# 구면(sphere)

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## 매개화

• 3차원상의 반지름이 R인 구면 $$x^2+y^2+z^2 = R^2$$
• 매개화$X(u,v)=R(\cos u \sin v, \sin u \sin v, \cos v)$$0<u<2\pi,0<v<\pi$
• 미분

$X_u=R(- \sin u \sin v , \cos u \sin v ,0)$$X_v=R( \cos u \cos v , \sin u \cos v ,-\sin v)$$N=(-\cos u \sin v, -\sin u \sin v, -\cos v)$$X_{uu}=R(-\cos u \sin v , -\sin u \sin v ,0)$$X_{uv}=R(-\cos v \sin u , \cos u \cos v , 0)$$X_{vv}=R(- \cos u \sin v , - \sin u \sin v , - \cos v )$

## 제1기본형식 (메트릭 텐서)

• $$E=R^2\sin^2 v$$
• $$F=0$$
• $$G=R^2$$

## 크리스토펠 기호

• 크리스토펠 기호 항목 참조$\Gamma^1_{11}=0$$\Gamma^1_{12}=\cot v$$\Gamma^1_{21}=\cot v$$\Gamma^1_{22}=0$$\Gamma^2_{11}=-\sin v \cos v$$\Gamma^2_{12}=0$$\Gamma^2_{21}=0$$\Gamma^2_{22}=0$

## 리만 곡률 텐서

• 리만 곡률 텐서$\begin{array}{ll} \begin{array}{ll} R_{111}^1 & 0 \\ R_{112}^1 & 0 \end{array} & \begin{array}{ll} R_{121}^1 & 0 \\ R_{122}^1 & 0 \end{array} \\ \begin{array}{ll} R_{211}^1 & 0 \\ R_{212}^1 & 1 \end{array} & \begin{array}{ll} R_{221}^1 & -1 \\ R_{222}^1 & 0 \end{array} \\ \begin{array}{ll} R_{111}^2 & 0 \\ R_{112}^2 & -\sin ^2(v) \end{array} & \begin{array}{ll} R_{121}^2 & \sin ^2(v) \\ R_{122}^2 & 0 \end{array} \\ \begin{array}{ll} R_{211}^2 & 0 \\ R_{212}^2 & 0 \end{array} & \begin{array}{ll} R_{221}^2 & 0 \\ R_{222}^2 & 0 \end{array} \end{array}$
• covariant tensor$\begin{array}{ll} \begin{array}{ll} R_{1111} & 0 \\ R_{1112} & 0 \end{array} & \begin{array}{ll} R_{1121} & 0 \\ R_{1122} & 0 \end{array} \\ \begin{array}{ll} R_{1211} & 0 \\ R_{1212} & R^2 \sin ^2(v) \end{array} & \begin{array}{ll} R_{1221} & -R^2 \sin ^2(v) \\ R_{1222} & 0 \end{array} \\ \begin{array}{ll} R_{2111} & 0 \\ R_{2112} & -R^2 \sin ^2(v) \end{array} & \begin{array}{ll} R_{2121} & R^2 \sin ^2(v) \\ R_{2122} & 0 \end{array} \\ \begin{array}{ll} R_{2211} & 0 \\ R_{2212} & 0 \end{array} & \begin{array}{ll} R_{2221} & 0 \\ R_{2222} & 0 \end{array} \end{array}$

## 측지선

• 측지선 이 만족시키는 미분방정식

$\frac{d^2\alpha_k }{dt^2} + \Gamma^{k}_{~i j }\frac{d\alpha_i }{dt}\frac{d\alpha_j }{dt} = 0$

• 풀어쓰면,

$\frac{d^2 u}{dt^2} + \Gamma^{1}_{~1 2 }\frac{du }{dt}\frac{dv }{dt} = 0$ $\frac{d^2 v}{dt^2} + \Gamma^{2}_{~1 1 }\frac{du }{dt}\frac{du }{dt} = 0$

## 가우스곡률

• 가우스곡률 항목 참조$K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right)$
• 반지름 R인 구면의 가우스곡률$K=\frac{1}{R^2}$

## 라플라시안

• 위의 좌표계에서 $$u=\phi,v=\theta$$ 로 생각하자.
• 라플라시안$\Delta f = {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}={1 \over r^2 }({\partial^2 f \over \partial \theta^2} +\cot\theta {\partial f \over \partial \theta} + \frac{1}{ \sin^2 \theta} {\partial^2 f \over \partial \phi^2})$

## 관련논문

• Neutsch, Wolfram. “Optimal Spherical Designs and Numerical Integration on the Sphere.” Journal of Computational Physics 51, no. 2 (August 1983): 313–25. doi:10.1016/0021-9991(83)90095-5.