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?
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.