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Help with ellipse geometry

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that the equations [itex]x=acos(\theta)[/itex] and [itex]y=bcos(\theta +\delta )[/itex] is the equation of an ellipse and what angle does this ellipse's major axis make with the x axis?


    2. Relevant equations

    Equation of an ellipse is [itex]x=acos\theta, y=asin\theta[/itex]
    Rotation matrix is for a rotation by [itex]\psi[/itex] is [tex]A=\begin{pmatrix} cos\psi & -sin\psi \\ sin\psi & cos\psi \end{pmatrix}[/tex]

    3. The attempt at a solution

    I know the special case of [itex]\delta = \pi/2[/itex] is easy but I cannot do it for arbitrary [itex]\delta[/itex]. I worked out what an ellipse whose major axis forms an angle [itex]\psi[/itex] with the x axis looks like. I did this by applying a rotation matrix to the standard equation [itex]x=acos\theta, y=bsin\theta[/itex]

    This gives [itex]x= acos\theta cos\psi - bsin\theta sin\psi[/itex] and [itex]y= acos\theta sin\psi + bsin\theta cos\psi [/itex]

    Now, what is the relation between [itex]\psi[/itex] and [itex]\delta[/itex] in general. And I need to show that the equations [itex]x=acos(\theta)[/itex] and [itex]y=bcos(\theta +\delta )[/itex] can be brought to the same form as [itex]x= acos\theta cos\psi - bsin\theta sin\psi[/itex] and [itex]y= acos\theta sin\psi + bsin\theta cos\psi [/itex]
     
  2. jcsd
  3. Feb 6, 2012 #2

    vela

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    This might not be the most efficient method. Use
    $$\cos \theta = \frac{x}{a}$$ to eliminate ##\theta## from
    $$\frac{y}{b} = \cos(\theta+\delta) = \cos \theta \cos \delta - \sin\theta \sin \delta$$ to get the equation of an ellipse in standard form.
     
  4. Feb 8, 2012 #3
    Thanks Vela. It took some doing but it worked!
     
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