Taxicab Metric on Real Number Plane is Translation Invariant

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Theorem

Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the translation of the Euclidean plane by the vector $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.

Let $d_1$ denote the taxicab metric on $\R^2$.


Then $d_1$ is unchanged by application of $\tau$:

$\forall x, y \in \R^2: \map {d_1} {\map \tau x, \map \tau y} = \map {d_1} {x, y}$


Proof

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

Then:

\(\ds \map {d_1} {\map \tau x, \map \tau y}\) \(=\) \(\ds \map {d_1} {x - \mathbf t, y - \mathbf t}\) Definition of Translation in Euclidean Space
\(\ds \) \(=\) \(\ds \size {\paren {x_1 - a} - \paren {y_1 - a} } + \size {\paren {x_2 - b} - \paren {y_2 - b} }\) Definition of $\mathbf t$, Definition of Taxicab Metric on Real Number Plane
\(\ds \) \(=\) \(\ds \size {x_1 - y_1} + \size {x_2 - y_2}\) simplification
\(\ds \) \(=\) \(\ds \map {d_1} {x, y}\) Definition of Taxicab Metric on Real Number Plane

$\blacksquare$


Sources

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