Structured discrete shape approximation: Theoretical complexity and practical algorithm

We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets node and edge types subject length orientation restrictions. show that already deciding feasibility such approximation problems is NP-hard, remains intractable even for very simple setups. then devise an algorithmic framework combines sampling with exact cardinality minimization obtain good approximations few components. As particular application showcase example, we discuss contours classical Zometool construction kit provide promising computational results, demonstrating our algorithm capable obtaining representations within reasonable time, in spite problem's general intractability. conclude paper outlook possible extensions developed methodology, regarding 3D tasks.