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四参数转换使用的公式

tk posted @ Feb 24, 2012 09:05:41 AM in 业余 with tags LaTeX Tex , 4022 阅读

设四参数转换 尺度 为 $k$,旋转角为$\theta$,坐标系位移量分别为$\Delta X$$\Delta Y$则单点转换公式为

\begin{bmatrix} X_1\\Y_1  \end{bmatrix} = \begin{bmatrix} \Delta X\\ \Delta Y  \end{bmatrix} +  \begin{bmatrix} k \cdot \cos\theta & - k \cdot \sin\theta \\ k \cdot \sin\theta & k \cdot \cos\theta \end{bmatrix} \begin{bmatrix} X_0\\Y_0  \end{bmatrix}

变换后的矩阵形式为

\begin{bmatrix}  1&0  & X_0 &-Y_0 \\   0&1  &Y_0  & X_0 \end{bmatrix} \begin{bmatrix} \Delta X \\ \Delta Y \\ k \cdot \cos\theta \\ k \cdot \sin\theta \end{bmatrix} = \begin{bmatrix}  X_1 \\ Y_1 \end{bmatrix}

这就是类似 \[BX=A\] 的平差式

所以可知

        \[X=\bigl(B^TB\bigr)^{-1}B^TA\]

 

PS:实验下学 \LaTeX{} 怎么写公式

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