in reply to Confused about 3D geometry / algebra

unclear ...

... how do you wanna place a line (or plane) between those two molecules?

    X  O       O
 
X       X  O
 
    X  O       O

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or in math lingo: convex or concave hull?

How is the "occupied space" defined?

Cheers Rolf

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Re^2: Confused about 3D geometry / algebra
by tomdbs98 (Beadle) on Jun 04, 2010 at 13:03 UTC
    The molecules would be, ideally, convex hulls. I would define occupied space as the hull in the case. At the moment I would consider the case you presented as an overlap of the two molecules. I merely wish to be able to see if a line (plane) like this is doable 
        x  |  o
  x   x|o   o
    x  | o o

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      IMHO¹ the surface of a convex hull already describes the (optimal) planes where all atoms of your molecule are on one side of the plane (or of course in the plane).

      So your problem reduces to calculate the margins of your convex hull:

      At least one edging plane should separate the atoms into distinct groups!

                |
   \      |
    x  /  o   o
  x   x   | o   
    x  \  o   o
   /      |
          |

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      Of course, when going into real world physics the distance between two molecules should be bigger than the distance between the atoms, to be considered different molecules. But this can be calculated with those surface planes by adding a tolerance.

      IMHO those are too close 

          x  |  o
  x   x|o   o
    x  | o o

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      I don't know if you need a simple or a fast algorithm, but in many cases you can shorten the cases by approximating a (minimal) sphere which includes all atoms of a molecule.

      Only if those spheres (maybe widened by the mentioned minimal distance) intersect you will need to calculate the surfaces of the molecule.

      And you will only need to calculate the surfaces close to this intersection.

      IMHO this should be considerably fast and intuitive, for further optimization better rely on the WP article on Support vector machines BrowserUK was linking to.

      Cheers Rolf

      ¹) prove should be trivial.

      UPDATE: It should be mentioned that IMHO physics knows examples of molecules which are distinct but intersect from a mathematical point of view. Something called bucketball rings a bell for me...

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        I have already implemented the fast sphere intersection test for my molecules, and I plan on making the acceptable distance between molecules user defined (so I can't be blamed as far as thats concerned;) ).

        I will be reading up convex hulls, SVMs and linear programming today and see if I can put 2+2(+2?) together. :P

        Thanks once again for your time,

        Tom

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