ZenoRogue | Hyperbolic analogs of spherical projections @ZenoRogue | Uploaded September 2020 | Updated October 2024, 19 minutes ago.
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 Lambert's 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 Lambert's 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. bulatov.org/math/1001 and 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: roguetemple.com/z/hyper/models.php
Wikipedia: en.wikipedia.org/wiki/Map_projection
TilingBot: twitter.com/TilingBot
A similar older video by David Madore: youtube.com/watch?v=xHvAqDuWG2M
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 Lambert's 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 Lambert's 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. bulatov.org/math/1001 and 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: roguetemple.com/z/hyper/models.php
Wikipedia: en.wikipedia.org/wiki/Map_projection
TilingBot: twitter.com/TilingBot
A similar older video by David Madore: youtube.com/watch?v=xHvAqDuWG2M
