Definition of normalized Euclidean distance

Recently I have started looking for the definition of normalized Euclidean distance between two real vectors u and v. So far, I have discovered two apparently unrelated definitions:


I am familiar with the context of the Wikipedia definition. However, I am yet to discover any context for the definition:

NormalizedSquaredEuclideanDistance[u,v] is equivalent to



The intuitive meaning of this definition is not very clear. Any help on this will be appreciated.


I find that the following intuitive explanation for the definition is given here

I am repeating that below:

Note that it is a DistanceFunction option for ImageDistance. Maybe
that helps some to see the context where it is used.

The relation to SquaredEuclideanDistance is:

NormalizedSquaredEuclideanDistance[x, y] == (1/2)
SquaredEuclideanDistance[x – Mean[x], y – Mean[y]]/
(Norm[x – Mean[x]]^2 + Norm[y – Mean[y]]^2)

So we see it is “normalized” “squared euclidean distance” between the
“difference of each vector with its mean”…

What is the meaning about 1/2 at the beggining of the formula?

The 1/2 is just there such that the answer is bounded between 0 and 1,
rather than 0 and 2.


The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is meaningful but the magnitude is not. It’s not related to Mahalanobis distance.

Source : Link , Question Author : PTDS , Answer Author : Aaron

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