Of course you can do it without remapping and reparameterizing your initial referenced surface.
The problem is the U & V directions of your initial referenced surface.(The bottom left corner is not {0,0}, in your case, Upper right corner is {0,0})
So, as you’ve noticed, I’ve reconstructed your surface U & V directions first. Check the domain difference between your initial referenced surface and recreated one.
When if you’d like to divide your surface from left to right, bottom to top, using Isotrim, you should take into account of your surface directions first.
There are other topics related to this including bellow…
