Taxicab distance

Taxicab geometry, considered by Hermann Minkowski in 19th-century Germany, is a form of geometry in which the 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 or ${\displaystyle \ell _{1}}$ norm (see L^p space), snake distance, city block distance, Manhattan distance or Manhattan length, with corresponding variations in the name of the geometry. The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two intersections in the borough to have length equal to the intersections' distance in taxicab geometry.

The taxicab distance, ${\displaystyle d_{1}}$, between two vectors ${\displaystyle \mathbf {p} ,\mathbf {q} }$ in an n-dimensional real vector space with fixed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. More formally,

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