4.2 Reflections, Rotations and Translations

4.9   Definition (Reflections and Rotations.) We now define a family of functions from $\mbox{{\bf R}}^2$ to $\mbox{{\bf R}}^2$. If $(x,y) \in \mbox{{\bf R}}^2$, we define

$$I(x,y) = (x,y) \mbox{(Identity function.)}$$ (4.10)

$$H(x,y) = (x,-y) \mbox{(Reflection of $(x,y)$ about the horizontal axis.)}$$

$$V(x,y) = (-x,y) \mbox{(Reflection of $(x,y)$ about the vertical axis.)}$$

$$D_+ (x,y) = (y,x) \mbox{(Reflection of $(x,y)$ about the line $y=x$.)}$$

$$D_- (x,y) = (-y,-x) \mbox{(Reflection of $(x,y)$ about the line $y=-x$.)}$$

$$R_{\pi /2}(x,y) = (y,-x) \mbox{(Clockwise rotation of $(x,y)$ by ${\pi\over 2}$.)}$$

$$R_{-{\pi\over 2}}(x,y)= (-y,x) \mbox{(Counter-clockwise rotation of $(x,y)$ by ${\pi\over 2}$.)}$$

$$R_\pi (x,y) = (-x,-y) \mbox{Rotation by $\pi$.}$$ (4.11)

Each of the eight functions just defined carries every box to another box with the same area. You should be able to see from the picture that

$$H\left(B(a,b\colon c,d)\right)=B(a,b\colon -d,-c).$$

We can see this analytically as follows:

$$\begin{eqnarray*} (x,y)\in B(a,b\colon c,d) & \iff & a\leq x\leq b \mbox{ and $c... ...\in B(a,b\colon -d,-c)\\ & \iff & H(x,y)\in B(a,b\colon -d,-c). \end{eqnarray*}$$

I will usually omit the analytic justification in cases like this.

Each of the eight functions described in definition 4.9 carries the square $B(-1,1\colon -1,1)$ to itself.

4.12   Definition (Symmetry of the square.) The eight functions defined in equations (4.10)-(4.11) are called symmetries of the square.

4.13   Exercise. Let $F$ be the set shown in the figure. On one set of axes draw the sets $F, R_{\pi /2}(F), R_{-{\pi\over 2}}(F)$ and $R_\pi (F)$ (label the four sets). On another set of axes draw and label the sets $V(F), H(F), D_+ (F)$ and $D_-(F)$.

4.14   Example. Let $a \in \mbox{${\mbox{{\bf R}}}^{+}$}$ and let

$$\begin{eqnarray*} S & = & \{ (x,y)\colon 0\leq x\leq\sqrt a \mbox{ and $0\leq y\... ...(x,y)\colon 0\leq x\leq a \mbox{ and $\sqrt x \leq y \leq a\}.$} \end{eqnarray*}$$

From the picture it is clear that $D_+(S)=T$. An analytic proof of this result is as follows:

$$(x,y)\in S \iff 0\leq x\leq\sqrt a \mbox{ and $0\leq y\leq x^2$}$$ (4.15)

$$\mbox{$\Longrightarrow$} 0\leq y\leq x^2\leq(\sqrt a)^2 \mbox{ and $0\leq \sqrt y\leq x \leq \sqrt a$}$$

$$\mbox{$\Longrightarrow$} 0\leq y\leq a \mbox{ and $\sqrt y\leq x\leq\sqrt a$}$$ (4.16)

$$\iff (y,x)\in T$$

$$\iff D_+(x,y)\in T.$$

To show that $D_+(x,y)\in T\mbox{$\Longrightarrow$}(x,y)\in S$, I need to show that (4.16) implies (4.15). This follows because

$$0\leq y\leq a \mbox{ and } \sqrt y\leq x\leq\sqrt a\; \mbox{$... ...\; 0\leq x\leq\sqrt a \mbox{ and } 0\leq y=(\sqrt y)^2\leq x^2.$$

In exercise 2.18 you assumed that $S$ and $T$ have the same area. In general we will assume that if $S$ is a set and $F$ is a symmetry of the square, then $S$ and $F(S)$ have the same area. (Cf. Assumption 5.11.)

4.17   Definition (Translate of a set.) Let $S$ be a set in $\mbox{{\bf R}}^2$ and let $\mbox{{\bf a}}\in\mbox{{\bf R}}^2$. We define the set $\mbox{{\bf a}}+S$ by

$$\mbox{{\bf a}}+S=\{\mbox{{\bf a}}+{\bf s}\colon{\bf s}\in S\}.$$

Sets of the form $\mbox{{\bf a}}+S$ will be called translates of $S$.

4.18   Example. The pictures below show some examples of translates. Intuitively each translate of $S$ has the same shape as $S$ and each translate of $S$ has the same area as $S$.

4.19   Example (Translates of line segments.) Let $\mbox{{\bf a}},\mbox{{\bf b}}\in\mbox{{\bf R}}^2$. If $\mathbf{{\bf c}}\in\mbox{{\bf R}}^2$, then

$$\begin{eqnarray*} \mathbf{{\bf c}}+[\mbox{{\bf a}}\mbox{{\bf b}}] &=& \mathbf{{\... ...thbf{{\bf c}}+\mbox{{\bf a}})(\mathbf{{\bf c}}+\mbox{{\bf b}})]. \end{eqnarray*}$$

In particular $-\mbox{{\bf a}}+[\mbox{{\bf a}},\mbox{{\bf b}}]=[\mbox{{\bf0}},\mbox{{\bf b}}-\mbox{{\bf a}}]$, so any segment can be translated to a segment with $\mbox{{\bf0}}$ as an endpoint.

4.20   Exercise. Let $a,b,c,d,r,s$ be real numbers with $a\leq b$ and $c\leq d$. Show that

$$(r,s)+B(a,b\colon c,d)=B(?,?;?,?)$$

if the four question marks are replaced by suitable expressions. Include some explanation for your answer.

4.21   Exercise. Let $P$ be the set shown in the figure below.

a) Sketch the sets $(-2,-2) + P$ and $(4,1) + P$.

b) Sketch the sets $R_{\pi\over 2}( (1,1) + P)$ and $(1,1) + R_{\pi\over 2}(P)$, where $R_{\pi\over 2}$ is defined as in definition 4.9