Taxicab Metric¶

Definition¶

The taxicab distance between points $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)$ and $\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$ in an $n$-dimensional real coordinate space $\mathbb{R}^n$ with fixed Cartesian coordinate system is:

The taxicab norm, or known as $\ell^1$ norm, of $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)$ is calculated as:

The coordinate sum defines a norm length, and subtracting two points gives a metric distance. Its triangle inequality follows by adding $|x_i-z_i|\le|x_i-y_i|+|y_i-z_i|$ over the coordinates.

Interpretation¶

The taxicab distance represents the distance a taxi would travel between two points when restricted to cardinal directions (north, south, east, west) without moving diagonally or in any other curved path. It adds the coordinate differences instead of combining them by Pythagoras, so it is the $p$-metric with $p=1$$1$.

The red staircase has taxicab length $d_1(\boldsymbol{x},\boldsymbol{y})=5+4=9$$9$. Any monotone staircase between the points has the same length, whereas the blue straight segment has Euclidean length $d_2(\boldsymbol{x},\boldsymbol{y})=\sqrt{5^2+4^2}\approx6.4$$6.4$.

The diamond-shaped unit circle shows how the taxicab norm weighs directions: the point $(\tfrac{1}{2},\tfrac{1}{2})$ has taxicab norm $1$, the same as $(1,0)$, although it is closer to the origin in the Euclidean sense. In general $d_2\le d_1\le\sqrt{n}\,d_2$ on $\mathbb{R}^n$, the upper bound by the Cauchy-Schwarz inequality. In the figure, $9\le\sqrt{2}\cdot\sqrt{41}\approx9.06$$9.06$, close to equality because the two coordinate differences are almost equal. These bounds make the two metrics strongly equivalent: they give different numbers but the same convergent sequences.

A common slip is to think that the taxicab distance depends only on the geometry of the plane. It depends on the chosen axes, which is why the definition fixes a Cartesian coordinate system: the vectors $(1,0)$ and $(\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{2}})$ have the same Euclidean length $1$, one being a rotation of the other, but taxicab norms $1$ and $\sqrt{2}$.

Key Results¶

• The closed unit ball in the taxicab norm is an $n$-dimensional cross-polytope, e.g., a diamond in $\mathbb{R}^2$ and a regular octahedron in $\mathbb{R}^3$.

Examples¶

• $d_1((0,1),\,(2,-1))=4$$4$.
• $\|(-3,1)\|_1=4$$4$.

Prerequisites¶

• Real Coordinate Space
• Metric and Metric Space
• Norm and Normed Vector Space
• Euclidean Norm

Related Concepts¶

• Euclidean Distance
• $p$-norm and $p$-metric
• Maximum Norm

Sources¶

• Math III: Analysis · Lecture 4

Aliases: Manhattan metric, taxicab distance, taxicab norm, rectilinear distance, L1 distance Keywords: Manhattan distance Symbols: real numbers, square root, sum

Last updated: 20 October 2026 03:02