Percentage Contribution To Euclidean Distance I am currently working with Euclidean Distances.
I am calculating the distance between two n-dimensional sets of data points, but I really want to know how much each point contributes to the final Euclidean Distance
For example, if we were considering a right triangle, I would want to know what percentage of the length of the hypotenuse came from each leg.
The problem is that I can't seem to find a way to relate the individual points to the amount of distance that they contribute.
Another example:
P1: 1, 2, 3
P2: 0, 0, 0
The total distance would be equal to the square root of 14.....but what percentage of that came from each point?
Is my only option for comparing lengths Squared Euclidean Distance?
 A: Suppose the lengths of the two legs are $a$ and $b$ with hypotenuse $c =\sqrt{a^{2} + b^{2}}$. In the limiting case, if $a = 0$ then $100\%$ of $c$ comes from $b$ and $0\%$ from $a$. Similarly if $b = 0$ then $100\%$ of the hypotenuse comes from $a$ and $0\%$ comes from $b$. If $a = b$ then $50\%$ of the hypotenuse comes from each of $a$ and $b$. One function, perhaps the simplest, that satisfies these conditions is the percent from $a$ is $\frac{a}{a + b}$ and the percent from $b$ is $\frac{b}{a + b}$. These functions have the additional properties that if $a$ is held fixed and $b$ increases the percent from $b$ will increase. Similarly if $b$ is fixed and $a$ increases.
A: The basic idea is flawed.  Let us take your example in 2D of a right triangle, specifically $5,12,13.$  You are asking what part of the $13$ came from the $5$?  If I decrease the short side to zero, the hypotenuse becomes $12$, so it seems only $1$ of the hypotenuse came from $5$.  If I decrease the long side to zero, the hypotenuse becomes $5$, so it seems $8$ came from the long side.  This leaves $4$ of the hypotenuse unaccounted for.  
Asking how much of the result comes from each input makes sense if the output is a linear function of the inputs.  If it is almost linear, you won't be far off.  But if it is far from linear, you can't get a reasonable answer.
