How to find local 2D polygon coordinates of a 3D object represented by TriRep

I am working with a large TriRep variable representing a 3D triangulation meshing.

Tri=object.

The Tri vertices are presented in global coordinates system.

given that:

a) polygons' unit normal in global coordinates is: n_local = [nx,ny,nz]

b) Polygon center in global coordinates C = [Cx,Cy,Cz]

c) Polygon vertices in global coordinates: V_global = [v1x,v1y,v1z;v2x,v2y,v2z;v3x,v3y,v3z]

I need to find the operator that transforms the global coordiantes of the polygon into local (in plane) coordinates.

given that n_global=(T*n_local')'; %T is 3x3 rotation transformation does V_local=(T*V_global ')' ? In my practice, it is not.

any suggestions?