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. ![Ellipse on a cylinder](

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 $$a R\cos\theta + b R\sin\theta + cz = d.$$