Comment by brosco
3 days ago
One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already.
Another is that when working with manifolds, you usually don't get a set of global coordinates. Manifolds are defined by various local coordinate charts. A smooth manifold just means that you can change coordinates in a smooth (differentiable) way, but that doesn't mean two people on opposite sides of the manifold will agree on their coordinate system. On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.
I'm not very well versed in the history, but the study of cartography certainly predates the modern idea of an abstract manifold. In fact, the modern view was born in an effort to unify a lot of classical ideas from the study of calculus on spheres etc.
Thanks. I've thought about those possibilites, but I really don't know the reasons.
> On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.
Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with distance from the poles (i.e., the distance is much greater at the equator than the poles), but there is a unique coordinate for every point on the sphere.
The problem is that this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). Manifolds are required to have an "atlas"[0]: a collection of coordinate systems ("charts") that cover the space and are continuous mappings from open subsets of the underlying topological space to open subsets of Euclidean space, with the overlaps between charts inducing smooth (i.e., infinitely differentiable) mappings in Euclidean space.
Colloquially, this means a manifold is just "a bunch of patches of n-dimensional Euclidean space, smoothly sewn together."
A sphere requires at least two charts for an admissible atlas (say two hemispheres overlapping slightly at the equator, or six hemispheres with no overlaps), otherwise you get discontinuities.
[0] https://en.wikipedia.org/wiki/Atlas_(topology)
This part I don't grasp:
> this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0).
I'm guessing that the issue is that I don't know your definition of 'continuous'.
I believe every point on the planet (sphere, for simplification) has unique corresponding coordinates on the map projection (chart). The only exceptions I can see are, A) surfaces perpendicular to the aspect (perspective) of the projection, which is usually straight down and causes points on exactly vertical surfaces to share coordinates; B) if somehow coordinates are limited in precision or to rational numbers; C) some unusual projection that does it.
> A sphere requires at least two charts for an admissible atlas (say two hemispheres overlapping slightly at the equator, or six hemispheres with no overlaps), otherwise you get discontinuities.
There are cartographic projections that use two charts. Regarding those with one, where is the discontinuity in a Mercator projection? I think when I understand your meaning, it will be clear ...
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