I’m still not sure I understand. In this drawing (please don’t judge), let’s say the node where the three arrows originate is the node in question. Asssume this is an interior node and that the mesh extends further in all directions. How would one choose any of the three (green, blue and red) tangent vectors as the “tangent vector of the node”?
Maybe you could provide an illustration of your problem?
Also, when you say your rotation matrix and growth tensor are defined pointwise on vertices, does that mean you actually have arrays with their values at the vertices? Or do you have finite element functions representing them, such that you have the degree of freedom/coefficient values at the vertices?