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.
Examples
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 \(\tau_{A, B}\big((1,2)\big).\)
\(\tau_{A,B}\) 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 \(\tau_{A, B}\tau_{B,C}\tau_{A,C}\big((1,2)\big).\)
The answer is \((9,10).\)
Find the equations for \(\tau_{A,B}\tau_{B,A}\).
They are \(x'=x\) and \(y'=y\).