【正文】 by shape B, or by sliding shape –B (., B rotated 180186。 (., –A) is slid around A, the reference vertex on –A will trace out the path shown as the heavy line in . This path is the Minkowski sum . Methods for calculating the Minkowski sum can be found in putational geometry texts such as [17, 18]. Sample Part A to be Nested.Minkowski Sum (heavy line) of sample Part (light line).The significance of this is that if the reference vertex on –A is on the perimeter of , A and –A will touch but not overlap. The two blanks are as close as they can be. Thus, for a layout of a pair of blanks with one rotated 180186。)1. Select the relative position of B with respect to A. The Minkowski sum defines the set of feasible relative positions ().2. ‘Join’ A and B at this relative position. Call the bined blank C.3. Nest the bined blank C on a strip using the Minkowski sum with the algorithm given in [14] or [15].4. Repeat steps 13 to span a full range of potential relative positions of A and B. At each potential position, evaluate if a local optima may be present. If so, numerically optimize the relative positions to maximize material uti lization. Layout Optimization of One Part Paired with ItselfThe first step in the above procedure is to select a feasible position of blank B relative to A. This position is defined by translation vector t from the origin to a point on , as shown in . During the optimization process, this translation vector traverses the perimeter of .Relative Part Translation Nodes on , showing Translation Vector t.Initially, a discrete number of nodes are placed on each edge of . The two parts are temporarily ‘joined’ at a relative position described by each of the translation nodes, then the bined blank is evaluated for optimal orientation and strip width using a singlepart layout procedure (., as in [14] or [15]). In this example, consists of 12 edges, each containing 10 nodes, for a total of 120 translation nodes. The position of each node is found via linear interpolation along each edge , where is vertex I on the Minkowski sum with a coordinate of ( , ). Defining a position parameter s such that s = 0 at and s = 1 at , coordinates of each translation node can be found as:(2)(3)If m nodes are placed on each edge, ,the position parameter values for the node, , are found as:Optimal Material Utilization for Various Translations Between Polygons A and –A.As a progression is made around , when local maxima are indicated, a numerical optimization technique is invoked. Since derivatives of the utilization function are not available(without additional putational effort),an intervalhalvingApproach was taken [19]. The initial interval consists of the nodes bordering the indicated local maximal point. Three equallyspaced points are placed across this interval (. at 1/4, 1/2 and 3/4 positions), and the utilization at each is calculated. By paring the utilization values at each point, a decision can be made as to which half of the interval is dropped from consideration and the process is repeated. This continues until the desired accuracy is obtained.