Isometry

DefinitionPermalink

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\to Y$ is called an isometry or distance-preserving if $\mathrm{\forall }a,b\in X:d_Y(f(a),f(b))=d_X(a,b)$.

Proposition: A isometry is automatically injective.

Proof: If not, there exist $a\ne b\in X$ such that $f(a)=f(b)$. Therefore $d_Y(f(a),f(b))=0$ but $d_X(a,b)\ne 0$. Contradiction. $◼$.

• 注意：有的定义会要求 isometry 是 bijective 的，但我们这里不采用

Isometric linear transformationPermalink

length-preserving 的 linear transformation 都是 isometry。

• 注意：length-preserving 意味着没有降维

这些 isometric linear transformation 包括：

• rotation：绕某个点旋转
• translation：平移
• reflection：对照某直线或某平面做镜像
• glides：先对照直线做 reflection，再沿着这条直线 translation
• identity

Two geometric figures related by an isometry are said to be geometrically congruent.

Isometry vs HomeomorphismPermalink

Proposition: Any bijective isometry is a homeomorphism between metric spaces.

Proof: 相当于要证明 isometry $f$ 和它的 inverse $f^{-1}$ 都是 continuous 的。

$\mathrm{\forall }ϵ>0:\mathrm{\exists }\delta =ϵ>0:d_X(a,b)<\delta \Rightarrow d_Y(f(a),f(b))=d_X(a,b)<\delta =ϵ$，所以 $f$ is continuous. (实际上 $f$ 还是 uniformly continuous, with $\delta =ϵ$)

同理可证 $f^{-1}$ 是 continuous。$◼$