# Rotate an ellipse around a certain point

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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.

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### Blax

Updated on December 05, 2020

• Blax over 1 year

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

• cgiovanardi over 5 years
There are a lot of questions and answers here in this site. Just write "rotated ellipse" in the Q&A box. For example math.stackexchange.com/questions/1477762/…
• Blax over 5 years
@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
• Hypergeometricx over 5 years
Did you mean $\alpha$ rather than $a$ in $x(t)$?
• Blax over 5 years
Yep, I forgot to change it sorry ;)
• cgiovanardi over 5 years
• Blax over 5 years
The problem is that I can't use matrixes
• Armen Avetisyan over 5 years
You don't have to use a matrix. Just do something like $x(t)' = x(t) \cdot cos(a) - y(t)*sin(a)$ and $y(t)' = x(t) \cdot sin(a) + y(t) \cdot cos(a)$.