From: edgar@math.ohio-state.edu (G. A. Edgar)
Newsgroups: sci.math.num-analysis,sci.math.research
Subject: Re: Stable formula for tetrahedron volume
Date: Thu, 24 Jul 1997 08:58:32 -0400
In article <33D58B1B.7787D08A@math.purdue.edu>, "Neil N. Carlson"
wrote:
> Herron's formula, A = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2, gives
> the area A of a triangle in terms of the lengths a, b, and c of its
> edges. (A careful organization of this formula is numerically stable.)
> I'm looking for an analogous formula for the volume of a =general=
> tetrahedon that involves only the areas of the faces and/or the
> lengths of the edges. Any references?
J. V. Uspensky, THEORY OF EQUATIONS (1948), p. 256...
The volume V of a tetrahedron in terms of the lengths of its
edges: If r12 is the square of the length of the edge from vertex 1
to vertex 2, and so on, then 288*V^2 is the determinant
0 r12 r13 r14 1
r21 0 r23 r24 1
r31 r32 0 r34 1
r41 r42 r43 0 1
1 1 1 1 0
--
Gerald A. Edgar edgar@math.ohio-state.edu