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Summary
Differential geometry is an actively developing area of modern mathematics. This volume presents a classical approach to the general topics of the geometry of curves, including the theory of curves in n-dimensional Euclidean space. The author investigates problems for special classes of curves and gives the working method used to obtain the conditions for closed polygonal curves. The proof of the Bakel-Werner theorem in conditions of boundedness for curves with periodic curvature and torsion is also presented. This volume also highlights the contributions made by great geometers. past and present, to differential geometry and the topology of curves.
Table of Contents
| Preface | p. vii |
| Definition of a Curve | p. 1 |
| Vector-valued Functions Depending on Numerical Arguments | p. 5 |
| The Regular Curve and its Representations | p. 9 |
| Straight Line Tangent to a Curve | p. 13 |
| Osculating Plane of a Curve | p. 17 |
| The Arc Length of a Curve | p. 21 |
| The Curvature and Torsion of a Curve | p. 27 |
| Osculating Circle of a Plane Curve | p. 37 |
| Singular Points of Plane Curves | p. 41 |
| Peano's Curve | p. 49 |
| Envelope of the Family of Curves | p. 51 |
| Frenet Formulas | p. 53 |
| Determination of a Curve with Given Curvature and Torsion | p. 59 |
| Analogies of Curvature and Torsion for Polygonal Lines | p. 63 |
| Curves with a Constant Ratio of Curvature and Torsion | p. 65 |
| Osculating Sphere | p. 69 |
| Special Planar Curves | p. 73 |
| Curves in Mechanics | p. 83 |
| Curve Filling a Surface | p. 89 |
| Curves with Locally Convex Projection | p. 91 |
| Integral Inequalities for Closed Curves | p. 97 |
| Reconstruction of a Closed Curve with Given Spherical Indicatrix of Tangents | p. 101 |
| Conditions for a Curve to be Closed | p. 103 |
| Isoperimetric Property of a Circle | p. 115 |
| One Inequality for a Closed Curve | p. 119 |
| Necessary and Sufficient Condition of the Boundedness of a Curve with Periodic Curvature and Torsion | p. 121 |
| Delaunay's Problem | p. 125 |
| Jordan's Theorem on Closed Plane Curves | p. 133 |
| Gauss's Integral for Two Linked Curves | p. 139 |
| Knots | p. 149 |
| Alexander's Polynomial | p. 157 |
| Curves in n-dimensional Euclidean Space | p. 169 |
| Curves with Constant Curvatures in n-dimensional Euclidean Space | p. 177 |
| Generalization of the Fenchel Inequality | p. 183 |
| Knots and Links in Biology and One Mystery | p. 187 |
| Jones' Polynomial, Its Generalization and Some Applications | p. 191 |
| References | p. 199 |
| Index | p. 203 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
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