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 three-dimensional 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 three-dimensional space \(\mathbb R^3\), objects always rotate around an imaginary line called a rotation axis.

Definitions: 1 2

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