The calculator below will calculate the image of the points in two-dimensional space after applying the transformation.

First, enter up to 10 points coordinates $\left(x,y\right)$

$A$
(, )
$B$
(, )
$C$
(, )
$D$
(, )
$E$
(, )
$F$
(, )
$G$
(, )
$H$
(, )
$I$
(, )
$J$
(, )

Then choose the transformation, enter any parameter if needed (angle, scale factor, etc), and choose the rounding option

°
decimal places

Please report any error to [email protected]

#### Transformation Matrices

The above transformations (rotation, reflection, scaling, and shearing) can be represented by matrices. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate.

The transformation matrices are as follows:

Type of transformation Transformation matrix
Clockwise rotation by an angle $\theta$ about the origin $\left(\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)$
Counter-clockwise rotation by an angle $\theta$ about the origin $\left(\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)$
Reflection against the $x$-axis $\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$
Reflection against the $y$-axis $\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$
Scaling (contraction or dilation) in both $x$ and $y$ directions by a factor $k$ $\left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right)$
Horizontal shear (parallel to the $x$-axis) by a factor $m$ $\left(\begin{array}{cc}1& m\\ 0& 1\end{array}\right)$
Vertical shear (parallel to the $y$-axis) by a factor $m$ $\left(\begin{array}{cc}1& 0\\ m& 1\end{array}\right)$

Examples:

• Rotate point $A\left(2,3\right)$ clockwise about the origin by an angle $90°$.

$\begin{array}{rl}\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{cc}\mathrm{cos}90°& \mathrm{sin}90°\\ -\mathrm{sin}90°& \mathrm{cos}90°\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}2\\ 3\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{c}3\\ -2\end{array}\right)\end{array}$

${A}^{\prime }\left(3,-2\right)$

• Reflect point $B\left(-3,4\right)$ against the $x$-axis.

$\begin{array}{rl}\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}-3\\ 4\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)& =\left(\begin{array}{c}-3\\ -4\end{array}\right)\end{array}$

${B}^{\prime }\left(-3,-4\right)$

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