Vector Calculus (2) - Coordinate Transformation

3. Coordinate Transformation

$ P(x_{1},x_{2}) \rightarrow P(x_{1}',x_{2}') $

$ x_{1}'=\overline{Oa}+(\overline{ab}+\overline{bc})=x_{1}\cos \Theta +x_{2}\sin \Theta $

$ x_{2}'=\overline{Od}-\overline{de}=-x_{1}\sin \Theta +x_{2}\cos \Theta $

 

$ \sin \Theta =\cos (\frac{\pi }{2}-\Theta) $ 를 이용하면, cos에 대한 식으로 만들 수 있다.

 

그리고 축이 회전하는 대신, 점 P가 반대 방향으로 회전한다고 생각해도 된다.

polar coordinate: $ P(r,\alpha ) $

$ x_{1}=r\cos \alpha , x_{2}=r\sin \alpha  $

$ x_{1}'=r\cos (\alpha-\Theta )=r\cos \alpha \cos \Theta + r\sin \alpha \sin \Theta=x_{1}\cos \Theta + x_{2}\sin \Theta $

$ x_{2}'=r\sin (\alpha-\Theta )=r\sin \alpha \cos \Theta - r\cos \alpha \sin \Theta=-x_{1}\sin \Theta + x_{2}\cos \Theta $

 

 

4. Direction Cosine, Transformation Matrix(Rotation Matrix)

 

$ (x_{i}',x_{j}) $ : angle between the $ x_{i}' $-axis and $ x_{j} $-axis

$ \lambda _{ij}\equiv \cos (x_{i}',x_{j}) $ : direction cosine

 

$ \lambda _{11}=\cos (x_{1}',x_{1})=\cos \Theta  $

$ \lambda _{12}=\cos (x_{1}',x_{2})=\cos (\frac{\pi }{2}-\Theta) =\sin \Theta  $

$ \lambda _{21}=\cos (x_{2}',x_{1})=\cos (\frac{\pi }{2}+\Theta) =-\sin \Theta  $

$ \lambda _{22}=\cos (x_{2}',x_{2})=\cos\Theta $

 

$ x_{1}'=x_{1}\cos (x_{1}',x_{1})+x_{2}\cos (x_{1}',x_{2})=\lambda _{11}x_{1}+\lambda _{12}x_{2} $

$ x_{2}'=x_{1}\cos (x_{2}',x_{1})+x_{2}\cos (x_{2}',x_{2})=\lambda _{21}x_{1}+\lambda _{22}x_{2} $

 

for three dimensions

$ x_{i}'=\sum_{j=1}^{3}\lambda _{ij}x_{j} (i=1,2,3) $

$ \bigl(\begin{smallmatrix}
x_{1}'\\ x_{2}'
\\ x_{3}'
\end{smallmatrix}\bigr)
=\lambda \bigl(\begin{smallmatrix}
x_{1} \\ x_{2}
\\ x_{3}
\end{smallmatrix}\bigr) $

 

$ \lambda =\bigl(\begin{smallmatrix}
\lambda _{11} & \lambda _{12} & \lambda _{13}\\ 
\lambda _{21} & \lambda _{22} & \lambda _{23}\\ 
\lambda _{31} & \lambda _{32} & \lambda _{33}
\end{smallmatrix}\bigr) $ : Transformation Matrix(Rotation Matrix)

 

 

5. Inverse Transformation

 

$ x_{i}=\sum_{j=1}^{3}\lambda _{ji}x_{j}' $