# 测地曲率

${\displaystyle k_{g}={\ddot {r}}\cdot \varepsilon }$

## 相关命题

• 曲面S上的曲线（C），它在P点的测地曲率的绝对值等于（C）在P点的切平面上的正投影曲线（C'）的曲率。
• ${\displaystyle k^{2}=k_{g}^{2}+k_{n}^{2}}$

## 二維曲面常用的測地曲率公式

${\displaystyle (g_{ij})={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}E&F\\F&G\\\end{pmatrix}}}$

### 二維曲面測地曲率之Beltrami公式

${\displaystyle C}$為曲面S上的一正則曲線，在此曲線上以其弧長${\displaystyle s}$為參數，則曲線${\displaystyle C}$的參數方程式為${\displaystyle C:r(s)=(u(s),v(s))}$，則它在P點的測地曲率${\displaystyle k_{g}}$可表為下列克氏符號（全稱克里斯多福符號Christoffel symbols）相關的表示式[1] [2] [3]

${\displaystyle k_{g}={\sqrt {EG-F^{2}}}\left[\Gamma _{11}^{2}\left({\frac {du}{ds}}\right)^{3}+\left(2\Gamma _{12}^{2}-\Gamma _{11}^{1}\right)\left({\frac {du}{ds}}\right)^{2}{\frac {dv}{ds}}+\left(\Gamma _{22}^{2}-2\Gamma _{12}^{1}\right){\frac {du}{ds}}\left({\frac {dv}{ds}}\right)^{2}-\Gamma _{22}^{1}\left({\frac {dv}{ds}}\right)^{3}+{\frac {du}{ds}}{\frac {d^{2}v}{ds^{2}}}-{\frac {d^{2}u}{ds^{2}}}{\frac {dv}{ds}}\right]}$

ij

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${\displaystyle (k_{g})_{u-line}=\Gamma _{11}^{2}{\dfrac {\sqrt {EG-F^{2}}}{E{\sqrt {E}}}}=\Gamma _{11}^{2}\left({\dfrac {g^{1/2}}{g_{11}^{3/2}}}\right)}$

${\displaystyle (k_{g})_{v-line}=-\Gamma _{22}^{1}{\dfrac {\sqrt {EG-F^{2}}}{G{\sqrt {G}}}}=-\Gamma _{22}^{1}\left({\dfrac {g^{1/2}}{g_{22}^{3/2}}}\right)}$

### 二維曲面測地曲率之Liouville公式

${\displaystyle C}$為曲面S上的一正則曲線，在此曲線上以其弧長${\displaystyle s}$為參數，則曲線${\displaystyle C}$的參數方程式為${\displaystyle C:r(s)=(u(s),v(s))}$，今其參數化是採正交座標系，換言之，第一基本形式的係數${\displaystyle F=0}$，又令曲線${\displaystyle C}$在P點與${\displaystyle u}$座標線的夾角為${\displaystyle \theta }$，則它在P點的測地曲率${\displaystyle k_{g}}$可表為下列與${\displaystyle \theta (s)}$夾角相關的Liouville公式[6] [7] [8]

{\displaystyle {\begin{aligned}k_{g}&={\dfrac {d\theta (s)}{ds}}-{\dfrac {1}{2{\sqrt {G}}}}{\dfrac {\partial \ln E}{\partial v}}\cos \theta +{\dfrac {1}{2{\sqrt {E}}}}{\dfrac {\partial \ln G}{\partial u}}\sin \theta \\&={\dfrac {d\theta (s)}{ds}}+(k_{g})_{u-line}\cos \theta +(k_{g})_{v-line}\sin \theta \\&={\dfrac {d\theta (s)}{ds}}+(k_{g})_{u-line}{\sqrt {E}}{\dfrac {du}{ds}}+(k_{g})_{v-line}{\sqrt {G}}{\dfrac {dv}{ds}}\end{aligned}}}

${\displaystyle (k_{g})_{u-line}=-{\dfrac {E_{v}}{2E{\sqrt {G}}}}}$

${\displaystyle (k_{g})_{v-line}={\dfrac {G_{u}}{2G{\sqrt {E}}}}}$

## 參考文獻

1. ^ Kreyszig, Erwin. Differential Geometry. Dover Publications, New York. 1991: 154-156. ISBN 978-0-486-66721-8.
2. ^ Patrikalakis, Nicholas M.; Maekawa, Takashi. Shape Interrogation for Computer Aided Design and Manufacturing. Springer, New York. 2002: 266-268. ISBN 978-3-642-04073-3.【推導過程見MIT線上開放課程 §10.2.1. Parametric surfaces】
3. ^ Blaga, Paul A. Lectures on the Differential Geometry of Curves and Surfaces. Napoca Press, Cluj-Napoca, Romania. 2005: 177-179. ISBN 9736568962.
4. ^ Nayak, Prasun Kumar. Textbook of Tensor Calculus and Differential Geometry. PHI Learning Pvt. Ltd., New Delhi. 2011: 364,369.
5. ^ Slobodyan, Yu.S., Geodesic curvature, (编) Hazewinkel, Michiel, 数学百科全书, Springer, 1989, ISBN 978-1-55608-010-4
6. ^ do Carmo, Manfredo P. Differential Geometry of Curves and Surfaces. Prentice-Hall. 1976: 253-254. ISBN 0-13-212589-7.
7. ^ Gray, Alfred; Abbena, Elsa; Salamon, Simon. Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition. Chapman & Hall/CRC. 2006: 904-905. ISBN 978-1584884484.
8. ^ Dube, K.K. Differential Geometry and Tensors. I. K. International Pvt Ltd. 2009: 200-201. ISBN 978-9380026589.
9. ^ Sigurd Angenent. A note and two problems on Liouville's formula. 這是介紹測地曲率之Liouville公式更加精簡形式的文件。