The first problem (the construction of F)
can be solved without having to
struggle with several algebraic equations by using
an auxiliary decomposition of 
into
an origin (arbitrary point of the simplex) and projections (local
coordinates) as shown in Figure 5.3 for the two-dimensional
case.
Figure 5.3: A two-simplex (triangle) and the decomposition of
using local coordinates
Using these substitutions one readily obtains F by comparison of coefficients
A reasonable method for the numerical computation of 
in both the two- and three-dimensional case is to explicitely determine the
eigenvalues and eigenvectors of 
and use the diagonal
transformation.
to compute
To this end we must determine the eigenvalues 
and eigenvectors of
F. For the two-dimensional case this leads to a quadratic
equation which has two real or conjugate complex solutions, depending
on the 
. The three-dimensional case is just slightly more
complex, as more cases
must be distinguished for the characteristic equation (which is a
cubic equation).
