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Math Help - Rotation of Ellipse to FIND the Bxy term, from horizontal to an angle of pi/4

  1. #1
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    Exclamation Rotation of Ellipse to FIND the Bxy term, from horizontal to an angle of pi/4

    In essence I'm trying to take a horizontal ellipse and put it at an angle, rotating it around the origin. It's for my conics project, I'm making the elliptical orbits of the planets.

    So I have the a, b, and c terms of the ellipse, (as in the distance from vertice to center and such) which are 28, 21, and 7\sqrt{7} respectively. When plugged into the equation they give:


    {x}^{2} / 784 +{y}^{2} /441 =1


    which turns into


    441 {x}^{2} + 784{y}^{2 } - 345,744 = 0


    from staring at my math book i've assumed that these x's and y's are actually x prime and y prime in the grand scheme of how we normally look at rotations.

    so actually


    441{x'}^{2 } + 784{y'}^{2 } - 345,744 = 0


    with this is mind I tried to apply the transformation equations knowing that I was trying to acheive a rotation of \pi / 4


    x = x' \cos \pi /4 - y' \sin \pi /4

    x = (x' - y')/\sqrt{2}

    \sqrt{2}x = x' - y'

    \sqrt{2}y = x' + y'



    But that's where I get stuck, because I can't think of how to substitute into my prime equation to get ride of the prime variable and create a Bxy term.

    So please, its due Wednesday... help?
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  2. #2
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    Quote Originally Posted by Lizziepaquette View Post
    I was trying to achieve a rotation of \pi / 4


    x = x' \cos \pi /4 - y' \sin \pi /4

    x = (x' - y')/\sqrt{2}

    \sqrt{2}x = x' - y'

    \sqrt{2}y = x' + y'



    But that's where I get stuck, because I can't think of how to substitute into my prime equation to get ride of the prime variable
    If you get from (x',y') to (x,y) by rotating through \pi/4, then you can get back from (x,y) to (x',y') by rotating through -\pi/4. Therefore if x = x' \cos \pi /4 - y' \sin \pi /4 then x' = x \cos(- \pi /4) - y \sin(- \pi /4) = \tfrac1{\sqrt2}x +\tfrac1{\sqrt2}y.
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  3. #3
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    Hmm let me plug it in
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  4. #4
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    I did the math out with that concept and my end result is

    597.5{x}^{2} -373xy + 597.5{y}^{2} -345,744 = 0

    I haven't done the forward math to check but i graphed it and it looks about right.

    Thanks
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