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Thread: Rotation in Cartesian Coordinates

  1. #1
    Mar 2010

    Question Rotation in Cartesian Coordinates

    Hi everyone,

    Suppose I have two sets of 4 points in cartesian coordinates:
    First set: {a={x1,y1,z1},b={x2,y2,z2},c={x3,y3,z3},d={x4,y4,z 4}}
    Secons set: {e={x5,y5,z5},f={x6,y6,z6},g={x7,y7,z7},h={x8,y8,z 9}}

    1) How can I check whether the second set can be obtained by translating and rotating the first set?

    2) Suppose I know that the second set is the translated and rotated version of the first set. I assume that after transformation "a" becomes "e", "b" becomes "f", "c" becomes "g" and "d" becomes "h". How can I find the translation and rotation matrices T and R such that e=T+R.a, f=T+R.b etc.

    Thanks for your help.

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  2. #2
    Senior Member
    Nov 2009
    1. You can check that the distances and angles are the same between the 2 sets. If not, then they aren't equivalent under translations and rotations. This is a necessary condition, not a sufficient one.

    2. You can get some information out by working with (f-e) = R (b-a). By choosing different pairs of e,f,g,h, you should be able to constrain R. Once you have R, getting T should be relatively easy.
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