Clamps... lots of clamps.
I just had my first experience bending lucite - just a simple 90 degree angle. Sandwiched one side between two pieces of wood. The wood was slightly bowed however, so there was a little give in the center (it was only clamped near the ends). This give was enough to make the curvature of the bend different in that area and the entire piece ended up crooked. ugh. Tried again, clamping more securely, and the results were adequate.
Your idea looks really cool. Good luck!