Circumscribing a convex polygon by a polygon of fewer sides with minimal area addition

An approach and an algorithm are introduced for circumscribing a given n-sided convex polygon P_n by an m-sided polygon P_m, (m < n), so that the added area is minimal. This algorithm constitutes one building block in an algorithm for efficient nesting of arbitrary geometric shapes in a given rectangular board. Flame cutting of steel sheets and laser cutting of textiles are two industrial situations where this problem is of great importance. The approach follows a top-down stepwise refinement and reduction of the original problem into simpler subproblems, the solutions of all of which permit the solution of the original problem. It is first shown that the optimal circumscription of P_n by P_m may be obtained by (n - m) iterative single side reductions. The solution of the single side reduction problem is then characterized, and an algorithm which is based on the triangle rotating side problem is proposed. This last problem is concerned with passing the third side of a triangle through a given point that lies within the area bounded by the two other given sides so that the triangle area is minimized. On the way to proving the optimality of the algorithms for the original problem and its subproblems, new concepts and theorems are introduced. The algorithm was tested on a very large number of polygons with varying numbers of sides and shapes, which were circumscribed by hexagons. The average efficiency-defined as the ratio of the area of P_n to that of P_m-was 96%. As n increases, efficiency reduces and approaches asymptotically the maximum achievable efficiency for circumscribing a circle ("infinite" sided polygon) by a regular hexagon: 90.69%. With n = 50 the average efficiency was 91.8%.