1次変換の解説

${\displaystyle \left(\begin{array}{c}x\\ y\end{array}\right)}$ を基本ベクトル表示すると，

${\displaystyle \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}x\\ 0\end{array}\right)+\left(\begin{array}{c}0\\ y\end{array}\right)}$ ${\displaystyle =x\left(\begin{array}{c}1\\ 0\end{array}\right)+y\left(\begin{array}{c}0\\ 1\end{array}\right)}$ ${\displaystyle =x{e}_1+y{e}_2}$

となる．

${\displaystyle \left(\begin{array}{c} x^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)}$ を以下のように式変形をする．

${\displaystyle \left(\begin{array}{c} x^{\prime }\\ y^{\prime }\end{array}\right)}$ ${\displaystyle =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)}$

${\displaystyle =\left(\begin{array}{c}ax+by\\ cx+dy\end{array}\right)}$

${\displaystyle =\left(\begin{array}{c}ax\\ cx\end{array}\right)+\left(\begin{array}{c}by\\ dy\end{array}\right)}$

${\displaystyle =x\left(\begin{array}{c}a\\ c\end{array}\right)+y\left(\begin{array}{c}b\\ d\end{array}\right)}$

${\displaystyle a=\left(\begin{array}{c}a\\ c\end{array}\right)}$  ， ${\displaystyle b=\left(\begin{array}{c}b\\ d\end{array}\right)}$ とおくと

下図のように，１次変換によって正方格子が平行四辺形の格子に変形する．

${\displaystyle =xa+yb}$

               

点 ${\displaystyle \text{P}}$ ， ${\displaystyle \text{Q}}$ ， ${\displaystyle \text{R}}$ ， ${\displaystyle \text{S}}$ は，１次変換によって，点 ${\displaystyle {\text{P}}^{\prime }}$ ， ${\displaystyle {\text{Q}}^{\prime }}$ ， ${\displaystyle {\text{R}}^{\prime }}$ ， ${\displaystyle {\text{R}}^{\prime }}$ に移る．

 

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最終更新日： 2025年4月22日