Remapping surface while retaining area

Hello everyone!

I've been tackling this project that I will do with wood, and I am working on a laser cut file using a pattern that has been turned to a surface, and since the morphed surface is what it should look like once compressed, I would like to "stretch it out" to a rectangle. The rectangle's dimensions are based on the unfolded circumference of the larger arc and the width of the surface.

The higher surface is the result of what I want to do with my sheet of wood, the lower left is the unrolled surface with the pattern on it. I created a grid of points that could help stretching it out, all it needs is to retrain its dimensions at those points. 

I've included the files, please ignore everything that is disabled/disconnected in the GH file, they are previous attempts that I might revisit. 

Thanks for the help!