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Lectures on Classical Differential Geometry
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Lectures on Classical Differential Geometry Contents Foreword 9 I Curves 11 1 Space curves 13 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Parameterized curves (paths) . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 The definition of the curve . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Analytical representations of curves . . . . . . . . . . . . . . . . . . . 26 1.4.1 Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4.2 Space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 The tangent and the normal plane . . . . . . . . . . . . . . . . . . . . . 32 1.5.1 The equations of the tangent line and normal plane (line) for different representations of curves . . . . . . . . . . . . . . . . 35 1.6 The osculating plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.7 The curvature of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.7.1 The geometrical meaning of curvature . . . . . . . . . . . . . . 44 1.8 The Frenet frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.8.1 The behaviour of the Frenet frame at a parameter change . . . . 47 1.9 Oriented curves. The Frenet frame of an oriented curve . . . . . . . . . 48 1.10 The Frenet formulae. The torsion . . . . . . . . . . . . . . . . . . . . . 50 1.10.1 The geometrical meaning of the torsion . . . . . . . . . . . . . 53 1.10.2 Some further applications of the Frenet formulae . . . . . . . . 54 1.10.3 General helices. Lancret’s theorem . . . . . . . . . . . . . . . 56 1.10.4 Bertrand curves . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.11 The local behaviour of a parameterized curve around a biregular point . 62 1.12 The contact between a space curve and a plane . . . . . . . . . . . . . 64 1.13 The contact between a space curve and a sphere. The osculating sphere 66 1.14 Existence and uniqueness theorems for parameterized curves . . . . . . 68 1.14.1 The behaviour of the Frenet frame under a rigid motion . . . . . 68 1.14.2 The uniqueness theorem . . . . . . . . . . . . . . . . . . . . . 70 1.14.3 The existence theorem . . . . . . . . . . . . . . . . . . . . . . 72 2 Plane curves 75 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2 Envelopes of plane curves . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2.1 Curves given through an implicit equation . . . . . . . . . . . . 77 2.2.2 Families of curves depending on two parameters . . . . . . . . 79 2.2.3 Applications: the evolute of a plane curve . . . . . . . . . . . . 79 2.3 The curvature of a plane curve . . . . . . . . . . . . . . . . . . . . . . 81 2.3.1 The geometrical interpretation of the signed curvature . . . . . 84 2.4 The curvature center. The evolute and the involute of a plane curve . . . 86 2.5 The osculating circle of a curve . . . . . . . . . . . . . . . . . . . . . . 91 2.6 The existence and uniqueness theorem for plane curves . . . . . . . . . 92 3 The integration of the natural equations of a curve 95 3.1 The Riccati equation associated to the natural equations of a curve . . . 95 3.2 Examples for the integration of the natural equation of a plane curve . . 96 Problems 103 II Surfaces 113 4 General theory of surfaces 115 4.1 Parameterized surfaces (patches) . . . . . . . . . . . . . . . . . . . . . 115 4.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 Representations of surfaces . . . . . . . . . . . . . . . . . . . . 116 4.3 The equivalence of local parameterizations . . . . . . . . . . . . . . . . 119 4.4 Curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.5 The tangent vector space, the tangent plane and the normal to a surface . 123 4.6 The orientation of surfaces . . . . . . . . . . . . . . . . . . . . . . . . 127 4.7 Differentiable maps on a surface . . . . . . . . . . . . . . . . . . . . . 130 4.8 The differential of a smooth map between surfaces . . . . . . . . . . . 134 4.9 The spherical map and the shape operator of an oriented surface . . . . 136 4.10 The first fundamental form of a surface . . . . . . . . . . . . . . . . . 139 4.10.1 First applications . . . . . . . . . . . . . . . . . . . . . . . . . 140 The length of a segment of curve on a surface . . . . . . . . . . 140 The angle of two curves on a surface . . . . . . . . . . . . . . . 141 The area of a parameterized surface . . . . . . . . . . . . . . . 142 4.11 The matrix of the shape operator . . . . . . . . . . . . . . . . . . . . . 144 4.12 The second fundamental form of an oriented surface . . . . . . . . . . 146 4.13 The normal curvature. The Meusnier’s theorem . . . . . . . . . . . . . 148 4.14 Asymptotic directions and asymptotic lines on a surface . . . . . . . . . 150 4.15 The classification of points on a surface . . . . . . . . . . . . . . . . . 152 4.16 Principal directions and curvatures . . . . . . . . . . . . . . . . . . . . 156 4.16.1 The determination of the lines of curvature . . . . . . . . . . . 159 4.16.2 The computation of the curvatures of a surface . . . . . . . . . 161 4.17 The fundamental equations of a surface . . . . . . . . . . . . . . . . . 162 4.17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.17.2 The differentiation rules. Christoffel’s coefficients . . . . . . . . 162 Christoffel’s andWeingarten’s coefficients in curvature coordinates...164 4.17.3 The Gauss’ and Codazzi-Mainardi’s equations for a surface . . 165 4.17.4 The fundamental theorem of surface theory . . . . . . . . . . . 167 4.18 The Gauss’ egregium theorem . . . . . . . . . . . . . . . . . . . . . . 172 4.19 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.19.2 The Darboux frame. The geodesic curvature and geodesic torsion 175 4.19.3 Geodesic lines . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Examples of geodesics . . . . . . . . . . . . . . . . . . . . . . 181 4.19.4 Liouville surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 182 5 Special classes of surfaces 185 5.1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.1.1 General ruled surfaces . . . . . . . . . . . . . . . . . . . . . . 185 The parameterization of a ruled surface . . . . . . . . . . . . . 186 The tangent plane and the first fundamental form of a ruled surface187 5.1.2 The Gaussian curvature of a ruled surface . . . . . . . . . . . . 189 5.1.3 Envelope of a family of surfaces . . . . . . . . . . . . . . . . . 190 5.1.4 Developable surfaces . . . . . . . . . . . . . . . . . . . . . . . 191 Developable surfaces as envelopes of a one-parameter family of planes.The regression edge of a developable surface ....193 5.1.5 Developable surfaces associated to the Frenet frame of a space curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 The envelope of the family of osculating planes . . . . . . . . . 197 The envelope of the family of normal planes (the polar surface) 198 The envelope of the family of rectifying planes of a space curves 199 5.2 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2.1 Definition and general properties . . . . . . . . . . . . . . . . . 200 5.2.2 Minimal surfaces of revolution . . . . . . . . . . . . . . . . . . 206 5.2.3 Ruled minimal surfaces . . . . . . . . . . . . . . . . . . . . . . 207 5.3 Surfaces of constant curvature . . . . . . . . . . . . . . . . . . . . . . 213 Problems 217 Bibliography 229 Index 233 [url][/url] [ Last edited by mainpro on 2007-3-27 at 00:25 ] |
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