Weighted Average Distance between 3 or more points

Question

I'm trying to find out the average point between 3 or more points, given each distance that the end point is away from each of the other points.

With 2 points it's easy, I believe the formula should be:

$x_p =\dfrac{(x_1 * d_2) + (x_2 * d_1)}{d_1 + d_2}$, $y_p =\dfrac{(y_1 * d_2) + (y_2 * d_1)}{d_1 + d_2}$

With 3 points and 3 distances I tried to follow that same logic and take it a step further by using:

$x_p =\dfrac{(x_1 * d_2 * d_3) + (x_2 * d_1 * d_3) + (x_3 * d_1 * d_2)}{d_1 + d_2 + d_3}$

$y_p =\dfrac{(y_1 * d_2 * d_3) + (y_2 * d_1 * d_3) + (y_3 * d_1 * d_2)}{d_1 + d_2 + d_3}$

However this doesn't end up giving me the same answer. Instead it gives me a point that ends up being nowhere near the center of the 3 initial points. Is there a way to take the weighted average between 3 points to find the end point? Any help would be great, thank you!

Answer 1

In the case of two points, the weights would be $w_1 = \dfrac{\frac{1}{d_1}}{\frac{1}{d_1}+\frac{1}{d_2}} = \frac{d_2}{d_1+d_2}$ and $w_2 = \frac{d_1}{d_1+d_2}$

This you got it correctly.

Now for the case of three points

$w_1 = \dfrac{\frac{1}{d_1}}{\frac{1}{d_1}+\frac{1}{d_2}+\frac{1}{d_2}} = \frac{d_2d_3}{d_1d_2+d_1d_3+d_2d_3}$ and $w_2 = \frac{d_1d_3}{d_1d_2+d_1d_3+d_2d_3}$$w_3 = \frac{d_1d_2}{d_1d_2+d_1d_3+d_2d_3}$

Your problem is you are dividing by $(d_1+d_2+d_3)$

If you try this it will work.

Good luckk

Satish

Answer 2

To be honest, I don't understand what are your distances $d_1, d_2, d_3$ and what is their role...

I assume two things in the following:

• First case: the distances $d_1, d_2, d_3$ means nothing at all, and are meaningless

• Second case: the distance $d_1, d_2, d_3$ are the weights associated to each of the point $P_1, P_2, P_3$

The general function to get the average point $A$ is

$x_A=\dfrac{x_1+x_2+x_3}{3}$ and $y_A=\dfrac{y_1+y_2+y_3}{3}$

However, IF you want to weight each point $P_1, P_2, P_3$, with what you call distance but is more a weight, you get the coordinates of the barycenter $B$:

$x_B=\dfrac{x_1*d_1+x_2*d_2+x_3*d_3}{d_1+d_2+d_3}$

$y_B=\dfrac{y_1*d_1+y_2*d_2+y_3*d_3}{d_1+d_2+d_3}$