Differential Geometry of Plane Curves

Description

This book features plane curves―the simplest objects in differential geometry―to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem. The last chapter connects the classical with the modern by giving an introduction to the curve-shortening flow that is based on original articles but requires a minimum of previous knowledge. Over 200 figures and more than 100 exercises illustrate the beauty of plane curves and test the reader's skills. Prerequisites are courses in standard one variable calculus and analytic geometry on the plane. Cover Contents Foreword Preface Background information Structure of the book Acknowledgments Chapter 1. Plane Curves 1.1. Continuous curves 1.2. Smooth curves, tangent vectors, and tangent lines 1.3. Reparametrization and arc length 1.4. Normal and tangent vector fields 1.5. Curvature and Frenet equations 1.6. Geometric interpretation of the curvature 1.7. Curves in the complex plane 1.8. The fundamental theorem of plane curves 1.9. Local canonical form 1.10. Parallel curves 1.11. Evolutes and involutes 1.12. Exercises Chapter 2. Winding Number 2.1. The angle function 2.2. Winding number of a closed curve 2.3. Properties of the winding number 2.4. Winding number of deformable curves 2.5. Calculation of winding and intersection numbers 2.6. Applications 2.7. Exercises Chapter 3. Rotation Index 3.1. Rotation index 3.2. The total curvature 3.3. Rotation indices of simple closed curves 3.4. The total absolute curvature 3.5. Exercises Chapter 4. Jordan Curve Theorem 4.1. Jordan curve theorem 4.2. Exercises Chapter 5. Isoperimetric Inequality 5.1. The classical isoperimetric inequality 5.2. Bonnesen isoperimetric inequality 5.3. Exercises Chapter 6. Convex Curves 6.1. Closed and convex curves 6.2. Schur theorem 6.3. Curves of constant width 6.4. Length and area of convex curves 6.5. Gage isoperimetric inequality 6.6. Exercises Chapter 7. The Four-Vertex Theorem 7.1. The four-vertex theorem for convex curves 7.2. A generalization of the four-vertex theorem 7.3. The converse of the four-vertex theorem 7.4. Exercises Chapter 8. Curve-Shortening Flow 8.1. Introduction and basic properties of the flow 8.2. Convex curves under the curve-shortening flow 8.3. The theorems of Gage and Hamilton 8.4. Convergence of the curvature functions of convex curves 8.5. Evolution of simple curves: Grayson theorem 8.6. Exercises Appendix A. The Class 𝒞^{∞} Convergence of the Curvature Function Under the Curve-Shortening Flow Appendix B. Answers to Selected Exercises B.1. Chapter 1 - Page 73 B.2. Chapter 2 - Page 136 B.3. Chapter 3 - Page 160 B.4. Chapter 4 - Page 179 B.5. Chapter 5 - Page 191 B.6. Chapter 6 - Page 243 B.7. Chapter 7 - Page 266 B.8. Chapter 8 - Page 359 Bibliography Index Back Cover