Isometry

Definition: Isometry

NOTE

An isometry is a transformation that preserves distance.

More formally, a function $f$ between metric spaces is an isometry if for any two points $x$ and $y$, there is $d(f(x),f(y))=d(x,y)$

This includes Translation, Rotation, Reflection, glided Reflection (the only four)

Cartan–Dieudonn´e theorem

Reflection representation of isometries

Any isometry of ${\mathbb{R}}^n$ that fixes $O$ is the product of at most $n$ reflections in hyperplanes through $O$

Proof by Induction For $n=1$: the map $x\mapsto -x$ is the only isometry that fixes $O$ in $\mathbb{R}$ For $n=k$: Suppose the statement is true for $k-1$ Let $f$ be an isometry of ${\mathbb{R}}^k$ fixing $O$ $f(v)=w$ for some $v$ and $w$ with $v\ne w$ Then we can use a hyper plane between $v$ and $w$ that reflects $v$ to $w$ (we let the axis on $w-v$ be $u$, and the reflection be $r_u$ ) The map $r_{u}f$ maps $w$ back to $v$ in the subspace $u\mathbb{R}$ (linear) and becomes a identity map in this subspace. Since the restriction of $r_{u}f$ on ${\mathbb{R}}^{k-1}$ can be represented as $k-1$ reflection (proved by induction), the isometry $r_u{r}_{u}f=f$ must can be represented as $k$ reflections.

Generalization

This theorem can be generalized to complex numbers and even finite fields

Special case: rotations as reflections

NOTE

Rotation about a point(2D)/an axis(3D (through O)) through angle $\theta$ can always be seen a composition of two reflections of any two axis that meet at angle $\frac{\theta }{2}$

This can be easily proved by plane geometry.

Observation

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The red reflections are the same as the blue reflections, since opposite reflections cancel out.