# Definition of normalized Euclidean distance

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

http://en.wikipedia.org/wiki/Mahalanobis_distance

and

http://reference.wolfram.com/language/ref/NormalizedSquaredEuclideanDistance.html

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

NormalizedSquaredEuclideanDistance[u,v] is equivalent to
1/2*Norm[(u-Mean[u])-(v-Mean[v])]^2/(Norm[u-Mean[u]]^2+Norm[v-Mean[v]]^2)

$$NED2[u,v]=0.5Var[u−v]Var[u]+Var[v] NED^2[u,v] = 0.5 \frac{ Var[u-v] }{ Var[u] + Var[v] }$$

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

Update:

I find that the following intuitive explanation for the Wolfram.com 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.