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Thread: find the parametric equation of an ellipse?

  1. #1
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    Question find the parametric equation of an ellipse?

    how to find the parametric equation of an ellipse?

    x^2/a^2 + y^2/b^2 =1

    thank you very much.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kittycat View Post
    how to find the parametric equation of an ellipse?

    x^2/a^2 + y^2/b^2 =1

    thank you very much.
    the ellipse $\displaystyle \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$ is given by the parametric equations: $\displaystyle x = a \cos t \mbox{ , } y = b \sin t$ for $\displaystyle 0 \le t \le 2 \pi$
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  3. #3
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    hi Jhevon,

    How do you work out this parametric equation?

    please teach me . thank you.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kittycat View Post
    hi Jhevon,

    How do you work out this parametric equation?

    please teach me . thank you.
    we know that $\displaystyle \cos^2 t + \sin^2 t = 1$

    and we also know from our knowledge of polar coordinates that it is more appropriate to relate the x to the cosine and the y to the sine, so

    $\displaystyle \frac {x^2}{a^2} + \frac {y^2}{b^2} = \left( \frac xa \right)^2 + \left( \frac yb \right)^2 = 1$

    thus we can let $\displaystyle \cos t = \frac xa$ and $\displaystyle \sin t = \frac yb$ (and so we would obtain $\displaystyle \cos^2 t + \sin^2 t = 1$)

    solving for $\displaystyle x$ and $\displaystyle y$ we get: $\displaystyle x = a \cos t$ and $\displaystyle y = b \sin t$ and we restrict $\displaystyle t$ to $\displaystyle 0 \le t \le 2 \pi$ since sine and cosine are periodic for that interval
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