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Plane Curves, Convex Curves, and Their Deformation

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37^ A/81 ~7S~// PLANE CURVES, CONVEX CURVES, AND THEIR DEFORMATION VIA THE HEAT EQUATION THESIS Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of MASTER OF ARTS By Johanna M. Debrecht, B.A. Denton, Texas August, 1998 Debrecht, Johanna M., Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation. Master of Arts (Mathematics), August, 1998, 93 pp., 24 figures, bibliography, 5 titles. We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple. Debrecht, Johanna M., Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation. Master of Arts (Mathematics), August, 1998, 93 pp., 24 figures, bibliography, 5 titles. We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple. 37^ A/81 ~7S~// PLANE CURVES, CONVEX CURVES, AND THEIR DEFORMATION VIA THE HEAT EQUATION THESIS Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of MASTER OF ARTS By Johanna M. Debrecht, B.A. Denton, Texas August, 1998 ACKNOWLEDGMENTS The writer gratefully acknowledges the patient and careful guidance of Dr. Joseph Iaia, who directed the work on this paper. Gratitude is also expressed for the many people who have given their encouragement, assistance, and support. These include her advisor, committee members, family, and friends. In particular, the writer wishes to express her deepest appreciation to her husband, David, whose tireless support and encouragement made this achievement possible. 111 TABLE OF CONTENTS LIST OF FIGURES vi 1 INTRODUCTION 1 2 LOCAL CURVE THEORY 5 2.1 Basic Definitions and Concepts 5 2.2 The Frenet-Serret Apparatus 15 3 PLANE CURVE THEORY 21 3.1 Basic Definitions and Results for Plane Curves 21 3.2 Rotation Index for Closed Curves 26 3.3 Rotation Index for Simple Closed Curves 37 4 CONVEX CURVES AND THE ISOPERIMETRIC INEQUALITY 47 4.1 Convex Curves 47 4.2 The Isoperimetric Inequality 53 5 THE HEAT EQUATION 60 5.1 Curves in the Plane 61 5.2 Evolution of Simple Curves with Bounded Curvature 71 5.3 Convex Curves in the Plane 81 5.4 An Application of the Isoperimetric Inequality 85 BIBLIOGRAPHY 93 iv LIST OF FIGURES 2.1 A Moving Frame: |t, N, 2?j 14 3.1 A Non-simple Curve 26 3.2 Two Simple Curves 26 3.3 Definition of 9 in the Plane 27 3.4 The Rotation Index for the Unit Circle 28 3.5 No Change in the Sign of y'/x' 31 3.6 y' Changes Sign an Odd Number of Times . 33 3.7 Points where y' = 0, x' 0 34 3.8 The Sign of y'/x' Changes from Positive to Negative 35 3.9 The Sign of y'/x' Changes from Negative to Positive 35 3.10 The Vector Valued Function p(u,v) 38 3.11 The Region of A 38 3.12 Case I and II Type Points 43 3.13 Case III and IV Type Points 44 4.1 A Convex Curve 47 4.2 The Tangent Lines at A, B, and C 48 4.3 Proof That Line I is Parallel to Line li 49 4.4 Coinciding Tangent Lines 51 4.5 A Straight Line from Si to S2 52 4.6 The Circle (3 and the Curve a Between Two Tangent Lines 54 4.7 The Circles O and O' and the Curve a 58 5.1 The Method of Deformation 61 5.2 Schur and Schmidt's Lemma 75 5.3 Convex Arc from a Circle 77 VI CHAPTER 1 INTRODUCTION The main focus of this paper is the study of convex, plane curves undergoing defor- mation by means of the heat equation. This topic was originally written about by M. Gage and R. S. Hamilton in three journal articles, [Ga, Ga2, GaHa]. In order to fully understand this topic, we begin with the basic theory of curves in R3. The majority of the material in chapters 2-4 is based on the book by Millman and Parker, [MiPa]. After defining what we mean by a curve, we discuss the importance and advantages of regular curves. The issue of how to represent the image of a curve is discussed next. This is referred to as the parametrization of the curve. It turns out that both the tangent vector field and the length of the curve are independent of the parametrization. In other words, the tangent vector field and the length of a curve are "geometric" quantities. Thus, we are able to represent it in any form we choose. There are definite advantages to using the arc length to parametrize the curve. We will largely assume that curves are parametrized by arc length, (we say such a curve has unit speed), and any exceptions will be so noted. Thereafter, we define the normal, and binormal vector fields, as well as the curvature and torsion of a curve. After the basic definitions have been made, we develop the Frenet-Serret apparatus for curves which have been parametrized by arc length. The Frenet-Serret apparatus is the basic tool used in the study of curves. It involves three vector fields along the given curve, namely the tangent, normal and binormal, and two real valued functions, the curvature and the torsion. In fact, the Frenet-Serret apparatus uniquely and completely determines the geometry of the curve. Having established the foundation for a study of curves, we restrict our focus to plane curves; i.e., curves which lie in a plane. It is not always clear from the parametrization of a curve whether or not the image lies in R3 or R2. (Though the curve may not technically lie in R2, by a suitable choice of coordinates, we can, without loss of generality, suppose that it does.) We will present an example in which this is the case. Then, we will develop a theorem which states that certain conditions in the Frenet-Serret apparatus are equivalent to ensuring that the curve lies in a plane. That is, by merely examining the Frenet-Serret apparatus of the curve, we can easily determine if it lies in a plane. We then proceed to make "modifications" to the definitions for the tangent and normal vector fields, and the curvature which are specific to plane curves. The ad- vantage to making these changes, is that we can make well-defined definitions; that is, the definitions are defined everywhere in the plane. (This is not the case for curves in R3; in particular, the normal vector field is not defined when the curvature (in R3) is zero.) From there, we continue by defining closed curves, simple curves, and the period of a closed curve. Our next focus is on the rotation index of a closed, plane curve. Roughly speaking, the rotation index is an integer which represents the number of times the curve makes a complete rotation (of 360°) before "closing up;" that is, before returning to an arbitrarily chosen starting point. As we will discover, the rotation index is dependent on the number of times the slope of the tangent line (to any point on the curve) changes sign, and upon the means by which the sign change occurs. Following this theorem, we are lead to discover that the rotation index for a simple, closed, plane curve is ±1. (A simple curve is one which does not intersect itself at any point between the starting point and the length of the curve.) The fourth chapter consists of two theorems, seemingly unrelated, and yet both intertwined in the results developed in the final chapter dealing with the deformation of convex curves via the heat equation. We begin by making a precise definition for what we mean by a convex curve. This definition automatically implies that convex curves are simple. Following this, we develop a sufficient and necessary condition for a simple, closed, plane curve to be convex.
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