ALG2_L16

Definition 1(Definition).

An isometry of ${\mathbb{R}}^n$ is a distance preserving map $f$ from ${\mathbb{R}}^n$ to itself, a map such that, for all $u$ and $v$ in ${\mathbb{R}}^n$,

$$|f(u)-f(v)|=|u-v|.$$

Artin composes isometries from the right.

Lemma 2(Lemma).

Let $\mathbf{x}$ and $\mathbf{y}$ be points in ${\mathbb{R}}^n$. If the three dot products $\mathbf{x}\cdot \mathbf{x}$, $\mathbf{y}\cdot \mathbf{y}$, and $\mathbf{y}\cdot \mathbf{y}$ are equal, then $\mathbf{x}=\mathbf{y}$.

Theorem 3(Theorem).

The following conditions on a map $\phi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ are equivalent:

1. $\phi$ is an isometry that fixes the origin: $\phi (0)=0$
2. $\phi$ preserves dot products: $\phi (\mathbf{v})\cdot \phi (\mathbf{u})=\mathbf{v}\cdot \mathbf{u}$ for all $\mathbf{v}$ and $\mathbf{u}$,
3. $\phi$ is an orthogonal linear operator.

Corollary

Every isometry of ${\mathbb{R}}^n$ is the composition of an orthogonal linear operator and a translation.

More precisely, if $f$ is an isometry and if $f(0)=a$, then $f=t_a\phi$ where $t_a$ is a translation and $\phi$ is an orthogonal linear operator. This expression for $f$ is unique.

Corollary

The set of all isometries of ${\mathbb{R}}^n$ forms a group that we denote by $M_n$, with the composition of functions as its law of composition

There is an important map $\pi :M_n\to O_n$, defined by dropping the translation part of an isometry $f$.

Theorem 4(Proposition).

The map $\pi$ is a surjective homomorphism. Its kernel is the set $T=\{t_v\}$ of translations, which is a normal subgroup of $M_n$.

Change of coordinates

Let $P$ denote an $n$-dimensional space. To analyze the effect of such a change, we begin with an isometry $f$, a point $p$ of $P$, and its image $q=f(p)$, without reference to coordinates. Now, suppose we introduce a coordinate system $V=(v_1,\dots ,v_n)$ of $P$. The space $P$ becomes identified with ${\mathbb{R}}^n$, and the points $p$ and $q$ have coordinates $[p{]}_V$ and $[q{]}_V$. The isometry $f$ will have a formula $t_a\phi$ in terms of the coordinates; call it $f_V$: $f_V([p{]}_V)=[q{]}_V$. We want to determine what happens to the coordinate vectors and to the formula when we change coordinates by an isometry.

Say we change our coordinates by some isometry $\eta$ of ${\mathbb{R}}^n$, such that $\eta ([p{]}_W)=[p{]}_V$, where $W$ would be the new basis. Substituting for $[p{]}_V$ and $[q{]}_V$ in the old formula, we get

$${\eta }^{-1}{f}_V\eta ([p{]}_W)=[q{]}_W,$$

so $f_W={\eta }^{-1}{f}_V\eta$.

Isometries of the plane

Theorem 5(Theorem).

Every isometry of the plane has one of the following forms:

(a) Orientation preserving symmetries:

1. Translation: A map $t_v$ that sends $p\mapsto p+v$.
2. Rotation: rotation of the plane through a nonzero angle $\theta$ about some point.

(b) orientation-reversing isometries:
3. Reflection: a bilateral symmetry about a line $l$
4. glide reflection: a reflection about a line $l$, followed by translation by a nonzero vector parallel to $l$.