# Isometry

## Definition

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 $\forall a,b \in X: d_Y \left( f(a),f(b) \right)= d_X(a,b)$.

**Proposition:** A isometry is automatically injective.

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

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

## Isometric linear transformation

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 Homeomorphism

**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。

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

同理可证 $f^{-1}$ 是 continuous。$\blacksquare$

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