구면(sphere)

개요

• 구면기하학의 모델

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\[\Delta f = {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}={1 \over r^2 }({\partial^2 f \over \partial \theta^2} +\cot\theta {\partial f \over \partial \theta} + \frac{1}{ \sin^2 \theta} {\partial^2 f \over \partial \phi^2})\]

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표
•  

 

 

메모

• http://users-phys.au.dk/fedorov/nucltheo/GTR/09/note6.pdf

 

 

관련된 항목들

• 미분기하학
• 지도와 수학
• n차원 공의 부피
• 구면좌표계

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxYWZjZjFkNWEtNTVlNC00OTVjLWJhMDMtODc1NjgyMGQ1OTA5&sort=name&layout=list&num=50
• http://www.wolframalpha.com/input/?i=sphere

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/sphere
• http://mathworld.wolfram.com/GreatCircle.html

리뷰, 에세이, 강의노트

• Distributing points on the sphere

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.