I have yet to build the 116 variant of the Burke chair. I had a tough time building the 115 you see above. Organic shapes are not my friend! I'm learning to be patient with them though.
Not sure how this has progressed since this post, however, I earn my living doing computer-aided design and have a pretty thorough understanding of the mathematics behind those organic shapes. If you like I can provide a very easy to understand tutorial that reveals the concept of tangency continuity, which is the mathematical requirement that is necessary to achieve a proper reflectivity on a shaded surface. Let me know if you're interested.
BTW, I really hope you'll be able to provide a walkthrough-capable version of your Enterprise someday. This stuff is way cool.