How to calculate weighted center point of 4 points?

Question

If I have 4 points

        var x1;
        var y1;
        var x2;
        var y2;
        var x3;
        var y3;
        var x4;
        var y4;

that make up a box. So

(x1,y1) is top left
(x2,y2) is top right
(x3,y3) is bottom left
(x4,y4) is bottom right

And then each point has a weight ranging from 0-522. How can I calculate a coordinate (tx,ty) that lies inside the box, where the point is closer to the the place that has the least weight (but taking all weights into account). So for example. if (x3,y3) has weight 0, and the others have weight 522, the (tx,ty) should be (x3,y3). If then (x2,y2) had weight like 400, then (tx,ty) should be move a little closer towards (x2,y2) from (x3,y3).

Does anyone know if there is a formula for this? Thanks

Answer 1

Assume w1, w2, w3, w4 are the weights. You can start with this (pseudocode):

M = 522
a = 1
b = 1 / ( (1 - w1/M)^a + (1 - w2/M)^a + (1 - w3/M)^a + (1 - w4/M)^a )

tx = b * (x1*(1-w1/M)^a + x2*(1-w2/M)^a + x3*(1-w3/M)^a + x4*(1-w4/M)^a)
ty = b * (y1*(1-w1/M)^a + y2*(1-w2/M)^a + y3*(1-w3/M)^a + y4*(1-w4/M)^a)

This should approximate the behavior you want to accomplish. For the simplest case set a=1 and your formula will be simpler. You can adjust behavior by changing a.

Make sure you use Math.pow instead of ^ if you use Javascript.