Structured discrete shape approximation: Theoretical complexity and practical algorithm

•Mathematical formalization of 2D shape approximation by discrete assembly systems.•Computational complexity analysis of associated shape approximation problems.•Algorithmic framework combining shape sampling and mixed-integer linear programming.•Practical application: freeform contour approximation with the Zometool system.•Numerical experiments show: high-quality solutions can be obtained in reasonable time. 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 of node and edge types subject to edge length and orientation restrictions. We show that already deciding feasibility of such approximation problems is NP-hard, and remains intractable even for very simple setups. We then devise an algorithmic framework that combines shape sampling with exact cardinality minimization to obtain good approximations using few components. As a particular application and showcase example, we discuss approximating shape contours using the classical Zometool construction kit and provide promising computational results, demonstrating that our algorithm is capable of obtaining good shape representations within reasonable time, in spite of the problem's general intractability. We conclude the paper with an outlook on possible extensions of the developed methodology, in particular regarding 3D shape approximation tasks.