Эквиаффинная геометрия

Пусть аффинное унимодулярное преобразование отображает три любые точки аффинной плоскости ${\displaystyle M_{1}(x_{1},y_{1}),}$ ${\displaystyle M_{2}(x_{2},y_{2})}$ и ${\displaystyle M_{3}(x_{3},y_{3})}$ в три точки соответственно ${\displaystyle M_{1}^{\prime }(x_{1}^{\prime },y_{1}^{\prime }),}$ ${\displaystyle M_{2}^{\prime }(x_{2}^{\prime },y_{2}^{\prime })}$ и ${\displaystyle M_{3}^{\prime }(x_{3}^{\prime },y_{3}^{\prime })}$. Прямым вычислением, используя свойства определителя, перепишем определитель:

${\displaystyle \Delta ^{\prime }={\begin{vmatrix}1&1&1\\x_{1}^{\prime }&x_{2}^{\prime }&x_{3}^{\prime }\\y_{1}^{\prime }&y_{2}^{\prime }&y_{3}^{\prime }\end{vmatrix}}=}$

${\displaystyle ={\begin{vmatrix}1&1&1\\a_{1}x_{1}+b_{1}y_{1}+c_{1}&a_{1}x_{2}+b_{1}y_{2}+c_{1}&a_{1}x_{3}+b_{1}y_{3}+c_{1}\\a_{2}x_{1}+b_{2}y_{1}+c_{2}&a_{2}x_{2}+b_{2}y_{2}+c_{2}&a_{2}x_{3}+b_{2}y_{3}+c_{2}\end{vmatrix}}=}$

${\displaystyle ={\begin{vmatrix}1&1&1\\c_{1}&c_{1}&c_{1}\\a_{2}x_{1}+b_{2}y_{1}+c_{2}&a_{2}x_{2}+b_{2}y_{2}+c_{2}&a_{2}x_{3}+b_{2}y_{3}+c_{2}\end{vmatrix}}+}$ ${\displaystyle {}+{\begin{vmatrix}1&1&1\\a_{1}x_{1}+b_{1}y_{1}&a_{1}x_{2}+b_{1}y_{2}&a_{1}x_{3}+b_{1}y_{3}\\a_{2}x_{1}+b_{2}y_{1}+c_{2}&a_{2}x_{2}+b_{2}y_{2}+c_{2}&a_{2}x_{3}+b_{2}y_{3}+c_{2}\end{vmatrix}}=}$

${\displaystyle ={\begin{vmatrix}1&1&1\\a_{1}x_{1}+b_{1}y_{1}&a_{1}x_{2}+b_{1}y_{2}&a_{1}x_{3}+b_{1}y_{3}\\a_{2}x_{1}+b_{2}y_{1}+c_{2}&a_{2}x_{2}+b_{2}y_{2}+c_{2}&a_{2}x_{3}+b_{2}y_{3}+c_{2}\end{vmatrix}}=}$

${\displaystyle ={\begin{vmatrix}1&1&1\\a_{1}x_{1}+b_{1}y_{1}&a_{1}x_{2}+b_{1}y_{2}&a_{1}x_{3}+b_{1}y_{3}\\a_{2}x_{1}+b_{2}y_{1}&a_{2}x_{2}+b_{2}y_{2}&a_{2}x_{3}+b_{2}y_{3}\end{vmatrix}}=}$

${\displaystyle =a_{1}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\a_{2}x_{1}+b_{2}y_{1}&a_{2}x_{2}+b_{2}y_{2}&a_{2}x_{3}+b_{2}y_{3}\end{vmatrix}}+{}}$
${\displaystyle {}+b_{1}{\begin{vmatrix}1&1&1\\y_{1}&y_{2}&y_{3}\\a_{2}x_{1}+b_{2}y_{1}&a_{2}x_{2}+b_{2}y_{2}&a_{2}x_{3}+b_{2}y_{3}\end{vmatrix}}=}$

${\displaystyle =a_{1}a_{2}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\x_{1}&x_{2}&x_{3}\end{vmatrix}}+a_{1}b_{2}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}+{}}$
${\displaystyle {}+b_{1}{\begin{vmatrix}1&1&1\\y_{1}&y_{2}&y_{3}\\a_{2}x_{1}+b_{2}y_{1}&a_{2}x_{2}+b_{2}y_{2}&a_{2}x_{3}+b_{2}y_{3}\end{vmatrix}}=}$

${\displaystyle =a_{1}b_{2}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}+{}}$
${\displaystyle {}+b_{1}{\begin{vmatrix}1&1&1\\y_{1}&y_{2}&y_{3}\\a_{2}x_{1}+b_{2}y_{1}&a_{2}x_{2}+b_{2}y_{2}&a_{2}x_{3}+b_{2}y_{3}\end{vmatrix}}=}$

${\displaystyle =a_{1}b_{2}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}+a_{2}b_{1}{\begin{vmatrix}1&1&1\\y_{1}&y_{2}&y_{3}\\x_{1}&x_{2}&x_{3}\end{vmatrix}}=}$

${\displaystyle =a_{1}b_{2}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}-a_{2}b_{1}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}=}$

${\displaystyle ={\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix}}{\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}=}$
${\displaystyle =\pm {\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}}=\pm \Delta .}$

Итак, ${\displaystyle \Delta ^{\prime }=\pm \Delta }$, то есть абсолютная величина определителя ${\displaystyle \Delta }$ матрицы ${\displaystyle 3\times 3}$ — инвариант трёх произвольных точек ${\displaystyle M_{1}(x_{1},y_{1}),}$ ${\displaystyle M_{2}(x_{2},y_{2})}$ и ${\displaystyle M_{3}(x_{3},y_{3})}$ унимодулярной аффинной группы.