文档介绍：Lectures on the Differential Geometry
of Curves and Surfaces
Paul A. Blaga
To Cristina, with love
Contents
Foreword 9
I Curves 11
1 Space curves 13
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Parameterized curves (paths) . . . . . . . . . . . . . . . . . . . . . . . 14
The definition of the curve . . . . . . . . . . . . . . . . . . . . . . . . 22
Analytical representations of curves . . . . . . . . . . . . . . . . . . . 26
Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
The tangent and the normal plane . . . . . . . . . . . . . . . . . . . . . 32
The equations of the tangent line and normal plane (line) for
different representations of curves . . . . . . . . . . . . . . . . 35
The osculating plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
The curvature of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 42
The geometrical meaning of curvature . . . . . . . . . . . . . . 44
The frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
The behaviour of the frame at a parameter change . . . . 47
Oriented curves. The frame of an oriented curve . . . . . . . . . 48
The formulae. The torsion . . . . . . . . . . . . . . . . . . . . . 50
6 Contents
The geometrical meaning of the torsion . . . . . . . . . . . . . 53
Some further applications of the formulae . . . . . . . . 54
General helices. Lancret’s theorem . . . . . . . . . . . . . . . 56
Bertrand curves . . . . . . . . . . . . . . . . . . . . . . . . . . 58
The local behaviour of a parameterized curve around a biregular point . 62
The contact between a space curve and a plane . . . . . . . . . . . . . 64
The contact between a space curve and a sphere. The osculating sphere 66
Existence