# How do I compose isometries algebraically?

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?

• You assume isometries of the plane $\Bbb{R}^2$? We can use matrices. – Dietrich Burde Aug 3 '18 at 20:55
• @DietrichBurde Yes, sorry I should have said that. It's all in $\mathbb{R}^2$. I'd love to know this matrices method. – Notsredt Aug 3 '18 at 20:57

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.
• Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful! – Notsredt Aug 3 '18 at 22:50
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$\left(\frac{p(a^{2}-b^{2})-2b(aq+c)}{a^{2}+b^{2}},\frac{q(b^{2}-a^{2})-2a(bp+c)}{a^{2}+b^{2}}\right).$$The rotation of $(p,q)$ with angle $\theta$ around $(a,b)$ is$$\bigl(a+(p-a)\cos(\theta )-(q-b)\sin(\theta),b+(p-a) \sin (\theta )+(q-b) \cos (\theta )\bigr).$$You can use these formulas to compute the composition of rotations or of reflections.