Translation

A translation $\alpha$ (also known as "slide") is a bijective mapping from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ that sends point $(x,y)$ to $(x',y')$ such that the $\alpha$ has equations $\small \begin{cases} x' = x+a \\ y'=y+b \end{cases}$ for some $a, b \in \mathbb{R}$. In other words, $\alpha\big((x,y)\big)=(x',y')$. Most of the time $\alpha$ (or any lower case greek letter) is used to represent any transformations (i.e. half turns, reflection, rotation, dilation, etc.). We shall use the greek letter $\tau$ for translations.

For example, let $A=(c,d)$ and $B=(e,f)$ be points in $\mathbb{R}^{2}$. Then there are unique numbers $a$ and $b$ such that $e=c+a$ and $f=d+b$. Thus, the unique translation that takes $A$ to $B$ has equations $x'=x+(e-c)$ and $y'=y+(f-d)$. We shall denote this unique translation by $\tau_{A,B}$.

A simple illustration would be this:

Given $A=(5,6)$ and $B=(7,10),$ find $\tau_{A, B}\big((1,2)\big).$

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$\tau_{A,B}$ has equations $\small \begin{cases} x'=x+(7-5) \\ y'=y+(10-6) \end{cases}$. Hence, $\tau_{A,B}\big((1,2)\big)=(3,6).\ _\square$

Given $A=(1,2), B=(3,4),$ and $C=(5,6),$ find $\tau_{A, B}\tau_{B,C}\tau_{A,C}\big((1,2)\big).$

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The answer is $(9,10).$

Find the equations for $\tau_{A,B}\tau_{B,A}$.

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They are $x'=x$ and $y'=y$.

• Linear Transformations