Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.
Author(s): Dirk J. Struik
Edition: 2ed.
Publisher: Dover
Year: 1988
Language: English
Pages: 243
Tags: Математика;Топология;Дифференциальная геометрия и топология;Дифференциальная геометрия;
Cover......Page 1
Title......Page 2
Copyright......Page 3
CONTENTS......Page 4
PREFACE......Page 6
BIBLIOGRAPHY......Page 8
1-1 Analytic representation......Page 12
1-2 Arc length, tangent......Page 16
1-3 Osculating plane......Page 21
1-4 Curvature......Page 24
1-5 Torsion......Page 26
1-6 Formulas of Frenet......Page 29
1-7 Contact......Page 34
1-8 Natural equations......Page 37
1-9 Helices......Page 44
1-10 General solution of the natural equations......Page 47
1-11 Evolutes and involutes......Page 50
1-12 Imaginary curves......Page 55
1-13 Ovals......Page 58
1-14 Monge......Page 64
2-1 Analytical representation......Page 66
2-2 First fundamental form......Page 69
2-3 Normal, tangent plane......Page 73
2-4 Developable surfaces......Page 77
2-5 Second fundamental form. Meusnier's theorem......Page 84
2-6 Euler's theorem......Page 88
2-7 Dupin's indicatrix......Page 94
2-8 Some surfaces......Page 97
2-9 A geometrical interpretation of asymptotic and curvature lines......Page 104
2-10 Conjugate directions......Page 107
2-11 Triply orthogonal systems of surfaces......Page 110
3-1 Gauss......Page 116
3-2 The equations of Gauss-Weingarten......Page 117
3-3 The theorem of Gauss and the equations of Codazzi......Page 121
3-4 Curvilinear coordinates in space......Page 126
3-5 Some applications of the Gauss and the Codazzi equations......Page 131
3-6 The fundamental theorem of surface theory......Page 135
4-1 Geodesic (tangential) curvature......Page 138
4-2 Geodesics......Page 142
4-3 Geodesic coordinates......Page 147
4-4 Geodesics as extremals of a variational problem......Page 151
4-5 Surfaces of constant curvature......Page 155
4-6 Rotation surfaces of constant curvature......Page 158
4-7 Non-Euclidean geometry......Page 161
4-8 The Gauss-Bonnet theorem......Page 164
5-1 Envelopes......Page 173
5-2 Conformal mapping......Page 179
5-3 Isometric and geodesic mapping......Page 186
5-4 Minimal surfaces......Page 193
5-5 Ruled surfaces......Page 200
5-6 Imaginaries in surface theory......Page 207
SOME PROBLEMS AND PROPOSITIONS......Page 212
APPENDIX: The method of Pfaffians in the theory of curves and surfaces.......Page 216
ANSWERS TO PROBLEMS......Page 228
INDEX......Page 237
