Lecturer: 蔡宜洵 I-Hsun Tsai
Textbook: Differential Geometry of Curves and Surfaces, Do Carmo
有人說幾何學是數學家的試金石，如果你不排斥，那麼你就可以當數學家。
大學幾何和高中幾何是完全不同的範疇，是高斯之所以被稱作數學王子的學問。
- Chp 1 ~ 3.3 + 4.2
Curve Theory
Arc length parameter and arbitrary parameter
Serret-Frenet Formula
Fundamental Theorem of Curve Local Theory
Turning Tangents Theorem
Osculating Plane
Center of Curvature
Huygens Pendulum and Envelopes
2-D Continuous Rigid Motion
Surface Theory
Regular Surface and Local Coordinates
Regular Value and Singularity
Change of Surface Coordinates
Differentiability of Surface
Surface of Revolution
Parameterized Surface
Differential
Tangent Plane
Inverse Function Theorem
First Fundamental Form
Isometric and Conformal Mapping
Orientation and Normal Vector Field
Second Fundamental Form
Normal Curvature
Gauss Curvature
Mean Curvature
Principal Direction
Umbilical points
Line of Curvature and Asymptotic Direction
Dupin Indicatrix
- Chp 3.4, 3.5, 4.3~4.4
Surface Theory
Developable Surface
Local Property of Tangent Plane
Conjugate Direction and Dupin Indicatrix
Vector Field, Trajectory and Integral Curve
Special coordinate curves
Ruled Surface
Doubly Ruling
Geometric Deduction of Line of striction and Second Defining Property
Distributed Parameter and Curvature of Ruling
3-D Continuous Rigid Motion
Theorema Egregium
Compatibility Equations and Christoffel Symbols
Fundamental Theorem of Surface Local Theory (Bonnet Theorem)
Covariant Derivatives and Parallel Transport
Geodesic and Geodesic Curvature
Lioville Formula
Geodesic Form of Surface of Revolution
Clairant Relation
- Chp 4.5
Surface Theory
Gauss-Bonnet Theorem
The Turning Tangent Theorem
Jacobi Theorem
Poincare Theorem
