Circumradius of a Simplex

[Robert May]

I am trying to find a formula for the circumradius of a 4-simplex in terms of its edge lengths.

I know how to find it if given coordinates of the vertices, but converting edge lengths to coordinates yields something rather unwieldy.

I know that the circumradius of a triangle is r=abc/4A, where a,b,c are the lengths of the sides and A is the area.

For a tetrahedron it is r=A/6V, where V is the volume and A is the area of a triangle with edges of lengths aa',bb',cc', where a,b,c are the lengths of the edges of one face and a',b',c' are the lengths of their opposite edges.

Does anyone know of any similar formula for a 4-simplex and/or any references that might be of help.

Thanks,
Robert May

[Thomas Mautsch]

In news:<3338477.11132513495...@nitrogen.mathforum.org>
schrieb Robert May <rm...@vhcc.edu>:

The buzzword is "Cayley-Menger determinant":
See
http://mathworld.wolfram.com/Cayley-MengerDeterminant.html
for a definition.

The idea is as follows:
The points of an n-simplex in n-space plus the circumcenter
form a degenerate (n+1)-simplex, whose (n+1)-dimensional volume is zero,
and for which all sides whose lengths you do not know yet
have length equal to the circumradius.

The Cayley-Menger determinant is a determinant
made up from the squared side lengths of a simplex,
and it vanishes iff the simplex has zero (hyper-)volume.

This vanishing of the Cayley-Menger determinant
yields a quadratic equation in the square of the circumradius,
which can be easily solved.
For triangles and tetrahedra, one obtains the formulas you cited above.

See also:

http://groups.google.de/groups?selm=7qpjvg%24pq1%241%40nntp.itservices.ubc.ca
http://groups.google.de/groups?selm=o9i4ca9snvhz%40legacy

[Robert May]

Thanks.

I know of the Cayley-Menger determinant, but I wasn't seeing how to connect it to the circumradius.

[Oleg]

Possible answer is
If the lengths of a tetrahedron's opposite edges are
m and n, u and v, p and q respectively and
the volume of this tetrahedron is V and the circumradius is R.
Then the following formula holds true:

(24VR)^2=2(upqv)^2-(mn)^4+2(mnuv)^2-(pq)^4+2(mpnq)^2-(uv)^4.

Given a tetrahedron. The lengths of its opposite edges are
m and n, u and v, p and q respectively.
The volume of this tetrahedron is V and the circumradius is R.
Then the following formula holds true:

(24VR)^2=2(upqv)^2-(mn)^4+2(mnuv)^2-(pq)^4+2(mpnq)^2-(uv)^4.