Move curve to rotate and match the direction of the surface

i am trying to create this definition for fabrication. i need to somehow cut the intersecting surfaces to make gaps for the joints both along the ribs and the beams. this way i might be able to cut them and put them together for my first fabrication project. i am not quiet sure how to do this. what i am aiming for is a final ribbed vault. among many other things i have also tried orienting it along along the surface but i cant seem to find the surface plane at these specific intersection points. (23.4 KB)

i have attached a sample image and my definition.