极值多边形问题（又称多边形近似问题）是计算几何领域的基础问题，在图像识别、运筹学等领域有许多实际应用，长期受到学者的关注和研究。申报人基于自己提出的一种名为Rotate-and-Kill的技术为三个典型的此类问题设计了线性时间算法，改进了其中一个问题现有的五个非线性算法（发表于STOC'82, Algorithmica'87, FOCS'88, DCG'94, SODA'95）且简化了另两个问题现有的线性算法（发表于JOA'86, IJCGA'92）。本质上，Rotate-and-Kill技术利用极值多边形问题局部最优解常常具有的某种单调性，给出了一种巧妙的方案从问题的一个局部最优解迅速计算下一个局部最优解，故它亦可解决许多其他同类问题。申报人计划继续发展和完善上述技术，用它解决计算几何领域更多的基础问题（含多边形近似问题的多个变种），尤其是一些长期开放的难题，从而引领本领域在未来的发展。

The problems of finding extremal polygons (also known as polygon approximation problems) are fundamental in computational geometry and have been studied extensively over the past years due to their enormous applications in operation research, shape recognition, and some other fields. Recently, the fund applicant proposed a so-called Rotate-and-Kill technique for solving these problems. Based on this new technique, he designs three linear-time algorithms for solving three typical polygon approximation problems. One of them improves over five non-linear algorithms that are respectively published in STOC'82, Algorithmica'87, FOCS'88, DCG'94, and SODA'95. His other two algorithms significantly simplify the existing linear algorithms published in JOA'86 and IJCGA’92. In fact, the Rotate-and-Kill technique is flexible and can be applied in many problems. It utilizes a monotonicity of the (locally-)optimal solutions which is admitted by almost all the polygon approximation problems and presents a framework for searching one optimal solution of the problem from another rapidly. Hopefully, this technique can lead to the resolution of other longstanding open problems in this filed..The fund applicant plans to mature the abovementioned technique in upcoming years. He has the ambition to solve plenty more polygon approximation problems, as well as other related problems in computational geometry, based on this technique, and thus lead the development of this field in the future.