### Euclidean vs Manhattan vs Chebyshev Distance

Reading time: 15 minutes

Euclidean distance, Manhattan distance and Chebyshev distance are all distance metrics which compute a number based on two data points. All the three metrics are useful in various use cases and differ in some important aspects which we bring out in this article.

### Visual difference

This image summarizes the difference in the three distance metrics:

### Computation

In a N dimensional space, a point is represented as (x1, x2, ..., xN).

Consider two points P1 and P2:

```
P1: (X1, X2, ..., XN)
P2: (Y1, Y2, ..., YN)
```

Then, the euclidean distance between P1 and P2 is given as:

$$ \sqrt{{(x1-y1)}^2\ +\ {(x2-y2)}^2\ +\ ...\ +\ {(xN-yN)}^2}

$$

The manhattan distance between P1 and P2 is given as:

$$ |x1-y1|\ +\ |x2-y2|\ +\ ...\ +\ |xN-yN|}

$$

The chebyshev distance between the two points P1 and P2 is:

```
Chebyshev distance = MAXIMUM (|xi - yi|) where i is 1 to N
```

### Usage in Chess

In chess, all the three distances are used as follows:

- the distance between squares on the chessboard for rooks is measured in
**Manhattan distance** - kings and queens use
**Chebyshev distance** - bishops use the
**Manhattan distance**(between squares of the same color) on the chessboard rotated 45 degrees, i.e., with its diagonals as coordinate axes. - To reach from one square to another, only kings require the number of moves equal to the distance (
**euclidean distance**) rooks, queens and bishops require one or two moves