Minkowski distance

The Minkowski distance is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

Definition

The Minkowski distance of order p between two points

is defined as:

For , the Minkowski distance is a metric as a result of the Minkowski inequality. When , the distance between (0,0) and (1,1) is , but the point (0,1) is at a distance 1 from both of these points. Since this violates the triangle inequality, for it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of . The resulting metric is also an F-norm.

Minkowski distance is typically used with p being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of p reaching infinity, we obtain the Chebyshev distance:

Similarly, for p reaching negative infinity, we have:

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between P and Q.

The following figure shows unit circles (the set of all points that are at the unit distance from the centre) with various values of p:

See also

Simple IEEE 754 implementation in C++

NPM JavaScript Package/Module

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