Inscribe a polyhedron in an ellipsoid

  • #1
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Main Question or Discussion Point

Hello mathematicians,

I'm a physics masters student and working on a subject where I have to create some random polyhedra for some purpose. I devised an algorithm to create polyhedra by assigning points on the surface of an ellipsoid, but someone told me that this causes a tough restriction on the formed polyhedra.

Actually I can't believe this, I also have a feeling that there's some theorem that would say that every polyhedron may be inscribed in an ellipsoid, is that true? could anyone please tell me where to read to understand whether this is a restriction or not?

Thank you,
Samer
 

Answers and Replies

  • #2
Galileo
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Here's an argument coming from a physicist.
For general polyhedra, there doesn't seem to be any severe restriction on the degrees of freedom required for their description (unless they are sufficiently regular).
I.e. for polyhedron with 30 edges. Any of those edges can be lengthened or shortened to some degree, given another (twisted version) of the original polyhedron. So it has at least 30 'degrees of freedom'. On the other hand, an ellipsoid has only 3 degrees of freedom (or 6 counting orientation).
 
  • #3
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Well I'm sorry, I forgot to mention that I mean convex polyhedra, does this still apply? and in a direct answer, does this mean that creating polyhedra with points on a surface of an ellipsoid isn't "custom" enough? and is there a better way or algorithm to generate random convex polyhedra taking into account that point order is important to define surface orientation?

Thank you :)
 
  • #4
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Come on man, tell me something! :)
 
  • #5
D H
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What's wrong with Galileo's degree of freedom argument? (BTW, I count nine degrees of freedom: Three for the center, three for the axes, and three for orientation.)

Simplify the problem a bit. Not even all quadrilaterals have a circumscribing circle. An ellipse just adds two degrees of freedom.
 
  • #6
Office_Shredder
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Thinking of it backwards makes it easier: The reason why every triangle can be inscribed in a circle isn't some magic property of triangles, it's because every circle is defined by three points, so given the vertices of a triangle you can find a circle going around it.

Every ellipsoid is the image of a sphere under an invertible linear transformation. Going backwards, from the ellipsoid to the sphere, we have that the polyhedron inscribed in the ellipsoid corresponds to a polyhedron inscribed in the sphere (linear transformations take polyhedra to polyhedra). Four points defines a sphere, so you probably can't do better than inscribing tetrahedra inside of ellipsoids in general
 
  • #7
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Thank you people :)
 

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