I kind of figured it was a poly hence saying your uvs will be condusive to the surface until editing it - unlike a nurb which will stay consistent.

If its a single polyplane then I suggest you hit auto mapping, it will find it and give you the exact shape. I think you are trying to look for something overly technical which it really doesnt need to be. Keep it simple the results happen faster.

See my result: I did a poly plane with ten divisions, extracted this one out and distorted it, hit automap and the uv on the left is the result, you can see theres no distortion on the chequer texture. Your representation of the poly is in the uv editor, you can snapshot that out
Jay