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.
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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.