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Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or metric) called the taxicab distance, Manhattan distance, or city block distance.
Taxicab distance (L 1 distance), also called Manhattan distance, which measures distance as the sum of the distances in each coordinate. Minkowski distance (L p distance), a generalization that unifies Euclidean distance, taxicab distance, and Chebyshev distance.
A sphere formed using the Chebyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using Manhattan distance is an octahedron: these are dual polyhedra, but among cubes, only the square (and 1-dimensional line segment) are self-dual polytopes.
The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point . The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1.
A distance transformation. Usually the transform/map is qualified with the chosen metric. For example, one may speak of Manhattan distance transform, if the underlying metric is Manhattan distance. Common metrics are: Euclidean distance; Taxicab geometry, also known as City block distance or Manhattan distance. Chebyshev distance
Manhattan distance is commonly used in GPS applications, as it can be used to find the shortest route between two addresses. [citation needed] When you generalize the Euclidean distance formula and Manhattan distance formula you are left with the Minkowski distance formulas, which can be used in a wide variety of applications. Euclidean distance
In the Euclidean TSP (see below), the distance between two cities is the Euclidean distance between the corresponding points. In the rectilinear TSP, the distance between two cities is the sum of the absolute values of the differences of their x- and y-coordinates. This metric is often called the Manhattan distance or city-block metric.
A metric or distance function is a function d which takes pairs of points or objects to real numbers and satisfies the following rules: The distance between an object and itself is always zero. The distance between distinct objects is always positive. Distance is symmetric: the distance from x to y is always the same as the distance from y to x.