We propose a new technique called Rotate-and-Kill for solving the polygon inclusion and circumscribing problems. By applying this technique, we obtain $O(n)$ time algorithms for computing (1) the maximum area triangle in a given $n$-sided convex polygon $P$, (2) the minimum area triangle enclosing $P$, (3) the minimum area triangle enclosing $P$ touching edge-to-edge, i.e. the minimum area triangle that is the intersection of three half-planes out of the $n$ half-planes defining $P$, and (4) the minimum perimeter triangle enclosing $P$ touching edge-to-edge. Our algorithm for computing the maximum area triangle is simpler than its alternatives given in [Chandran and Mount, IJCGA'92] and [Kallus, arXiv'17]. Our algorithms for computing the minimum area or perimeter triangle enclosing $P$ touching edge-to-edge improve the $O(n\log n)$ or $O(n\log^2n)$ time algorithms given in [Boyce \emph{et al.}, STOC'82], [Aggarwal \emph{et al.}, Algorithmica'87], [Aggarwal and J. Park., FOCS'88], [Aggarwal \emph{et al.}, DCG'94], and [Schieber, SODA'95].
翻译:我们提出一个新的技术,叫做“旋转和Kill”解决多边形包容和限制问题。通过应用这一技术,我们获得了美元(n)美元的时间算法,用于计算:(1) 某个给定的美元正方形多边形的最大区域三角($1美元)、(2) 最小区域三角($P美元),(3) 最小区域三角($P$),其中含有触摸边缘到边缘的最小区域三角($P美元),即最小区域三角(美元半平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。我们计算最大区域三角的最大区域三角的算法比[Chandran和Mount,IJJCGGGA'92]和[Kllus,arph'17]提供的替代方平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面,即平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。