@ZenoRogue

@ZenoRogue

ZenoRogue | Imprecise Birds @ZenoRogue | Uploaded April 2021 | Updated October 2024, 1 hour ago.
A rainbow of birds is flying in a hyperbolic manifold. Each one flies in a straight line, and initially, they form a tight single file.

However, because of numerical precision errors, they quickly end up flying all over the place.

This is because of negative curvature: when two birds fly "parallel" with a short distance between them, new space grows exponentially between them.

0:00 The video start 20s after the start of the simulation -- you need only this much time for the birds to scatter (in the current implementation of RogueViz engine).

0:20 the same animation on a "hypersian rug" (which shows the negative curvature)

0:40 a three-dimesional version -- unfortunately it is more difficult to see what's going on here

Discuss this stuff and non-Euclidean geometry in general in the #geometry channel in the HyperRogue discord: discord.gg/8G44XkR

Music: "Caribbean" by Will Savino, from the HyperRogue soundtrack (CC BY-SA 3.0)

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$Hyperbolic analogs of spherical projections Cartographers need to project the surface of Earth to a flat paper. However, since the surface of Earth is curved, there is no perfect way to do this. Some projections will be conformal (map angles and small shapes faithfully), equidistant (map distances along SOME lines faithfully), equal-area (map areas proportionally), etc., but no map will be all at once. Cartographers use many projections. Interestingly, most of them have natural analogs in hyperbolic plane H²! This video shows 22 projections and their H² analogs. 0:00 sphere and hyperboloid If you think S²={(x,y,z): x²+y²+z²=1}, then you should think H²={(x,y,t):x²+y²-t² 1, t≥0}. The picture may be a bit confusing: this is Minkowski space, so squared distance is x²+y²-t²! 0:15 stereographic projection Project the S²/H² from (0,0,-1) onto an OXY plane. Conformal, maps circles to circles, great when working with Delaunay triangulations. If you think of H² in the Poincaré model, you should think of S² in the stereographic projection. 0:30 gnomonic projection (Beltrami-Klein model) Project the S²/H² from (0,0,0) onto a plane parallel to OXY. Maps straight lines to straight lines. Only half of S² is visible. 0:45 orthographic projection (Gans model) Project the S²/H² orthogonally onto OXY. 1:00 Lamberts azimuthal equidistant projection Azimuthal means that a point in direction α and distance d (from some chosen central point) will be mapped to an Euclidean point in direction α and distance f(d); f usually does not depend on α. 1:15 Lamberts azimuthal equal-area projection 1:30 Equirectangular projection (Lobachevsky coordinates) Every point has a latitude φ (distance from equator) and longitude λ (closest point on equator). In this projection, we map (λ,φ) to Euclidean (x,y) = (λ,φ). The H² analog is called Lobachevsky coordinates. Pick a line as the equator, geodesics orthogonal to the equator are meridians, and curves equidistant to the equator are parallels. 1:45 Mercator projection (band model) Cylindrical: (λ,φ) mapped to (λ,f(φ)). Choose f is to make this conformal. See e.g. http://bulatov.org/math/1001/ and https://github.com/zenorogue/newconformist . 2:00 Cylindrical equal-area 2:15 Central cylindrical projection Meridians mapped like in the gnomonic projection. 2:30 Gall stereographic projection Meridians mapped like in the stereographic projection. 2:45 Miller cylindrical projection Scale φ by 4/5, use Mercator, scale y by 5/4. 3:00 Loximuthal projection Like the azimuthal equidistant projection, but we use loxodromes rather than geodesics, and distances along them. Loxodromes are lines which go in a constant geographic direction (in H², directions are defined by Lobachevsky coordinates). 3:15 Sinusoidal projection We stretch the equirectangular projection along the parallels so they are mapped in an equidistant way. Should be named cosinusoidal the hyperbolic sinusoid and the hyperbolic cosinusoid are very different! 3:30 Mollweide projection We map (λ,φ) to (λf(φ),g(φ)), where f and g are chosen to get an equal-area projection where the parallels become ellipses, or hyperbolas in H². 3:45 Collignon projection Like Mollweide, but f and g are chosen so the the parallels are mapped to straight lines. 4:00 Two-point equidistant We pick 2 points, and map every point in such a way that the distances from these two points are correct. The resulting map is correct close to these 2 points. 4:15 Two-point azimuthal Pick 2 points, and map every point in such a way that the angles from these 2 points are correct. Actually a horizontally stretched gnomonic projection. Useful as a simulation of binocular vision. 4:30 Aitoff projection Halve λ, use the azimuthal equidistant projection, double x. 4:45 Hammer projection Halve λ, use the azimuthal equi-area projection, double x. 5:00 Winkel tripel projection Average of Aitoff and equirectangular. 5:15 Werner projection Correct distances from the center; circles are mapped to circular arcs of the same length, making it equidistant along these circular arcs and along a chosen parallel. In S², the circle is shorter than the Euclidean circle, so the model is interrupted into a heart shape; in H², the circle is longer, so some Euclidean points represent multiple points. *** Not all projections/models of S²/H² models have analogs in the other geometry. There are also projections of S² based on interruptions, where the projection is broken along some lines, since there is less space in S² than in the Euclidean plane (we have not enough sphere to draw anything there). In the hyperbolic case, there is more space, so we get a map that covers itself. This tends to work badly (see the Werner projection). See also: HyperRogue: http://www.roguetemple.com/z/hyper/models.php Wikipedia: https://en.wikipedia.org/wiki/Map_projection TilingBot: https://twitter.com/TilingBot/ A similar older video by David Madore: https://www.youtube.com/watch?v=xHvAqDuWG2M$

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Imprecise Birds @ZenoRogue