Catenary Mesh Color Mapping

Hello,

Here is the gist of what I would like to do:

(a) Based on an input curve, generate a mesh that it is then simulated as a catenary arch using Kangaroo. (b) During the simulation, I would like to color the mesh using a gradient based on local curvature. (c) I would like to then remap the 3d vertex colors back onto the 2d mesh in order to use the gradient as a variable in (d) generating a pattern on the 2d form.

I am a noob in Grasshopper so not sure how ambitious this is, but at this point, I have created a definition that satisfies part (a) and - kind of - satisfies (b) -- the definition I have colors the mesh based on z-height, not curvature. I have no idea how to go about part (c) and am currently working on (d) with slow success. 

I have attached the definition that goes along with parts (a), (b) and (c). Any help would be greatly appreciated!!