Question: I have a volume enclosed by an orientable closed surface. I have a function that gives me the cross sectional area of this volume in any plane specified. I think I can reconstruct the convex hull of the surface from output from my function. Can I reconstruct my surface from output from my function?
I study mechanical engineering and my knowledge of topology is very limited. Any help that might set me in the right direction is welcome.
On
In general, I believe the answer is no. If the shape of each cross section is known, then you should be able to.
Here is why it is no: Imagine your function says the area of a cross section is 1. Well there is a circle with area 1 and a square with area 1.
Which do we choose? Either would be valid, so without more information, we can't make a decision.
If we were to assume that every cross section was a circle though, we could make decent guess, but there is still ambiguity. There is a process called shearing which lets you slide layers around in a solid and not change the volume. So you could end up with something like the following two very reasonable guess, one using circles all centered around the same axis, the other all squares, with one corner on the same axis. And each layer has the same area. There could also be some twisting or other weird things you do in individual layers, which can add more variety.
Hope this helps. Good luck.
The answer to this question is YES, such surface is reconstructable from cross sectional areas! Here is a very recent paper about Dip Transform:
http://irc.cs.sdu.edu.cn/3dshape/
http://irc.cs.sdu.edu.cn/3dshape/files/paper.pdf