# Why metric defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2|$ is known as taxicab metric?

$\mathbb{R}^2$ with the function defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2|$ where $a = (a_1, a_2)$ and $b = (b_1, b_2)$ is a metric. I wonder why it is known as taxicab metric. Could anyone explain me?

Thank you very much

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I know the metric as the Manhatten metric, but it's the same. Thing is you only get these kind of streets afaik only in big American cities. Do you know the Paris metric? –  Simon Markett Jun 7 '12 at 11:16
@SimonMarkett I have never heard before about this metric but just now I goggled and got to know. Thank you. –  srijan Jun 7 '12 at 11:20

Imagine a big city laid out so that the streets all run either north-south or east-west and are spaced at regular intervals. In order to go from one point to another by taxicab, you must travel along the streets, so you can only travel north-south or east-west. The distance travelled is therefore the sum of the north-south separation and the east-west separation between you and your destination. The taxicab metric is the distance as the cab would travel instead of the straight-line, or Euclidean distance distance. (The latter is sometimes described in English as as the crow flies, so we could call the Euclidean metric the crow metric, but no one does!)

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Dear sir, I didn't this point;The distance travelled is therefore the sum of the north-south separation..... –  srijan Jun 7 '12 at 10:51
@srijan: Look at this picture: the green line shows the straight-line distance, and the red, blue, and yellow lines all show the taxicab distance, following the streets. Note that all three cover the same distance, $6$ blocks north-south and $6$ blocks east-west: the total is $12$ blocks, a north-south separation of $6$ blocks plus an east-west separation of $6$ blocks. –  Brian M. Scott Jun 7 '12 at 11:00
Wow that pic tells everything. I understand now. –  srijan Jun 7 '12 at 11:09

A picture is definitely the way to see this most easily. From a top down view the metric looks like a city, and the paths an object in this metric must take are the same as a taxicab must take in such a city (not being able to fly or drive through buildings for instance).

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Thank you very much. –  srijan Jun 7 '12 at 11:12

The idea is that it reflects the distance between two points if you only travel parallel to the axes. So to get from $(7,3)$ to $(4,8)$ you first travel 3 units left to $(4,3)$ then 5 up to $(4,8)$, for a total distance of eight. Hence the distance is the sum of the vertical distance and the horizontal distance.

It's meant to be evocative of the distance a taxicab would travel in a city where streets are in a grid pattern, so that you can only get places by going, say, east-west or north-south, rather than travelling directly towards them.

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Thank you very much. I got your point. –  srijan Jun 7 '12 at 10:56

See

how the streets of Manhattan look, as opposed to other places nearby...

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Thank you very much. I never thought of getting such a clear insight og taxicab metric. Thanks again.:) –  srijan Jun 7 '12 at 13:44