Taxicab geometry

A taxicab geometry 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, L1 distance or ℓ 1 {displaystyle ell _{1}} norm (see Lp 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. A taxicab geometry 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, L1 distance or ℓ 1 {displaystyle ell _{1}} norm (see Lp 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 geometry has been used in regression analysis since the 18th century, and today is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. The taxicab distance, d 1 {displaystyle d_{1}} , between two vectors p , q {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, where ( p , q ) {displaystyle (mathbf {p} ,mathbf {q} )} are vectors For example, in the plane, the taxicab distance between ( p 1 , p 2 ) {displaystyle (p_{1},p_{2})} and ( q 1 , q 2 ) {displaystyle (q_{1},q_{2})} is | p 1 − q 1 | + | p 2 − q 2 | . {displaystyle |p_{1}-q_{1}|+|p_{2}-q_{2}|.} 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 happen to be parallel. A circle is a set of points with a fixed distance, called the radius, from a point called the center. In taxicab geometry, distance is determined by a different metric than in Euclidean geometry, and the shape of circles changes as well. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. The image to the right shows why this is true, by showing in red the set of all points with a fixed distance from a center, shown in blue. As the size of the city blocks diminishes, the points become more numerous and become a rotated square in a continuous taxicab geometry. While each side would have length 2 r {displaystyle {sqrt {2}}r} using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to π {displaystyle pi } is 4 in this geometry. The formula for the unit circle in taxicab geometry is | x | + | y | = 1 {displaystyle |x|+|y|=1} in Cartesian coordinates and in polar coordinates. A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center.