Taxicab geometry
Taxicab geometry, considered by
Hermann Minkowski in the
19th century, is a form of
geometry in which the usual
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 coordinates. More formally, we can define the
Manhattan distance, also known as the
L1-distance is the
distance between two points measured along axes at right angles. In a
plane with p
1 at (x
1, y
1) and p
2 at (x
2, y
2), the Manhattan distance is:
- , when m = 1.
(One can note that the L
2-distance is the normal
Euclidean distance.)
Manhattan distance is also known as city block distance. It is so named because it is the distance a car would drive in a city laid out in square blocks, like Manhattan (discounting the facts that in Manhattan there are one-way and oblique streets and that real streets only exist at the edges of blocks - there is no 3.14th Avenue). Any route from a corner to another one that is 3 blocks East and 6 blocks North, will cover at least 9 blocks.
See also: