# Definition: Euclidean Movement - Isometry

In a Euclidean metric space $$(\mathbb R^n,d)$$ with the Euclidean distance

$d(A,B):=\sqrt{(x_1+y_1)^2+(x_2+y_2)^2+\ldots+(x_n+y_n)^2}$

for any two points $$A,B\in\mathbb R^n$$:

$\begin{array}{rcl} A&=&(x_1,x_2,\ldots,x_n),\\ B&=&(y_1,y_2,\ldots,y_n) \end{array}$

a Euclidean movement is an automorphism (i.e. reversible function of $$\mathbb R^n$$ to itself)

$f :\mathbb R^n \mapsto \mathbb R^n,$

which is also an isometry, i.e. which preserves the distance

$d(f(A),f(B))=d(A,B).$

There are four types of Euclidean movements:

1. Identity - maps every point $$A$$ to itself (no movement at all)
2. Translation - moves every point $$A$$ by the same amount in a given direction.
3. Rotation - circular movement of any point $$A$$ around a center . In the plane $$\mathbb R^2$$, this center is called the point of rotation. In the three-dimensional space $$\mathbb R^3$$, objects always rotate around an imaginary line called a rotation axis.
4. Rotation - circular movement of any point $$A$$ around a center . In the plane $$\mathbb R^2$$, this center is called the point of rotation. In the three-dimensional space $$\mathbb R^3$$, objects always rotate around an imaginary line called a rotation axis.

Definitions: 1 2

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