User:Egm6322.s09.Three.ge/MyReport3

J is a constant matrix for linear and affine maps.

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

Where:

$$\underline{J}=\underline{E}$$

$$\alpha=\begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix} \left \lfloor \partial_{x} \ \partial_{y} \right \rfloor$$

$$\begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix}\left ( \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor  \underline{J}(x,y))$$

If J is not a constant, take the product rule of the above function.

$$\underline{J}^{T}\begin{Bmatrix} \partial_{\underline{x}}\\ \partial_{\underline{y}} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}+ \begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}$$

$$Let \ \beta=\underline{J}^{T}\begin{Bmatrix} \partial_{\underline{x}}\\ \partial_{\underline{y}} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}; \; and \  \gamma= \begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}$$

In $$\beta$$ J is treated as a constant while in $$\gamma$$ the operator $$\left \lfloor \partial_{\overline{x}} \;  \partial_{\overline{y}} \right \rfloor$$ is treated as a constant. This results in a non-zero $$\gamma$$ matrix of:

$$\gamma= \begin{bmatrix} a & b\\ c & d \end{bmatrix}$$

If J is a constant then the $$\gamma$$ matrix is equal to zero.

Example of affine map that is not rotation.

(Picture of coordinate transform.)

$$\begin{matrix} \widehat{\underline{i}}=m \widehat{i}+n \widehat{j}\\ \widehat{\underline{j}}=p \widehat{i}+q \widehat{j} \end{matrix}$$

How to get the transformation matrix.

$$\overrightarrow{OP}=\overrightarrow{O\overline{O}}+\overrightarrow{\overline{O}P}$$

Noting that:

$$\begin{matrix} \overrightarrow{OP}=x\widehat{i}+y\widehat{j}\\ \overrightarrow{O\overline{O}}=x_{o}\widehat{i}+y_{o}\widehat{j}\\ \overrightarrow{\overline{O}P}=\overline{x}\widehat{\underline{i}}+\overline{y}\widehat{\underline{j}} \end{matrix}$$

We get ... (Homework Problem)