Geometric Transformations (Afbeeldingen - Wiskunde Klas 4)

Vocabulary practice flashcards defining coordinate rules, compositions, and notations for 2D geometric transformations including translations, reflections, rotations, stretches, and homotheties.

Translation $T_{\begin{pmatrix} a \\ b \end{pmatrix}}$

A translation by vector $\begin{pmatrix} a \\ b \end{pmatrix}$, defined by the coordinate transformations $x_1 = x + a$ and $y_1 = y + b$.

Reflection $S_{x=0}$

A reflection in the line $x = 0$ (the y-axis), defined by the coordinate transformations $x_1 = -x$ and $y_1 = y$.

Reflection $S_{y=0}$

A reflection in the line $y = 0$ (the x-axis), defined by the coordinate transformations $x_1 = x$ and $y_1 = -y$.

Reflection $S_{x=a}$

A reflection in the vertical line $x = a$, defined by the coordinate transformations $x_1 = 2a - x$ and $y_1 = y$.

Reflection $S_{y=b}$

A reflection in the horizontal line $y = b$, defined by the coordinate transformations $x_1 = x$ and $y_1 = 2b - y$.

Point Reflection $S_{(0,0)}$

A reflection in the origin $(0,0)$, defined by the coordinate transformations $x_1 = -x$ and $y_1 = -y$.

Point Reflection $S_{(a,b)}$

A reflection in the point $(a, b)$, defined by the coordinate transformations $x_1 = 2a - x$ and $y_1 = 2b - y$.

Reflection $S_{y=x}$

A reflection in the line $y = x$, defined by the coordinate transformations $x_1 = y$ and $y_1 = x$.

Reflection $S_{y=-x}$

A reflection in the line $y = -x$, defined by the coordinate transformations $x_1 = -y$ and $y_1 = -x$.

Reflection $S_{y=x+b}$

A reflection in the line $y = x + b$, defined by the coordinate transformations $x_1 = y - b$ and $y_1 = x + b$.

Reflection $S_{y=-x+b}$

A reflection in the line $y = -x + b$, defined by the coordinate transformations $x_1 = -y + b$ and $y_1 = -x + b$.

Composition $S_{x=a} \circ S_{x=b}$

The composition of two reflections in parallel vertical lines $x = b$ and $x = a$, which equals the translation $T_{\begin{pmatrix} 2a - 2b \\ 0 \end{pmatrix}}$.

Composition $S_{y=a} \circ S_{y=b}$

The composition of two reflections in parallel horizontal lines $y = b$ and $y = a$, which equals the translation $T_{\begin{pmatrix} 0 \\ 2a - 2b \end{pmatrix}}$.

Composition $S_{(a,b)} \circ S_{(c,d)}$

The composition of two point reflections in points $(c, d)$ and $(a, b)$, which equals the translation $T_{\begin{pmatrix} 2a - 2c \\ 2b - 2d \end{pmatrix}}$.

Rotation $R_{(0,0), 90^\circ}$

A rotation of $90^\circ$ about the origin $(0,0)$, defined by the coordinate transformations $x_1 = -y$ and $y_1 = x$.

Rotation $R_{(0,0), -90^\circ}$

A rotation of $-90^\circ$ about the origin $(0,0)$, defined by the coordinate transformations $x_1 = y$ and $y_1 = -x$.

Rotation $R_{(a,b), 90^\circ}$

A rotation of $90^\circ$ about the point $(a, b)$, defined by the coordinate transformations $x_1 = -y + a + b$ and $y_1 = x - a + b$.

Rotation $R_{(a,b), -90^\circ}$

A rotation of $-90^\circ$ about the point $(a, b)$, defined by the coordinate transformations $x_1 = y + a - b$ and $y_1 = -x + a + b$.

Half-Turn Equivalence ($R_{(a,b), 180^\circ}$)

A rotation of $180^\circ$ (or $-180^\circ$) about the point $(a, b)$, which is equivalent to the point reflection in $(a, b)$: $R_{(a,b), 180^\circ} = R_{(a,b), -180^\circ} = S_{(a,b)}$.

Stretch $L_{x=0, k}$

A stretch relative to the line $x = 0$ by factor $k$, defined by the coordinate transformations $x_1 = kx$ and $y_1 = y$.

Stretch $L_{y=0, k}$

A stretch relative to the line $y = 0$ by factor $k$, defined by the coordinate transformations $x_1 = x$ and $y_1 = ky$.

Homothety $H_{(0,0), k}$

A dilation (homothety) centered at the origin $(0,0)$ with scale factor $k$, defined by the coordinate transformations $x_1 = kx$ and $y_1 = ky$.

Homothety $H_{(a,b), k}$

A dilation (homothety) centered at the point $(a, b)$ with scale factor $k$, defined by the coordinate transformations $x_1 = k(x - a) + a$ and $y_1 = k(y - b) + b$.