Taxicab geometry

Taxicab geometry versus Euclidean distance: In taxicab geometry, the red, yellow, and blue paths all have the same shortest path length of 12. In Euclidean geometry, the green line has length ${\displaystyle 6{\sqrt {2}}\approx 8.49}$ and is the unique shortest path.

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 ${\displaystyle \ell _{1}}$ norm (see L^p space), snake distance, city block distance, Manhattan distance or Manhattan length, with corresponding variations in the name of the geometry.^[1] 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^{[citation needed]}, 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.

In two dimensions, the taxicab distance between two points ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$ is ${\displaystyle |x_{1}-x_{2}|+|y_{1}-y_{2}|}$. That is, it is the sum of the absolute values of the differences between both sets of coordinates.

Formal definition[]

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,

${\displaystyle d_{1}(\mathbf {p} ,\mathbf {q} )=\|\mathbf {p} -\mathbf {q} \|_{1}=\sum _{i=1}^{n}|p_{i}-q_{i}|,}$

where ${\displaystyle (\mathbf {p} ,\mathbf {q} )}$ are vectors

${\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})\,}$

For example, in the plane, the taxicab distance between ${\displaystyle (p_{1},p_{2})}$ and ${\displaystyle (q_{1},q_{2})}$ is ${\displaystyle |p_{1}-q_{1}|+|p_{2}-q_{2}|.}$

Properties[]

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.

Balls[]

Circles in discrete and continuous taxicab geometry

A topological ball is a set of points with a fixed distance, called the radius, from a point called the center. In n-dimensional Euclidean geometry, the balls are spheres. In taxicab geometry, distance is determined by a different metric than in Euclidean geometry, and the shape of the ball changes as well. In n dimensions, a taxicab ball is in the shape of an n-dimensional orthoplex. In two dimensions, these 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 ${\displaystyle {\sqrt {2}}r}$ 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 ${\displaystyle \pi }$ is 4 in this geometry. The formula for the unit circle in taxicab geometry is ${\displaystyle |x|+|y|=1}$ in Cartesian coordinates and

${\displaystyle r={\frac {1}{|\sin \theta |+|\cos \theta |}}}$

in polar coordinates.

A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center.

A circle of radius r for the Chebyshev distance (L_∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L₁ and L_∞ metrics does not generalize to higher dimensions.

Whenever each pair in a collection of these circles has a nonempty intersection, there exists an intersection point for the whole collection; therefore, the Manhattan distance forms an injective metric space.

Applications[]

Compressed sensing[]

In solving an underdetermined system of linear equations, the regularization term for the parameter vector is expressed in terms of the ${\displaystyle \ell _{1}}$-norm (taxicab geometry) of the vector.^[2] This approach appears in the signal recovery framework called compressed sensing.

Differences of frequency distributions[]

Taxicab geometry can be used to assess the differences in discrete frequency distributions. For example, in RNA splicing positional distributions of hexamers, which plot the probability of each hexamer appearing at each given nucleotide near a splice site, can be compared with L1-distance. Each position distribution can be represented as a vector where each entry represents the likelihood of the hexamer starting at a certain nucleotide. A large L1-distance between the two vectors indicates a significant difference in the nature of the distributions while a small distance denotes similarly shaped distributions. This is equivalent to measuring the area between the two distribution curves because the area of each segment is the absolute difference between the two curves' likelihoods at that point. When summed together for all segments, it provides the same measure as L1-distance.^[3]

History[]

The L¹ metric was used in regression analysis in 1757 by Roger Joseph Boscovich.^[4] 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, (Minkowski 1910). The formalization of L^p spaces is credited to (Riesz 1910).

See also[]

• Fifteen puzzle
• Hamming distance – Number of bits that differ between two strings
• Manhattan wiring
• Mannheim distance
• Metric – Mathematical function defining distance
• Minkowski distance
• Normed vector space – Vector space on which a distance is defined
• Orthogonal convex hull – Minimal superset that intersects each axis-parallel line in an interval
• Random walk – Mathematical formalization of a path that consists of a succession of random steps

Notes[]

References[]

• Krause, Eugene F. (1987). Taxicab Geometry. Dover. ISBN 978-0-486-25202-5.
• Minkowski, Hermann (1910). Geometrie der Zahlen (in German). Leipzig and Berlin: R. G. Teubner. JFM 41.0239.03. MR 0249269. Retrieved October 6, 2019.
• Riesz, Frigyes (1910). "Untersuchungen über Systeme integrierbarer Funktionen". Mathematische Annalen (in German). 69 (4): 449–497. doi:10.1007/BF01457637. hdl:10338.dmlcz/128558. S2CID 120242933.

External links[]

• Weisstein, Eric W. "Taxicab Metric". MathWorld.
• Malkevitch, Joe (October 1, 2007). "Taxi!". American Mathematical Society. Retrieved October 6, 2019.