On Finding a Better Position of a Convex Polygon Inside a Circle to Minimize the Cutting Cost

The problem of cutting a convex polygon P out of a planar piece of material Q (P is already drawn on Q) with minimum total cutting cost is a well studied problem in computational geometry that has been studied with several variations such as P and Q are convex or non-convex polygons, Q is a circle, and the cuts are line cuts or ray cuts. In this paper, we address this problem without the restriction that P is fixed inside Q and consider the variation where Q is a circle and the cuts are line cuts. We show that if P can be placed inside Q such that P does not contain the center of Q, then placing P in a most cornered position inside Q gives a cutting cost of 6.48 times the optimal. We also give an O(n)-time algorithm for finding such a position of P , a problem that may be of independent interest. When any placement of P must contain the center of Q, we show that P can be cut of Q with cost 6.054 times the optimal.