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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.

jazzcool
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2 Answers2

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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

jazzcool
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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.

A square and a circle that have equal areas

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.

enter image description here

Hope this helps. Good luck.

N. Owad
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  • Thank you for your response. In your explanation you assume we know cross sectional area on planes normal to a single line. We can measure cross sectional areas on any plane. Though two examples you have shares the same cross sections over a single line, they do not share same cross sectional areas over every plane. – jazzcool Mar 24 '17 at 20:49
  • Ah, I see. So you know the cross sectional area from every possible orientation of the line? – N. Owad Mar 24 '17 at 21:51
  • Yes. Using cross sectional area along any plane is fair. Can you give any recommendations for reading material for this question? Anything that you might "feel" related to the question is fine. – jazzcool Mar 24 '17 at 21:58