This file describes formulas for measuring the length of edges in metric space and the quality of tetrahedra in metric space. They may look different from what certain authors use, but they should be mathematically equivalent to what most authors use. You may need to do some algebra to confirm whether or not your method agrees with these methods.
For an edge with endpoint metric tensors M_a
and M_b, and vector v from one endpoint to
another, its metric length l is:
l = { if |l_a - l_b| > eps: (l_a - l_b) / log(l_a / l_b)
{ else: (l_a + l_b) / 2.
l_a = sqrt(v^T * M_a * v)
l_b = sqrt(v^T * M_b * v)
Where eps is some arbitrary tolerance (Omega_h uses 1e-3).
For a tetrahedron K=(E,C) with corner vertices C
and edges E, real space volume V,
edge vectors v(e), e in E, and corner metric tensors M(c), c in C,
its mean ratio in metric space Q is:
Q = ((V_m / V_eq)^(2/3))/l_msq
V_m = V * sqrt(det(M_max))
M_max = argmax over M(c), c in C: det(M(c))
l_msq = (sum over e in E: v(e)^T * M_max * v(e)) / 6
Where V_eq is the volume of a tetrahedron with all edge lengths
equal to one.
Notice that this uses a simpler formula for the
tetrahedron's edge lengths than what was done for
individual edge lengths, so that may be different
from what other authors do.