User:Egm6322.s09.bit.sahin/r3-2

$$\overrightarrow{\overline{O}P}=\overline{x}\left (m\widehat{i}+n\widehat{j} \right )+\overline{y}\left (p\widehat{i} +q\widehat{j}\right )$$

$$x\widehat{i}+y\widehat{j}=x_{0}\widehat{i}+y_{0}\widehat{j}+\overline{x}\left (m\widehat{i}+n\widehat{j} \right )+\overline{y}\left (p\widehat{i}+q\widehat{j} \right )$$

then,

$$x=x_{0}+\overline{x}m+\overline{y}p$$

$$y=y_{0}+\overline{x}n+\overline{y}q$$

from this equation system $$\overline{x}$$ and $$\overline{y}$$ are obtained as

$$\overline{x}=\frac{q}{\left (mq-np \right )}x-\frac{p}{\left (mq-np \right )}y-\frac{q}{\left (mq-np \right )}x_{0}+\frac{p}{\left (mq-np \right )}y_{0}$$

$$\overline{y}=\frac{m}{\left (mq-np \right )}y-\frac{n}{\left (mq-np \right )}x+\frac{n}{\left (mq-np \right )}x_{0}-\frac{m}{\left (mq-np \right )}y_{0}$$

we can rewrite these equations

$$\overline{x}=xcos\alpha +ysin\alpha -x_{0}cos\alpha -y_{0}sin\alpha $$

$$\overline{y}=ysin\theta +xcos\theta -x_{0}cos\theta -y_{0}sin\theta$$

where $$\alpha$$ is an angle between $$x$$ and $$\overline{x}$$, $$\theta$$ is an angle between $$y$$ and $$\overline{y}$$.

Recall:

$$\begin{Bmatrix} \underline{x}\\ \underline{y} \end{Bmatrix} = \underline{E} \begin{Bmatrix} x\\ y \end{Bmatrix} + \underline{F}$$

$$\begin{Bmatrix} \overline{x}\\ \overline{y}

\end{Bmatrix} =\begin{bmatrix} e_{11} &e_{12} \\ e_{21} &e_{22} \end{bmatrix}\begin{Bmatrix} x\\ y

\end{Bmatrix}+\begin{Bmatrix} f_{11}\\ f_{21}

\end{Bmatrix}$$

therefore,

$$\overline{x}=e_{11}x+e_{12}y+f_{11}$$

$$\overline{y}=e_{21}x+e_{22}y+f_{21}$$

Eventually we obtain $$\underline{E}$$ and $$\underline{F}$$ as following

$$\underline{E}=\begin{bmatrix} cos\alpha &sin\alpha \\ cos\theta&sin\theta \end{bmatrix}$$

$$\underline{F}=\begin{Bmatrix} \left (-x_{0}cos\alpha-y_{0}sin\alpha \right )\\ \left (-x_{0}cos\theta-y_{0}sin\theta \right ) \end{Bmatrix}$$