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  1. Newsgroups: comp.graphics
  2. Path: sparky!uunet!news.smith.edu!orourke
  3. From: orourke@sophia.smith.edu (Joseph O'Rourke)
  4. Subject: Re: polyhedron/polyhedron intersection
  5. Message-ID: <1992Nov13.023548.19446@sophia.smith.edu>
  6. Keywords: maximum volume inscribed convex polyhedron
  7. Organization: Smith College, Northampton, MA, US
  8. References: <1992Nov12.011952.1154@cis.uab.edu> <1992Nov12.091352.5323@leland.Stanford.EDU> <1992Nov12.161117.24398@cs.wisc.edu>
  9. Date: Fri, 13 Nov 1992 02:35:48 GMT
  10. Lines: 28
  11.  
  12. In article <1992Nov12.161117.24398@cs.wisc.edu> seitz@cs.wisc.edu (Steve Seitz) writes:
  13. >In article <1992Nov12.091352.5323@leland.Stanford.EDU>, ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
  14. >|> In article <1992Nov12.011952.1154@cis.uab.edu>, sloan@cis.uab.edu (Kenneth Sloan) writes:
  15. >|> |> 
  16. >|> |> GIVEN an arbitrary, simple polyhedron, P.
  17. >|> |> 
  18. >|> |> FIND the largest (greatest volume) convex polyhedron completely
  19. >|> |> contained in P.
  20. >|> 
  21. >|> Interesting question ... well it would seem that you'd have to split the
  22. >|> polyhedra with planes containing each concave edge. 
  23. >
  24. >I don't think this will work in general (if I understand you correctly).
  25. >Try it with this example:
  26. >                                ____________________
  27. >                                |                   |
  28. >                                |                   |
  29. > minute sharp indentation -->   >                   |  
  30. >                                |                   |
  31. >                                |___________________|
  32. > 
  33.  
  34. Yes, I agree.  But this raises an interesting question.  Call a 
  35. largest inscribed convex polyhedron Q.  Is there a P with a unique Q 
  36. such that a face of Q intersects the surface of P in an edge of P only?
  37. In other words, might a face of Q pivot on a reflex (=concave) edge
  38. of P, being neither flush with a face of P nor touching at three
  39. points forming a triangle?  My guess is yes. 
  40.