Maximal Area Triangles in a Convex Polygon

The widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect recently by Keikha et. al.(arXiv:1705.11035). We present an alternative algorithm in this paper. Comparing to the only previously known correct solution, ours is much simpler and more efficient. More importantly, our new approach is powerful in solving related problems

Stock cutting to minimize cutting length

In this paper we investigate the following problem: Given two convex Pin and Pout, where Pin is completely contained in Pout, we wish to find a sequence of ‘guillotine cuts’ to cut out Pin, from Pout such that the total length of the cutting sequence is minimized. This problem has applications in stock cutting where a particular shape or design (in this case the polygon Pin) needs to be cut out of a given piece of parent material (the polygon Pout) using only guillotine cuts and where it is desired to minimize the cutting sequence length to improve the cutting time required per piece. We first prove some properties of the optimal solution to the problem and then give an approximation scheme for the problem that, given an error range &#x03B4, produces a cutting sequence whose total length is at most &#x03B4 more than that of the optimal cutting sequence. Then it is shown that this problem has optimal solutions that lie in the algebraic extension of the field that the input data belongs to — hence due to this algebraic nature of the problem, an approximation scheme is the best that can be achieved. Extensions of these results are also studied in the case where the polygons Pin and Pout are non-convex