Ellipsoids

Date: 12 Jan 1995 02:15:41 -0500
From: Anonymous
Subject: Ellipsoids

Dear Math SWAT Team,

I am looking for an analytical method, or a proof that an analytical method
does not exist, for the following:

        1.      Partial Surface Area of an Ellipsoid

        2.      Partial Volume of an Ellipsoid

        3.      The Intersection Volume of Two or More Ellipsoids

Can anyone provide the above, or point me in the right direction to find it?

Thank you.

tedson@neosoft.com

Date: 12 Jan 1995 11:05:23 -0500
From: Walter Whiteley
Subject: Re: Ellipsoids

. . . For the 'volume' issue (piece of an ellipsoid) I think it is clear 
that the direct way to do this is:

   (a) take an affine transformation T of the ellipsoid to
       a sphere.

   (b) Find the corresponding volume of the portion of
      the sphere.

   (c) Then multiply by det(T^-1), the ratio by which
     all volumes change going back from the sphere to the ellipsoid.

   I don't know for the intersection of two ellipsoids unless
either one transformation makes them each into spheres, or
the intersection can be decomposed (as a Boolean sum and difference) 
into computable 'pieces' of single ellipsoids.

  Walter Whiteley
__________

Date: 13 Jan 1995 01:02:30 -0500
From: John Conway
Subject: Re:  Ellipsoids

"Elliptic Functions" are so named because they first arose in the
problem of determining the arc length of an ellipse (or a part
thereof).  It follows easily from their double-periodicity that
they are not expressible in terms of the elementary functions,
that is, algebraic, logarithmic, exponential, trigonometric, and
inverse trigonometric functions.

   The problem of finding the surface area of an ellipsoid (or
part thereof) is similar but harder, so I presume it is also
fairly easily provable that it is not solvable in terms of the
elementary functions (though I can't see the proof right at
this moment).

    The volume of an ellipsoid is another matter.  Since every
ellipsoid is affinely equivalent to a sphere, the volume of any
chunk of it is proportional (with an obvious constant) to that
of the corresponding chunk of a sphere.  So one is elementary
if and only if the other is.  So for instance there are
elementary formulae for the volume of the part on any side of
any given plane, or the part inside any three planes through
the origin, etc.

    I think however that the volume of the intersection of
two general ellipsoids will be inexpressible by elementary
functions.

    John Conway