[mmooss]

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.

[senderista]

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)

[mmooss]

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 ...

[senderista]

Continuity is fundamentally a topological property of a mapping. It just means that for a mapping F and a point p, for any neighborhood del of F(p), we can find a neighborhood eps of p such that F(eps) is contained entirely in del. In simpler terms, if you draw a little ball around F(p), I can find a little ball around p whose image under F is contained in the little ball you drew around F(p). If I have coordinates on the sphere that suddenly jump between 0 and 2*pi, I can’t satisfy this property, because points that are arbitrarily close on the sphere will be mapped to opposite sides of the “coordinate square” with sides [0,2*pi).

The Mercator projection is obtained by removing two points from the sphere (both poles) and stretching the hole at each pole until the punctured sphere forms a cylinder, then cutting the cylinder along a line of longitude. So you can see that the 3 discontinuities in the Mercator projection correspond to the top and bottom edges (where we poked a hole at each pole) and the left/right edges (where we cut the cylinder). (Note that stretching the sphere at the poles changes the curvature, but cutting the cylinder does not. The projection would have the same properties on a cylinder.)

It is possible to continuously map the sphere to the entire (infinite) plane if you just remove a single point (the north pole): place the sphere so the south pole is touching the origin of the plane and for any point on the sphere, draw a line from the north pole through that point. Where that line intersects the plane is that point’s image under this mapping (called the Riemann sphere).

[mmooss]

That makes sense. As I thought, I just needed to understand continuity in this context. That also helps address my original question - why manifolds aren't widely used in cartography. Thank you.