EllipsoidsDate: 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 |
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