闭平面 meaning in English

close plane

First according to the fact that tangential components of the evolution do not affect the geometric shape of the evolving curves , we introduce the evolution equation of geometric quantities for the general planar curves . then we describe the work of gage - hamilton briefly . last we consider a special curvature flow of curve which evolves with speed function of the principal curvatures along the inner norm and show that convexity of the curve is kept and its length and area are contracted if the initial closed curve is smooth and convex . so the final shape of the curve will be a point in finite time
首先根据曲线在切向分量上发展是不影响曲线的发展形状，我们引入了曲线的一些几何变量的发展方程；其次我们简要地回顾gage - hamilton研究曲线发展的一般步骤；最后我们考虑沿曲线的内法线以曲率的函数为发展速度的一类特殊的曲线族，证明了在初始曲线为凸的闭平面简单曲线条件下，曲线将保持凸的，并且它的面积和周长将同时收缩，并在有限时间内成为一个点。