I have 2 triangles ABC and PMN

I want to generalize the triangle PMN from points to lines, where any choice along that line will produce the same UV mapping for the triangle ABC

Basically i want to find all triangles that project to a given set of UV coordinates

And as the projection throws away one dimension that should be infinitely many, but they should have some ordering

So i should be able to start with one and then walk along a line to find all

i am not sure on how i could do that, any advice would be appreciated.

Formula that creates UV mapping from triangle PMN & reverse: reversing plane projection texture mapping algorithm