I used this equation for rotated 2D ellipse:
$$\frac{(x\cos\theta+y\sin\theta)^2}{a^2} +\frac{(x\sin\theta-y\cos\theta)^2}{b^2} =1$$
$a$, $b$ are major-axis and minor-axes length of the ellipse and $\theta$ is the rotation angle.
what is this kind of equation for the rotated ellipsoid in 3D? we have $a$, $b$ and $c$ that are 3 axes length of ellipsoid and $\theta , \phi$ which are rotation angle in horizontal plane and vertical plane respectively. Can anyone help me please?
Thank you in advance.
The equation of unrotated the ellipsoid in 3D is $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ Write the rotation matrices (you can look it up on wikipedia). You have $(x',y',z')^T=R(\theta,\phi)\cdot(x,y,z)^T$ This yield each of $x',y',z'$ as linear combinations of $x,y,z$. Now just plug in $x',y',z'$ instead of $x,y,z$ in the first equation, and you are done.