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I have two ellipses each centered at the origin and defined by:

  • Semi major axis $a$
  • Semi minor axis $b$
  • Angle $\psi$, which is the angle of $a$ counterclockwise from the positive $x$ axis

I add them together such that, if you parametrize an ellipse as $r(\theta)$, then $r_{sum}(\theta)=r_1(\theta)+ r_2(\theta)$.

Is it possible to express the new ellipse as a function of the parameters of the other two, so a function of $(a_1, a_2, b_1, b_2, \psi_1, \psi_2)$?

I hope this makes sense.

J. doe
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  • I believe you are looking for the "implicit form" of the ellipse described above. Not sure how to get it (I'm sure it will depend on where they are with respect to each other etc), but that's the term you need – Michael Stachowsky Nov 23 '18 at 21:27
  • I would think so, even without working it out completely myself. Here's how I would set it up: The equation for the non-rotated ellipse is $$\frac{r^2}{a^2}\cos^2(\theta) + \frac{r^2}{b^2}\sin^2(\theta) = 1$$. To rotate this ellipse clockwise by angle $\psi$, you can just do a phase shift: $$\frac{r^2}{a^2}\cos^2(\theta+\psi) + \frac{r^2}{b^2}\sin^2(\theta+\psi) = 1$$. You could try isolating $r(\theta)$ and using trig identities to simplify the sum. – D.B. Nov 23 '18 at 21:30
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    Please include the specific parameterizations that you’re using. How do you know that the result is also an ellipse? – amd Nov 23 '18 at 21:40
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    Which parametrization are you thinking of? We have e.g. $(x,y)=(a\cos u, b\sin u)$ with the center of the ellipse in the origin. Or $r=\frac{b^2}{a-\cos\theta}$ in polar coordinates with a focal point in the origin. – Klaas van Aarsen Nov 23 '18 at 21:54
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    Addition of the radius $r(\theta)$ in polar coordinates does not give an ellipse, as you can check by considering two ellipses $(a=1,b=\epsilon,\phi=0)$ and $(a=\epsilon,b=1,\phi=0)$ where $\epsilon\ll1$. –  Nov 23 '18 at 22:17
  • @Rahul I'm afraid this is the right answer, because I mistakenly assumed that addition would give me another ellipse. Thanks to all. – J. doe Nov 23 '18 at 22:53
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    The problem is that the angle $\theta$ is a poor parameter for choosing corresponding points from the ellipses. It's worth noting that, because there's a linear transformation from any ellipse to another, you use that transformation to establish the correspondence between points. In that case, the "sum" of two ellipses is again an ellipse. – Blue Nov 23 '18 at 23:16

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