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:
 Identity  maps every point \(A\) to itself (no movement at all)
 Translation  moves every point \(A\) by the same amount in a given direction.
 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 threedimensional space \(\mathbb R^3\), objects always rotate around an imaginary line called a rotation axis.
 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 threedimensional space \(\mathbb R^3\), objects always rotate around an imaginary line called a rotation axis.
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