why hyperbolic geometry is used in data visualization is its exponential growth, which makes it a much better fit at visualizing hierarchical data than the Euclidean plane. Large trees (e.g., the "tree of life" of all the species of Earth, or the infinite tree arising from the Collatz conjecture) are basically impossible to fit in the Euclidean plane so that all edges are of equal length, but they fit great in the hyperbolic plane. Once the geometry is appropriate for our data, the choice of the model is secondary -- the Poincaré model is great though because of symmetry and conformality.
After playing some games which showed a close-up view of the player character's neighborhood and a "mini-map" of the whole level about 20 years ago, I wondered whether it is possible to make a single view which combines both. I tried some kind of fisheye perspective, but I was not satisfied with it -- the distortion was too big to read anything at distance. We really need something similar to conformality. HyperRogue, using hyperbolic geometry, works great in comparison, the Poincaré model does not seem distorted.
There is a game "Birdsong" by Managore (2014) which manages to do this in a more successful way -- it seems to use some kind of weird projection though, which gets low distortion at low ranges and low distortion at high ranges, but high distortion an middle ranges (and it does not seem to be generalizable to show the whole plane).
I think I need to experiment with some (animated) images of euclidean and spherical tilings in these models!
There are also many ways to do the projection from the hemisphere to the disk (perpendicular, perspective, azimuthal equidistant, azimuthal equal-area). I think there's no conformal way to do it; that would just explode the map back out to the whole plane.
https://photos.google.com/share/AF1QipPZtKw6FGptHrg6wPyeljfpVImk24fYxN3R1PWLNVNEHrX0NPBy3MsIjaSPvPmfBQ?key=eXcwRVFBY2labFA2NWdzYUdhQmdQOHlFNVo1c1B3
Fisheye is a perfect term. These reminded me of honeycomb renderings I did a while back using an actual fisheye lens. See for example:
plus.google.com/+RoiceNelson/posts/aQsKkeecN8e
plus.google.com/+RoiceNelson/posts/RoujAUi4zZ7
...and various other fisheye images of rectified honeycombs here:
plus.google.com/photos/+RoiceNelson/albums/5997889591534804353
(aside: I already had functions for transforming between stereographic and gnomonic models of spherical geometry on the plane. Transforming from stereographic to gnomonic tends to stretch everything way away from the origin. To get these images, it turns out all I had to do was apply the gnomonic to stereographic function. This pulls in all the material far out from the unit disk.)
In any case, I've seen the complete Euclidean plane. It's what is commonly known as Möbius geometry, the plane where every circle represents a straight line. It's really the epitome of 0/0, actually.
It came to me by a rather unusual route: the horizon or infinity of the Euclidean and of the Hyperbolic planes must be a horocycle, and the apparent differences is that in the former, we see eighty digits, while in the latter, just eighty.
Even more extraordinary, is you set the point U on the plane, and straight lines are the intersections of E2 and the S2 sphere, where the E2 contains U, give the conformal mapping of inversion.
We can move U inwards or outwards slightly to get H2 or S2 conformal mappings, which aligns with the notion the universe might be curved one way or the other, without shifting from one geometry to another.
Likewise, we see if we take a plane M, and project the E2 that contain the point U, give straight lines on M, represent a projective model of the geometries on the plane M, the 'line at infinity' is a tangent to the sphere at U.
The main reason why hyperbolic geometry is used in data visualization is its exponential growth, which makes it a much better fit at visualizing hierarchical data than the Euclidean plane. Large trees (e.g., the "tree of life" of all the species of Earth, or the infinite tree arising from the Collatz conjecture) are basically impossible to fit in the Euclidean plane so that all edges are of equal length, but they fit great in the hyperbolic plane. Once the geometry is appropriate for our data, the choice of the model is secondary -- the Poincaré model is great though because of symmetry and conformality.
After playing some games which showed a close-up view of the player character's neighborhood and a "mini-map" of the whole level about 20 years ago, I wondered whether it is possible to make a single view which combines both. I tried some kind of fisheye perspective, but I was not satisfied with it -- the distortion was too big to read anything at distance. We really need something similar to conformality. HyperRogue, using hyperbolic geometry, works great in comparison, the Poincaré model does not seem distorted.
There is a game "Birdsong" by Managore (2014) which manages to do this in a more successful way -- it seems to use some kind of weird projection though, which gets low distortion at low ranges and low distortion at high ranges, but high distortion an middle ranges (and it does not seem to be generalizable to show the whole plane).