All I can say (again), is WOW. This community is truly awesome. Instead of commenting / replying separately:
@DanielPiker - I see now that Kangaroo wasn’t needed, but if anything, your swift response is even more impressive. And the process of going through the “economy” of your definition and C# is truly an edifying experience!
@Joseph_Oster - Thank you for your non-Kangaroo solution! Seeing it done in a different way, and incorporating useful components I didn’t know existed, like Seam and Region Union, is a wonderful gift.
@diff-arch - Thanks for the link to Paul Gould’s stuff. Love it!
@martynjhogg - The “end actuator” affixed to the carriage will be a magnet (traveling under a pan holding a field of sand and a steel ball). This is not really a “new” project - I’ve been using a polar mech to do sand-plotting for over 20 years, and continue to do so commercially. While I remain firmly biased toward circles over rectangles, fabricating large round things is harder. And evidently, not everyone wants a round coffee table . As for the inputs to the motors - that’s what led to my post. I want to be able to use our existing control-ware / mobile app platform with a Cartesian mech by modifying my original code so it can use existing “tracks” (nearing 1000 downloadable continuous line drawings created by our community of users) which are all in the polar “thr” format (Theta radians, Rho 0 to 1 for each vertex). Conversion of each polar vertex to a corresponding XY (and from there to AB) is trivial - but for long travel between distant polar vertices, I need to calculate intermediate points along the Archimedes spiral that is the “natural” path between those two points using a polar mech. I was hoping to use GH to guide / confirm my strategy - and thanks to you all, am on my way…