Ellipse on a Circular Cylinder in Cylindrical Coordinates

Question

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.

Answer 1

For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and $$a R\cos\theta + b R\sin\theta + cz = d.$$

Answer 2

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

$$ \varphi = \frac{1}{2} {\rm atan}\left( \frac{2 a b}{a^2-b^2} \right) + n \frac{\pi}{2} $$