I'm not a mathematician much beyond AP Calc in high school (almost 40 years ago). I am deeply fascinated by Bézier curves and geometric continuity. I've spent a

I'm not a mathematician much beyond AP Calc in high school (almost 40 years ago). I am deeply fascinated by Bézier curves and geometric continuity. I've spent a lot of time digging up research papers and references about this and related Computer Aided Geometric Design mathematics. Mostly I skim them for the illustrations and more geometric relations. For several years I've been trying to understand how to make sure that a Bézier curve is G3 to an adjoining curve, given the tangent direction, and first and second curvature derivatives. I've tried a variety of ways to ask various LLMs to help solve this. Finally with access to ChatGPT o1-preview I was able to get a good answer. The first answer was wrong, but with a little more prompting and clarification I was able to get the answer I wanted to relate the positions of P0, P1, P2 and P3 so that a Bézier curve could be G3. This isn't something that is unknown because there are many CAD programs which can do this already, but I had not been able to find the answer I was looking for in a form that was useful to me. I don't really know where that puts o1-preview relative to a math grad student, but after spending tons of time over a couple years on this pet project, getting an answer from a chat bot was one of the more magical moments I've had with technology in a long time.