# Taxicab geometry

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A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 distance, L1 distance or ${\displaystyle \ell _{1}}$ norm (see Lp space), snake distance, city block distance, Manhattan distance or Manhattan length.[1] The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets.
In ${\displaystyle \mathbb {R} ^{2}}$, the taxicab distance between two points ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$ is ${\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}$. That is, it is the sum of the absolute values of the differences in both coordinates.