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Given a set of planes having normals facing away from a point, it seems possible to carve out a region of space and call it a shape (even if part of the shape would be infinite). For example, 6 planes could be used to construct a cuboid.

What is the method for finding the minimum bounding volume represented by the planes? Also, whether the shape has an infinite dimension?

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What do you mean by “minimum bounding volume”, as opposed to simply “volume”? –  MvG Oct 17 '12 at 20:27
    
I think you need to narrow your focus, here. You are asking a lot of very different questions. You are allowed to ask multiple questions, but it's better to ask them in multiple posts :) –  rschwieb Oct 18 '12 at 13:52
    
Oh, sorry, maybe a better way to say it would be the smallest volume containing the point. –  Jake Oct 18 '12 at 14:02
    
I asked both in the same place since they're closely related. For example, two planes could describe an infinite wedge which would have infinite volume. –  Jake Oct 18 '12 at 14:09

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What is the method for finding the minimum bounding volume represented by the planes?

For lack of a better method, I suggest triangulating the surface of your shape and iterating over those triangles, computing a determinant for each, and adding all of these. Make sure to use consistent orientation for the triangles, so that they are all oriented the same way when seen from the inside. Divide the result by 6 and you have the volume. The reason is that you're adding a number of oriented pyramid volumes, which cancel outside your shape and eventually describe exactly the interior.

Also, whether the shape has an infinite dimension?

Imagine a sphere of infinite radius enclosing your space. Every plane you add will intersect that sphere in a great circle, and cutting space in two corresponds to cutting the sphere into two halves. In this fashion, you could keep track of the polygon the planes cut from your sphere. Every plane you add might either not remove any more infinite points, leaving the polygon as it is. Or it might intersect the polygon, adding some new vertices and probably removing others. Or it might cut away the polygon as a whole. In that last case, you can be sure that your shape is compact, i.e. has no infinite points.

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