Transforming 3D matrix into 2D matrix?

Question

So I have no idea if this is even remotely possible, but I thought the mathematics forum was the place to answer this question.

My problem is that I'm trying to get a 2D surface map of a 3D matrix. So in one case I have a cylinder. The outer surface of this cylinder (matrix) has different values indicating, let's say temperature at the voxel level. In matrix form each "slice" of the matrix is just a ring of the cylinder. If I then wish to make a 2D representation of this I can actually just cut open the cylinder, and it would unfold to a flat square 2D surface without any distortion in any voxel.

But can the same thing be done with a sphere like object (in this case it is not a 100% sphere, but it is closed like a sphere) ? I know I could probably do a projection of some sort onto a cylinder and then unfold it again. But if I do the projection from the center-of-mass of the "sphere" there will indeed get distortions during the projection. So again. Is there some clever math way (I'm not a certified mathematician) where this can actually be done without distorting the values from the sphere ?

I hope I have explained it properly. If not, please let me know.

Answer 1

I think you are asking about representing a three dimensional surface as a plane without distortions.

That is possible for surfaces that are developable, like cylinders and cones. See https://en.wikipedia.org/wiki/Developable_surface .

The sphere is not developable, which is why making a map of the world is hard. You have to choose among distortions. You can preserve areas, or angles, but not both. You may have to choose a projection that best suits your particular purpose. See https://en.wikipedia.org/wiki/Map_projection .

Your use of the word "matrix" is quite confusing. I may have misread the question on that account.

Answer 2

What you are facing is known as the "mapmaker's dilemma", and the general answer is "no, there will always be (some kind of) distortion."

If you want great circles to become lines, one projection will do that. If you want distances approximately preserved, it's pretty much hopeless. If you want angles preserved, so-called stereographic projection will do the job. But the overall fact is that it can't generally be done to anyone's satisfaction (and Gauss proved it!)

Sorry to be a Debbie Downer about this, but them's the facts.

Answer 3

Is there some clever math way (I'm not a certified mathematician) where this can actually be done without distorting the values from the sphere?

Remember maps of the earth? They all have some kind of distortion.

The reason seems to be that the Gaussian curvature is an invariant (stays the same) between isometries (transformations which preserve the distance between points). And that curvature is different between a sphere and a plane.

E.g. see here and here.