STOCK CUTTING OF COMPLICATED DESIGNS BY COMPUTING MINIMAL NESTED POLYGONS
Engineering optimization (Print)
This paper studies the following problem in stock cutting: when it is required to cut out complicated designs from parent material, it is cumbersome to cut out the exact design or shape, especially if the cutting process involves optimization. In such cases, it is desired that, as a first step, the machine cut out a relatively simpler approximation of the original design, in order to facilitate the optimization techniques that are then used to cut out the actua1 design. This paper studies this
... roblem of approximating complicated designs or shapes. The problem is defined formally first and then it is shown that this problem is equivalent to the Minima1 Nested Polygon problem in geometry. Some properties of the problem are then shown and it is demonstrated that the problem is related to the Minimal Turns Path problem in geometry. With these results, an efficient approximate algorithm is obtained for the origina1 stock cutting problem. Numerica1 examples are provided to illustrate the working of the algorithm in different cases. Key Words: stock cutting, design approximation, minima1 nested polygon, approximate algorithm. Article: Lemma 3: If is convex, consider any point .x that is in the complete visibility polygon of P in in P out . It is always true that MT(x) can have at most two more edges than P * . Proof. In the appendix. Lemma 4: If P in is non-convex and CH(P in )is contained in P out , for any point x that is in the annulus between CH(P in ) and the complete visibility polygon of CH(P in ) in P out , MT(x) has at most two more edges than P * . If CH (P in ) is not contained in P out , then there always exists a point x on the boundary of CH(P in ) for which the above is true. Proof. In the appendix. Lemma 5: For a given pair of polygons P in and P out if P * is convex and if x is a point in the annulus outside CH(P in ) such that it is possible to draw both clockwise and anticlockwise tangents from x to CH(P in ), then MT(x) is also convex. Proof. In the appendix. Hence by lemmas 3, 4 and 5, if one can choose a point x that is outside CH(P in ) and from which it is possible to draw both the clockwise and the anti-clockwise tangents to CH (P in ), then the Minimal Turns Polygon of x can be used as an approximation of P * as it will have the same property of convexity as P * and have at most two more edges. This will form the basis of the algorithm given in the next section.