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The regular ellipse formula in 2D is $x^2/a^2 + y^2/b^2 = 1$ but how can it be transformed into a 3D formula including the parameter of $r, \theta$ and $z$? enter image description here

Zander
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Norman
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1 Answers1

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To translate to cylindrical coordinates you use the following.

$$x = r\cos\theta$$

$$y = r\sin\theta$$

This translates $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ into $$\frac{r^2\cos^2\theta}{a^2}+\frac{r^2\sin^2\theta}{b^2} = 1$$

Notice that $z$ does not enter into the transformation. That is because in three dimensions the equation for an ellipse describes an elliptical cylinder.

John Douma
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  • so any point on the ellipse can we write like this $(r\cos\theta, r\sin\theta,z)$? – Norman May 16 '13 at 05:44
  • Yes, but $(r\cos\theta, r\sin\theta, z)$ is rectangular coordinates. You would write $(r,\theta, z)$ for cylindrical coordinates. The $r$ and $\theta$ values would be the solutions to the fourth equation in the answer. – John Douma May 16 '13 at 05:49