structural:scholar_edition
Isometries

An isometry between two metric spaces is a bijection preserving the distance, that is

d (f (x), f (y)) = d (x, y). {\displaystyle d(f(x),f(y))=d(x,y).}

In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm

‖ f (x) ‖ = ‖ x ‖, {\displaystyle \|f(x)\|=\|x\|,}

since the norm of a vector is its distance from the zero vector. It preserves also the inner product

f (x) ⋅ f (y) = x ⋅ y, {\displaystyle f(x)\cdot f(y)=x\cdot y,}

since

x ⋅ y = 1 2 (‖ x + y ‖ 2 − ‖ x ‖ 2 − ‖ y ‖ 2). {\displaystyle x\cdot y={\tfrac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}

An isometry of Euclidean vector spaces is a linear isomorphism.

An isometry f: E → F {\displaystyle f\colon E\to F} of Euclidean spaces defines an isometry f →: E → → F → {\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if E and F are Euclidean spaces, O ∈ E, O ′ ∈ F, and f →: E → → F → {\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} is an isometry, then the map f: E → F {\displaystyle f\colon E\to F} defined by

f (P) = O ′ + f → (O P →) ({\displaystyle f(P)=O'+{\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr)}{\vphantom {\frac {(}{}}}}

is an isometry of Euclidean spaces.

It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.