I wonder if the good people here can help me out.
I want to know what are the steps for calculating the distance to a centroid of points in which I do not know the co-ordinates of the points only the distance between them. The space has high unknown dimensionality.
For example I have point A which is 10 from point B, 12 from point C and 14 from point D.
Point B is 6 from point C, 8 from point D
and point C is 14 from point D.
I then have a new point Z which is 8 from point A, 11 from point B, 14 from point C and 12 from point D.
I want to calculate the distance of the new point Z to the centroid (or the average ) of points A, B, C and D.
Any help appreciated.
The centroid of a set of points is the point whose coordinates are the arithmetic average of the cooresponding coordinates of the various points. If you don't even know the dimension of the space, I have no idea what you are supposed to do here!
I think I might be trying to find the minimum distance, for the minimum number of dimensions as you are right to point out, if we don't know the dimensions then a new point could be an infinite distance away in some dimension.
My thinking so far is to set one point as the origin Say A and to have 4 dimensions of space (one for each point) as I think that is the smallest number of dimensions for it to definitely work( I could be wrong here..)
So we would have A[0,0,0,0], then we would have some simultaneous equations to try and work out the coordinates of each other point, B=[a1,b1,c1,d1], C=[a2,b2,c2,d2] and D =[a3,b3,c3,d3]
Then we know that
||A-B|| = 10
||A-C|| = 12
Would we be able to get the co-ordinates of the points from this?