Geometry / Interactive experiment
Cylinder intersections.
A cylinder, a plane, and the ellipse where they meet. Adjust the controls to see how the cross-section moves and changes shape.
Center (Xc, Yc): (0.00, 0.00)Semi-major axis (a): 0.00Semi-minor axis (b): 0.00
01 / Cylinder & plane
02 / Top view & orbit
What am I looking at?
The intersection of a tilted circular cylinder and a horizontal plane is an ellipse. Increasing the tilt stretches its major axis; raising the plane moves its center away from the origin. Global rotation turns the ellipse and its orbit together.
For a cylinder of radius \(r\), tilted by \(\alpha\) from vertical, the semi-axis lengths are:
\[a = \frac{r}{\cos\alpha}, \qquad b = r.\]At plane height \(h\), the center follows a circle of radius \(h\tan\alpha\):
\[X_c = h\tan\alpha\cos\theta, \qquad Y_c = h\tan\alpha\sin\theta.\]The dashed circle shows that orbit. When present, the shaded blue disk in the top view has radius \(r-h\tan\alpha\).