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I have a question on parametric equation of ellipses.

I would like to rotate an ellipse around a certain point. I managed to find the half of the equation but something is missing...

$$x(t) = 3\cos(α)\cos(t) - 2\sin(α)\sin(t) + u$$

$$y(t) = 3\sin(α)\cos(t) + 2\cos(α)\sin(t) + v$$

where $C(u,v)$ is the center of the ellipse ,$P(h,k)$ is the certain point and $α$ is the angle of the rotation.

I tried many things but nothing worked...

Thanks Blaxou

Blax
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  • There are a lot of questions and answers here in this site. Just write "rotated ellipse" in the Q&A box. For example http://math.stackexchange.com/questions/1477762/how-to-find-the-equation-of-an-ellipse-rotated-by-25 – cgiovanardi Dec 04 '16 at 14:54
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    @cgiovanardi I'm searching for rotated ellipse around a certain point ;) If you find the answer of the question on an another question, don't hesitate to let me know :D – Blax Dec 04 '16 at 14:58
  • Did you mean $\alpha$ rather than $a$ in $x(t)$? – Hypergeometricx Dec 04 '16 at 15:31
  • Yep, I forgot to change it sorry ;) – Blax Dec 04 '16 at 15:54
  • @Blaxou Maybe this http://math.stackexchange.com/questions/426150/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-and-rotate/426164#426164 ? – cgiovanardi Dec 04 '16 at 16:16

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Do a rigid body transformation with a rotation matrix for instance. That means use a rotation matrix $R$ that does the job.

Then you simply have to do the following

$$ x' = R \cdot x $$

where $x$ is the position vector [x,y] and $x'$ is the new position vector after rotation.

Armen Avetisyan
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