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Coordinate transformation parameterization

  1. Oct 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose two observers O and O', whose positions coincide , each sets up a set of 2D cartesian coordinates (x,y) and (x',y') respectively to describe the position of a certain object at a fixed point . Derive a set of formulae for one observer to convert the other observer's coordinates into his own.

    2. Relevant equations


    3. The attempt at a solution
    Assuming linearity ,
    x'=ax+by
    y'=cx+dy

    since the distance between that object and either observer is the same ,

    d^2=x^2+y^2=x'^2+y'^2

    0=(a^2+c^2-1)x^2+(b^2+d^2-1)y^2+2xy(ab+cd)

    then , since the formulae must hold for all (x,y) ,

    a^2+c^2-1=0
    b^2+d^2-1=0
    ab+cd=0

    the standard parameterization gives
    a=cosA ; b=sinB ; c=sinA ; d=cosB

    so sin(A+B)=0
    which has distinct solutions A=-B and A+B=pi
    the first solution is relatively easy to understand , as it just yields a standard rotation of coordinate axes .

    however, with A=pi-B ,

    I get

    x'=-xcos(A)+ysinA

    y'=xsinA+ycosA

    When I try to picture it , it's a transformation where the original x-axis 'rotates' about the original y-axis by 180 degrees , and the whole coordinate system rotates about the common origin by A radian in the clockwise direction . This is where I get uncomfortable and unsure of whether I'm right
     
  2. jcsd
  3. Oct 8, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    It is a mirrored coordinate system. You get it from the regular solution by the substitution x' -> -x'.
     
  4. Oct 8, 2014 #3
    thanks. that's a helpful insight
     
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