Study On Geometric Evolution Properties Of Planar Closed Curve Flow

In this paper,the properties of the convexity of the plane simple curve are obtained by studying the geometric evolution properties of the plane simple closed curve.The geometric evolution characteristics of the plane closed curve are discussed by using the governing equations of the geometric variables of the curve.The description and characterization;the overall evolution law and characteristics of the plane closed curve flow are characterized by the distance from the curve to the curve,and the finite result of the overall evolution speed of the plane simple closed curve is obtained.By studying the curvature of the plane closed convex curve,the research is obtained.The monotonicity of the curvature velocity?_?in the time interval and the concavity and convexity of the curvature?in the time interval.At the same time,the geometric evolution characteristics of the traditional plane curve contraction flow and a new type of plane curve contraction flow are discussed.According to the evolution equation of the curve flow,the arc length,the enclosed area and the related properties of the curvature are discussed.The main content of the full text is divided into six parts:The first part introduces the research background and current status of this paper as well as the main results;The second part lists some basic concepts,several inequalities and related theorem lemmas involved in this paper;The third part uses the governing equations of the curve geometric variables to explore the geometric evolution characteristics of the plane curve flow;In the fourth part,the geometrical estimation of the curvature of the plane closed convex curve is carried out,and the monotonicity of the curvature velocity?_?in the time interval and the concavity and convexity of the curvature?in the time interval are discussed.In the fifth part,the contraction flow of two kinds of plane curves is studied.Through the evolution control equations of several geometric variables,the geometric evolution properties of these two kinds of plane curves are discussed.The sixth part summarizes the research contents and prospects for the future research direction and existing problems.