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Math Help - Centroids

  1. #1
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    Centroids

    Hello,

    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.

    Sam
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  2. #2
    MHF Contributor

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    Re: Centroids

    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!
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  3. #3
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    Re: Centroids

    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
    ||A-D||=14
    ||B-C||=6
    ||B-D||=8
    ||C-D||=14

    Would we be able to get the co-ordinates of the points from this?
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