Analytical and Numerical Aspects on Motion of Polygonal Curves with Constant Area Speed

The first purpose of this paper is to propose a formulation of general area-preserving motion of polygonal curves by using a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to admissible class used in the so-called crystalline curvature flow. Actually, if the initial curve is a convex polygon, then our polygonal flow is nothing but the crystalline curvature flow. However, if the initial polygon is not convex and does not belong to any admissible class, then the polygonal flow cannot be regarded as a crystalline curvature flow. Because the prescribed polygonal class is determined by the initial polygon and one can take any polygon as the initial data. On the other hand, in the framework of the crystalline curvature flow, the initial polygon should be taken from the admissible class. The second purpose is to discretize the ODEs implicitly in time keeping a given constant area speed, while the solution polygonal curve exists in the prescribed polygonal class. The organization of this paper is as follows. In the next section, we will introduce notion of polygonal motion and a polygonal class will be given. In section 3, our problem will be formulated and some examples will be given. In the last section, we will propose a scheme which achieves the second purpose and show convergence of the scheme.