Ellipse on a Circular Cylinder in Cylindrical Coordinates Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r?  For simplicity, I envision the cylinder to be coincident with the x-axis.
I am aware that the cylinder could be "unwrapped" into a plane, which would result in the ellipse becoming a sine curve.  I am just not sure how that information ties into cylindrical coordinates.  
EDIT: I have realized that I am looking for a parametric equation.  For an ellipse on the surface of a cylinder of radius r which has a certain angle of inclination, is there a parameterization where I can calculate the axial coordinate seperately from the azimuth angle for a given t?
Thank you.


As stated before assume the (normalized) equation of the plane is
$$ \frac{a x + b y + c z}{\sqrt{a^2+b^2+c^2}} = d $$
and the parametric equation of the cylinder
$$ \pmatrix{x & y & z}= f(\varphi,z) = \pmatrix{ R \cos\varphi, R\sin\varphi, z} $$
Where the two intersect you have your ellipse in cartesian coordinates
$$ z(\varphi) = \frac{d \sqrt{a^2+b^2+c^2}-R (a \cos\varphi+b \sin\varphi) }{c} $$
or
$$\vec{r}_{\rm curve}(\varphi) = \pmatrix{x\\y\\z} = \pmatrix{ R \cos\varphi \\ R \sin\varphi  \\ \frac{d \sqrt{a^2+b^2+c^2}-R (a \cos\varphi+b \sin\varphi) }{c} } $$
Now let's find the properties of this ellipse.
The center of the ellipse is at $\vec{r}_{\rm cen} = \pmatrix{0 & 0 & \frac{d \sqrt{a^2+b^2+c^2}}{c}} $
The ellipse in polar coordinates is
$$ r(\varphi) = \| \vec{r}_{\rm curve}-\vec{r}_{\rm cen} \| = \sqrt{R^2 + \frac{R^2}{c^2} \left( \frac{a^2+b^2}{2} + a b \sin(2 \varphi) + \frac{a^2-b^2}{2} \cos(2\varphi) \right)} $$
This allows us to find the major and minor radii
$$ \begin{aligned}
  r_{\rm major} & = R \frac{ \sqrt{a^2+b^2+c^2}}{c} \\
  r_{\rm minor} & = R 
\end{aligned} $$
The principal axes of the ellipse are on
$$ \varphi = \frac{1}{2} {\rm atan}\left( \frac{2 a b}{a^2-b^2} \right) + n \frac{\pi}{2} $$
