# Constructing the original function using its (low dimensional) projections

So, I confess that the question is quite vague. And I have absolutely no clue as to where to begin.

Here is, roughly, what I have in mind: given the analytic form (or an approximation to the analytic form) of all 2-dimensional projections of a surface, can I reconstruct the analytic form of the original surface?

So if I tell you that the original surface is in 3 dimensions, and its projections are: $x^2 + y^2 \leq 1, x^2 + z^2 \leq 1, z^2 + y^2 \leq 1$, then will it be possible to infer that the projections could have come from $x^2 + y^2 + z^2 = 1$ ?

How about if I had all 3-D projections? Or is there a measurable/definable correlation between the dimensionality of the projections provided and accuracy of the inference i.e. can we claim that given all $k_1$ dimensional projections we are in a better position to infer the original surface than when given $k_2$ dimensional projections, where $k_1 > k_2$? Some sort of a quantification of the information loss. At the boundaries this notion seems to obviously hold - a trivial 0-dimensional projection tells us nothing about the original surface, while, for a surface in n-dimensions, a n-dimensional projection gives us the equation for the surface.

I may not require exactly the equation of original surface back - some approximation would do. Or a suggestion of a family of surfaces might work too.

Where do I start looking?

Thanks!

PS: Just to be clear, I know that in a general case reconstructing the original surface is not possible from its various projections. I want to know what is the closest to it we can get to.

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Firstly, the projection of $x^2 + y^2 +z^2 = 1$ on the $xOy$ plane is $x^2 + y^2 \leq 1$. Secondly, $x^2 + y^2 + z^2 = 1$ is not a function. Did you mean surfaces or any subsets? Is there any smoothness involved? Either way, the statement is, clearly false, in general case. There is no way to distinguish a cylinder from a pot from their projections. –  Karolis Juodelė Sep 2 '12 at 16:46
Also, even a convex surface would take an infinite number of projections to represent exactly. As for accuracy, you'll have to think of a metric for it. –  Karolis Juodelė Sep 2 '12 at 17:08
@Karolis Juodelė, thanks for your response, and pointing out errors in my question! Yes, what I had meant was a surface, and instead of the projection I had (inaccurately) mentioned the intersection with the x-y plane However,I don't understand your pot/cylinder example - given one projection, say, the top-view, they are indistinguishable, but with all given projections a cylinder and pot cannot be discriminated? Anyway, in the general case, as mentioned, I want to know how close can we get to the original surface and with what error bound? I don't have a metric, I am asking for help. –  abhgh Sep 3 '12 at 4:07
Consider paulbourke.net/fun/planelev. More generally, you could look up visual hulls. –  Rahul Sep 3 '12 at 4:29
@abhgh, when I said "pot", I meant a cylinder without the top. For any concave shape there is at least one convex shape with the same projection (of course, there are many). –  Karolis Juodelė Sep 3 '12 at 6:17