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New coordinates from rotation of axis

  1. May 3, 2015 #1
    1. The problem statement, all variables and given/known data
    There is a point P(x,y) and now I rotate the x-y axis, say by θ degree. What will be the coordinates of P from this new axis.

    I have google but found formula for new coordinates when the points is rotated by θ degree. So I tried my own. So is there other simplified formula for the above situation.

    2. Relevant equations


    3. The attempt at a solution
    Plz see the attached figure.
    AC-AD is the new axis and (x',y') are the new coordinates of point P.

    y'=PC
    cos θ= PC/PB = y'/PB
    y'= PB cos θ

    PB= y- BE
    tan θ= BE/x , BE = x tanθ
    PB = y - xtanθ
    y'= (y- xtanθ) cosθ
    = ycosθ - xsinθ --------------eq(1)


    x'= AB + BC
    sinθ=BE/AB
    AB=BE/sinθ, tanθ=BE/x
    BE=xtanθ
    AB=xtanθ/sinθ = x/cosθ

    tanθ = BC/y'
    BC= y'tanθ
    = (ycosθ-xsinθ)tanθ , ( y' from eq(1) )
    = ysinθ - x sinθtanθ
    x' = AB + BC
    = x/cosθ + ysinθ - x sinθtanθ

    So the new coordinates are
    x'= x/cosθ + ysinθ - x sinθtanθ
    y'= ycosθ - xsinθ
     

    Attached Files:

  2. jcsd
  3. May 3, 2015 #2

    Orodruin

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    Can you think of a way of rewriting x(1/cosθ - sinθ tanθ) to a form which is slightly more pleasant to look at? Otherwise this formula is correct.
     
  4. May 3, 2015 #3
    Thanks.
    x'= x/cosθ + ysinθ - x sinθtanθ
    x'= x/cosθ - xsin2θ/cosθ + ysinθ
    x'= x(1-sin2θ)/cosθ) + ysinθ
    x'= x(cos2θ/cosθ) + ysinθ
    x'= xcosθ + ysinθ

    Now it is pleasant to look at.

    This looks very much similar to, when a points is rotated by θ. The new coordinates are,

    x' = x cos θ - y sin θ
    y' = y cos θ + x sin θ

    Only difference is the -/+ sign. Seems like the difference is because as we increase the θ(when a points is rotated), x' gets shorter. Thats just a guess with a first look.
     
  5. May 3, 2015 #4

    Orodruin

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    Yes, the sign depends on the active-vs-passive transformation, i.e., if you rotate the points or the coordinate system.
     
  6. May 3, 2015 #5

    Ray Vickson

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    If you fix the point and rotate the coordinate system, the new coordinates ##(x',y')## are given by
    [tex] x' = \cos(\theta) x + \sin(\theta) y \\
    y' = -\sin(\theta) x + \cos(\theta) y
    [/tex]
    If you fix the coordinate system and rotate the point, the new coordinates ##(x',y')## are given by
    [tex] x' = \cos(\theta) x - \sin(\theta) y\\
    y' = \sin(\theta) x + \cos(\theta) y
    [/tex]
     
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