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city-block metric (Definition)

The city-block metric, defined on $ \mathbb{R}^n$, is

$\displaystyle d(a,b) = \sum_{i=1}^n \vert b_i-a_i\vert $

where $ a$ and $ b$ are vectors in $ \mathbb{R}^n$ with $ a=(a_1,\ldots,a_n)$ and $ b=(b_1,\ldots,b_n)$.

In two dimensions and with discrete-valued vectors, when we can picture the set of points in $ \mathbb{Z} \times \mathbb{Z}$ as a grid, this is simply the number of edges between points that must be traversed to get from $ a$ to $ b$ within the grid. This is the same problem as getting from corner $ a$ to $ b$ in a rectilinear downtown area, hence the name “city-block metric.”



"city-block metric" is owned by akrowne.
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Other names:  city-block distance, taxicab metric
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Cross-references: area, edges, number, grid, points, dimensions, vectors
There are 4 references to this entry.

This is version 5 of city-block metric, born on 2002-01-23, modified 2003-02-23.
Object id is 1552, canonical name is CityBlockMetric.
Accessed 16986 times total.

Classification:
AMS MSC54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

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