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Information and research about the geometry invented by Herman Minkowski
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<spacer> <spacer> Taxicab Geometry > Definitions

As with many mathematical concepts, there have been several variations of taxicab geometry discussed over the years. In this section we will define the original version of taxicab geometry and discuss the most common variations.

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Traditional Definition

At its simplest, traditional taxicab geometry changes the Euclidean distance formula to the metric proposed by Herman Minkowski where the distance between two points (x1,y1) and (x2,y2) is


The idea behind this distance formula is that the distance between two points is not measured on a straight line, but on horizontal and vertical lines as if traveling the streets of a perfect city - hence the name "taxicab geometry". (See The Basics for more information about the basic interpretation of the taxicab distance formula.)

This definition leaves other geometric features such as points, lines, and angles as Euclidean. Until 1996, this was the form in which the geometry was investigated, discussed, and used. It was around this time that Thompson and Kaya independently began research into angles that natively belong to taxicab geometry thus launching investigations into a purer form of taxicab geometry.

Note: This website will rarely discuss traditional taxicab geometry (see "Pure Taxicab Geometry" below).

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Pure Taxicab Geometry

Traditionally, taxicab geometry has included elements that are not native to the geometry. The primary example is Euclidean angles. Since angles are defined as arc length along a circle and the taxicab circle is quite different than the Euclidean circle, native taxicab angles are not Euclidean. Pure taxicab geometry uses angles that are native and natural to the geometry.

Note: Unless stated otherwise, this website will always deal with pure taxicab geometry.

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Other Names

The taxicab metric often occurs in other contexts and has therefore been given various names over the years. The following are the most common names you may see for the taxicab metric in other contexts or fields of mathematics.
  • rectilinear distance
  • Manhattan distance
  • the L1 norm - the taxicab metric is another name for the norm of the Lp space when p = 1:
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