This seems to be very hard to find. I am looking for the parametric equation of an ellipse that is parallel to the x-y plane and tilted wrt the z-axis.
In 2D the parametric equations are x = acos(u) and y = bsin(u), where 'u' is the parameter and 'a' and 'b' are the semi-major and semi-minor axes respectively.
In 3D x, y, and z will be functions of the parameters 'u' and 'v', and 'a' and 'b' are as above. What will the parametric equations be?
On
Using matrices, you can rotate the ellipse if you think of the parametric equations as a vector, $\vec{r}(t)=\begin{pmatrix} {a}\cos{t} \\ {b}\sin{t} \\ 0 \end{pmatrix}$ for $0 \leq t \leq 2\pi$. For example, to rotate the ellipse by an angle $\theta$ about the x-axis, you compute the following: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos{\theta} & -\sin{\theta} \\ 0 & \sin{\theta} & \cos{\theta} \end{pmatrix} \begin{pmatrix} {a}\cos{t} \\ {b}\sin{t} \\ 0 \end{pmatrix}\text{.}$$ See https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations for more info.
Suppose you take an ellipse that is parallel to the $xy$-plane and tilt the ellipse along one of its semi-axes.
Notice that if you look at the ellipse from above (i.e. project it down to the $xy$-plane), the ellipse will look like an ellipse (possibly a circle).
By looking at the ellipse from the front ($xz$-plane) or the side ($yz$-plane) (depending on which semi-axis you tilted with respect to), the ellipse will still look like an ellipse.
However, when looking at from the other perspective, the ellipse will look just like a straight diagonal line.
Thus, when you create your parametrization for the entire ellipse, when you look at the ways of pairing the parametrizations of $x, y, z$, two of the pairs should be the parametrization of an ellipse and one of the pairs should be the parametrization of a line.