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Math Help - Corners of a Cube

  1. #1
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    Corners of a Cube

    G'day, I've had this problem latly that has been driving me crazy, any help on what to do would be appreciated.

    basicly i've been given a cube with the top like this (the slopy drawring attached) and i've been told the 3D coordinates of A and C. What is the best method of calculating points D and B with the only other information given is that we're talking about a pefect cube placed on a slant, so only the Z coordinates of B and D are the same.

    I have calculated the vector between A and C, and also the midpoint of the line (also the midpoint between D and B) however i now have no idea how to proceed.

    ANY help appreciated greatly.
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  2. #2
    Member alunw's Avatar
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    Suppose you were given
    A= (x1,y1,z1), C = (x2,y2,z2)

    Then O the centre of the face is at 0.5* (x1+x2,y1+y2,z1+z2)

    And you must have D = ((x1+x2)*0.5+a,(y1+y2)*0.5+b,(z1+z2)*0.5)
    and B = ((x1+x2)*0.5-a,(y1+y2)*0.5-b,(z1+z2)*0.5)

    since you were told the Z coordinates were equal.

    Now OA = ((x1-x2)*0.5,(y1-y2)*0.5),(z1-z2)*0.5)
    while OD = (a,b,0)

    But OD.OA = 0 so (x1-x2)*a+(y1-y2)*b = 0;
    Also |OA|=|OD] so aa+bb = ((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)*0.25

    Eliminate either a or b using the equation coming from OD.OA = 0 and then
    solve the quadratic to find a or b.
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