We know that we can transform the traditional Mohr-Coulomb system to the p-q system by using the equations
Stated formally, this means that, for every set of principal stresses, there is a unique pair of p and q values.
But did you know you can go the other way, if you need to? Let’s start by putting these equations into matrix format, which yields
Inverting the matrix and premultiplying the right hand side yields
The inversion is the key step. The fact that the matrix is invertible, square and of the same rank as the vectors means that the transformation is linear, one-to-one and onto. We can also say that, for every set of p and q values, there is a unique set of principal stresses.
Those principal stresses are
As an example, consider the first set of p and q values computed in my original post on the subject. Substituting those into the last equation yields
which of course are the original values given.