I've learned the general gist of isometries: reflections, translations, rotations, and glide reflections.
However, I've been taught this geometrically. So when I want to find what $\text{ref}_{l}(\text{ref}_{m})$ (composition of two reflections at different lines $m$ and $l$) is for example, I can only draw certain shape like a triangle and see that it can be a rotation. However, apparently it can also be a translation (no idea how).
I'm asking if there are rules that can algebraically help me determine that and similar things like:
(a) a reflection in some line $l$ followed by a rotation about a point on $l$, (b) rotation about a point followed by a rotation about a different point. Are there algebraic rules of composition that can help me determine what type of isometries that these can represent?
Best Answer
The orientation perserving isometry group of the plane is $SO_2(\mathbb{R})\rtimes\mathbb{R}^2$ (the orthogonal group $SO_2(\mathbb{R})$ are the rotations and $\mathbb{R}^2$ are the translations). This can be embedded in $SL_3(\mathbb{R})$ as follows: $$ \left( \begin{array}{ccc} 1&0&0\\ a&\cos\theta&-\sin\theta\\ b&\sin\theta&\cos\theta\\ \end{array} \right) \text{first rotation by } \theta \text{ then translation by } (a,b). $$ This acts on a point $(x,y)$ of the plane by $$ \left( \begin{array}{ccc} 1&0&0\\ a&\cos\theta&-\sin\theta\\ b&\sin\theta&\cos\theta\\ \end{array} \right) \left( \begin{array}{c} 1\\ x\\ y\\ \end{array} \right). $$ For instance if $\theta=0$ we get the translation $$ (x,y)\mapsto(a+x,b+y) $$ or if $a=b=0$ we get the rotation $$ (x,y)\mapsto(x\cos\theta+y\sin\theta,x\sin\theta-y\cos\theta). $$ Together we get a rotation by $\theta$ followed by a translation by $(a,b)$: $$ (x,y)\mapsto(a+x\cos\theta+y\sin\theta,b+x\sin\theta-y\cos\theta). $$ This also extends as you might think to the full isometry group $O_2(\mathbb{R})\rtimes\mathbb{R}^2$ by throwing in a reflection, e.g. $$ \left( \begin{array}{ccc} 1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{array} \right), \ (x,y)\mapsto(-x,y), $$ and it also extends to higher dimension in the obvious way.