This post follows on from the previous post, which described the cubed sphere mapping which preserves areas in taking a surface grid from cube to sphere. I should apologise here for messing up the links for the associated WebGL plot for that post. I had linked to a local file version of the master JS file, so while it worked for me, I now realise that it wouldn't work elsewhere. I've fixed that.

If you have an area preserving plot onto the flat surfaces of a (paper) cube, then you only have to unfold the cube to get an equal-area map of the world on a page. It necessarily has distortion, and of course the cuts you make in taking apart the cube. But the area preserving aspect is interesting. So I'll show here how it works.

I've repeated the top and bottom of the cube, so you see multiple poles. Red lines are latitudes, green longitudes. The blue lines indicate the cuts in unfolding the cube, and you should try to not let your eye wander across them, because there is confusing duplication. And there is necessarily distortion near the ends of the lines. But it is an equal area map.

Well, almost. I'm using the single parameter tan() mapping from the previous post. I have been spending far too much time developing almost perfectly 1:1 area mappings. But I doubt they would make a noticeable difference. I may write about that soon, but it is rather geekish stuff.

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What's interesting about these kind of spherical transformations is that they can significantly simplify problem solving. I have an example of simplifying Laplace's derivation of the primitive equations along the equator here:

ReplyDeletehttp://contextearth.com/compact-qbo-derivation/

The realization of the earth's equatorial latitude (phi) varying due to the mutual interaction with the nodal orbit of the moon and sun provides a huge simplification to the equations. This is what amounts to a clever ansatz defining the time-varying equatorial path with the greatest lines of attractive force. If one applies this simplification, Laplace's tidal equations reduce to a closed-form analytical solution. If one doesn't simplify, the equations remain underdetermined and difficult to deal with. And I think that's why they can't make sense of behaviors such as QBO.

That being said, the issue that I am certain people will have with this formulation is that it appears that it's fiddling with the space-time continuum by having one of the spherical rotational axes (the latitude) vary with time. That makes it qualitatively similar to the Lorentz transformation. I don't necessarily have a ideal answer for this other than how it may help to understand the observed dynamics. By allowing the equatorial latitude to slightly vary in cyclic fashion, is the simplification worth the understanding we get from this formulation?

Yes, that is a good way of doing it. Anything that is topologically equivalent to a sphere can more easily have the right natural frequency behaviour.

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