Evolutions of convex plane curves

This study consist of five chapters.In the first chapter, we give an historical development of the evolution problem and search through the works have been done before in literature and then describe the evolution problem on which we study.In the second chapter, we state the fundamental theorem of the local theory of curves, different definitions of curvature of a curve, Frenet-Serre equations of the curves and further more we investigate the motion of a plane curve with respect to time variable and obtain its evolution equations.In the third chapter, we dealt with the general evolution equation of embedded planar curves and obtain the evolution equations for the length of the curve and the area it bounds and prove that the tangential componenets of the evolution vector does not effect the length and area during the evolution process.Fourth chapter is devoted to proving that the evolving curve remain convex during the evolution process and the final shape is a circle in the Haussdorff metric if the initial curve is convex.Finally, the last chapter covers general remarks and an evaluation of the thesis