A Classification Method of Simple Polygons Based on Dihedral Group

In this paper,a new classification method of simple polygons is presented.Being different from the old method,we take the polygons and their mirror reflections into one class in the new classification method,which will reduce the number of classification.The process is as follows.First,the concave-convex property of simple polygons is considered in order to establish an identification matrix which is used to represent a simple polygon.Then we transform the classification problem of simple polygons into computing the number of orbits when the dihedral group acts on the state set that consists of all of the identification matrices.Finally,the number of group obits can be solved by Pólya enumeration theorem.The tables in the end of paper show that the classification number computed by the new classification method is less than the one computed by the old method when the sides of a polygon is 6.Also,with the increase of the number of sides,the gap between the two classification numbers is gradually larger.