Taxicab geometry

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 norm (see Lp space), snake distance, city block distance, Manhattan distance or Manhattan length. 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. The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In , the taxicab distance between two points and is . That is, it is the sum of the absolute values of the differences in both coordinates. The taxicab distance, , between two vectors 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,For example, in , the taxicab distance between and is The L1 metric was used in regression analysis in 1757 by Roger Joseph Boscovich. The geometric interpretation dates to the late 19th century and the development of non-Euclidean geometries, notably by Hermann Minkowski and his Minkowski inequality, of which this geometry is a special case, particularly used in the geometry of numbers, . The formalization of Lp spaces is credited to . Taxicab distance depends on the rotation of the coordinate system, but does not depend on its reflection about a coordinate axis or its translation. Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry) except for the side-angle-side axiom, as two triangles with equally "long" two sides and an identical angle between them are typically not congruent unless the mentioned sides are parallel.