Their justification for no boundary cut error is kinda dodgy.. they say they have none because this projection is really two discs back to back, 'you can just stretch a string over the edge of the disk'.
That's cool but by that argument can't i just fold a Mercator map in half and also have no boundary cut?
I would say worse than "kinda dodgy" - it's completely intellectually dishonest, and their paper should not have got past peer review if it claims this has no boundary cut but the Mercator projection has a big one.
Equally, one could just glue together the edges of a butterfly map, eliminating the boundary penalty. I think this is cheating.
The problem statement is: find a mapping from the surface of a sphere to ℝ² that minimizes a particular penalty function. This paper maps each hemisphere to ℝ², and then argues that the normal boundary penalty term can be ignored.
However, if you just look at what the map does to South America and Africa, where there's a massive discontinuity at the equator, it's absurd to argue that the boundary penalty should be ignored. This map is useless for equatorial regions, and the penalty function should reflect that.
Yes, dodgy. Same argument could be made for the dymaxion map, which can be folded into an icosahedron, then you can easily stretch the string over the polyhedron.
Comments
Their justification for no boundary cut error is kinda dodgy.. they say they have none because this projection is really two discs back to back, 'you can just stretch a string over the edge of the disk'.
That's cool but by that argument can't i just fold a Mercator map in half and also have no boundary cut?
Google maps just repeats if you dont set the limits:
https://i.stack.imgur.com/UhosY.jpg
I would say worse than "kinda dodgy" - it's completely intellectually dishonest, and their paper should not have got past peer review if it claims this has no boundary cut but the Mercator projection has a big one.
Equally, one could just glue together the edges of a butterfly map, eliminating the boundary penalty. I think this is cheating.
The problem statement is: find a mapping from the surface of a sphere to ℝ² that minimizes a particular penalty function. This paper maps each hemisphere to ℝ², and then argues that the normal boundary penalty term can be ignored.
However, if you just look at what the map does to South America and Africa, where there's a massive discontinuity at the equator, it's absurd to argue that the boundary penalty should be ignored. This map is useless for equatorial regions, and the penalty function should reflect that.
In the online version you can click and it moves the map so it doesn't have to cut things anywhere specifically - but it will always cut somewhere.
https://imgur.com/rgeg1Lc
You can always make good local maps of a sphere. I think that's just a basic property of Riemannian manifolds: they locally look like flat space.
The problem that this paper is trying to solve, however, is the creation of a minimally distorted (according to some penalty function) global map.
As far as I can tell its not published anywhere nor received any peer review.
https://xkcd.com/2304/
Yes, dodgy. Same argument could be made for the dymaxion map, which can be folded into an icosahedron, then you can easily stretch the string over the polyhedron.
https://en.wikipedia.org/wiki/Dymaxion_map
You need to both fold it in half and glue the ends together, basically creating a torus (or two-sided cylinder) shape