Euclidean Metric on Real Number Plane is Rotation Invariant

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Theorem

Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.

Let $d_2$ denote the Euclidean metric on $\R^2$.


Then $d_2$ is unchanged by application of $r_\alpha$:

$\forall x, y \in \R^2: \map {d_2} {\map {r_\alpha} x, \map {r_\alpha} y} = \map {d_2} {x, y}$


Proof

Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.

Note that $\paren {\map {d_2} {x, y} }^2$ can be expressed as:

$\paren {\map {d_2} {x, y} }^2 = \paren {\mathbf x - \mathbf y}^\intercal \paren {\mathbf x - \mathbf y}$

where:

$x$ and $y$ are expressed in vector form: $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \end {pmatrix}, y = \begin {pmatrix} y_1 \\ y_2 \end {pmatrix}$
$\paren {\mathbf x - \mathbf y}^\intercal$ denotes the transpose of $\paren {\mathbf x - \mathbf y}$


Then:

\(\ds \map {d_2} {\map {r_\alpha} x, \map {r_\alpha} y}\) \(=\) \(\ds \map {d_2} {\begin {pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end {pmatrix} \begin {pmatrix} x_1 \\ x_2 \end {pmatrix}, \begin {pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end {pmatrix} \begin {pmatrix} y_1 \\ y_2 \end {pmatrix} }\) Matrix Form of Plane Rotation
\(\ds \) \(=\) \(\ds \map {d_2} {\begin {pmatrix} x_1 \cos \alpha + x_2 \sin \alpha \\ -x_1 \sin \alpha + x_2 \cos \alpha \end {pmatrix}, \begin {pmatrix} y_1 \cos \alpha + y_2 \sin \alpha \\ -y_1 \sin \alpha + y_2 \cos \alpha \end {pmatrix} }\)
\(\ds \) \(=\) \(\ds \sqrt {\begin {pmatrix} \paren {x_1 - y_1} \cos \alpha + \paren {x_2 - y_2} \sin \alpha \\ -\paren {x_1 - y_1} \sin \alpha + \paren {x_2 - y_2} \cos \alpha \end {pmatrix}^\intercal \begin {pmatrix} \paren {x_1 - y_1} \cos \alpha + \paren {x_2 - y_2} \sin \alpha \\ -\paren {x_1 - y_1} \sin \alpha + \paren {x_2 - y_2} \cos \alpha \end {pmatrix} }\) Definition of Euclidean Metric on Real Number Plane
\(\ds \) \(=\) \(\ds \sqrt {\paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2} \paren {\cos^2 \alpha + \sin^2 \alpha} + 2 \paren {x_1 - y_1} \paren {x_2 - y_2} \cos \alpha \sin \alpha - 2 \paren {x_1 - y_1} \paren {x_2 - y_2} \cos \alpha \sin \alpha}\) multiplying out and gathering terms
\(\ds \) \(=\) \(\ds \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}\) Sum of Squares of Sine and Cosine and simplifying
\(\ds \) \(=\) \(\ds \map {d_2} {x, y}\) Definition of Euclidean Metric on Real Number Plane

$\blacksquare$


Sources

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