(from E. B. Vinberg, Volumes of non-Euclidean Polyhedra,
Russian Math. Surveys 48:2 (1993) 15-45.)
Let x0 , x1 , x2 , x3
be the four vertices of a spherical tetrahedron
in the unit S3 in R4;
The length of the edge (xi , xj)
is the angle rij = arccos(xi .
xj). Consider the
four faces obtained by excluding x0 , x1 , x2 , x3
in turn.
Each of these faces is the
intersection of S3 with a unique 3-plane through the origin;
let e0 be the inward-pointing
normal to the 0-th plane, etc., scaled so that
(*) ei . xi =1
(the other ei . xj
are 0).
Then the dihedral angle dij between the
i-th and j-th
face is given by
cos(dij) = - ei .
ej / sqrt((ei . ei)
(ej . ej)).
For a non-degenerate simplex, the sets (x1 ... x4) and
(e1 ... e4) are
both bases of R4. Writing xi =
\sum aij ej and
ei = \sum bij xj,
it is clear that the matrices (aij) and
(bij) are inverse to each
other. Dotting the first equation with xk and the second
with ek
and invoking (*) yields
xi . xk = aik
ei . ek = bik.
So the calculation goes
{rij} --> {xi . xj}
--> {ei . ej} --> {dij}.
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