Calculating dihedral angles from edge lengths for spherical tetrahedra

(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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