Is it possible to create a linear map that goes from the surface of the Earth to a 2-d screen?

Question

There are many ways to map the surface of the Earth to a literal map one can see on a screen. The projections have pros and cons. It is well known that the Mercator projection distorts areas. But there are equal area projections that preserve areas at the expense of angles.

In linear algebra, we have the concept of a linear map. It satisfies: $$f(v+u) = f(v)+f(u)$$ $$f(c.v) = c. f(v)$$

My question - is it possible to project the surface of a 2-d sphere to a plane in a way that it is a linear map? And is that also possible for hyperbolic surfaces?

Answer 1

A sphere $\mathcal S^2$ is not a vector space so it does not make sense to speak of a linear map

$$ f : \mathcal S^2 \to \mathbb R^2.$$ Hence it is not possible to find such a linear map.

Answer 2

Without further qualification, one could say this:

Any bijection from $\mathbb R^2\to \mathcal S^2$ can be used to transport a vector space structure to $\mathcal S^2$, and doing that makes the original map a linear isomorphism.

This somewhat disappointing answer highlights the fact that one can't really say anything interesting without first specifying what linear structure one wants to place on $\mathcal S^2$.

Now, a better way to go about this is to check out what a manifold is. From that perspective, one would say that "$\mathcal S^2$ looks like $\mathbb R^2$ when you are up close to it." From far away, though, the manifolds $\mathcal S^2$ and $\mathbb R^2$ are globally not alike.