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Simple Ellipse Question

  1. Jun 27, 2011 #1

    WannabeNewton

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    Can the semi - major and semi - minor axes of an ellipse be time dependent? More specifically, can you have time dependent semi - major and semi - minor axes present in the standard form of the ellipse? I have an equation of the form [tex]\frac{(\xi ^{1}(t))^{2} }{a^{2}} + \frac{(\xi ^{2}(t))^{2}}{b^{2}} = 1 [/tex] where [itex]\xi ^{\alpha }[/itex] are components of a separation vector, [itex]a^{2} = [2 + \frac{1}{2}sin^{2}\omega t](\xi ^{1}(0))^{2}[/itex], and [itex]b^{2} = [2 + \frac{1}{2}sin^{2}\omega t](\xi ^{2}(0))^{2}[/itex] but I don't know if the standard form can actually have time dependent semi - major and minor axes.
     
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  3. Jun 28, 2011 #2

    HallsofIvy

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    Yes, of course. However what you are writing does NOT.
    [tex]\frac{\xi^1(t))^2}{a^2}+ \frac{\xi^2(t))^2}{b^2}= 1[/tex]
    is a single ellipse with axes of length a and b for all t. If t is "time", this could be interpreted as an object moving around that fixed ellipse with changing speed.

    An ellipse with "time dependent sem- major and minor axes" would be
    [tex]\frac{x^2}{(a(t))^2}+ \frac{y^2}{(b(t))^2}= 1[/tex]

    You can add "time dependence" or dependence on any other variable at will, just by making some parameters function of that variable.
     
  4. Jun 28, 2011 #3

    WannabeNewton

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    So if in the original equation, If I had the usual X and Y instead of the time dependent separation vector but with the same form of the semi - major and minor axes that I posted above then it would be allowed? Thanks for the reply.

    EDIT: Never mind I get what you are saying. I think I should have had the initial coordinate separations on the top and the time variance of the separation plus the sin terms on the bottom.
     
    Last edited: Jun 28, 2011
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