cross section of a cylinder

Thanks for all the help to my beginner's questions. This time, I have a question regarding practical use of geometry. Say I have a cylinder of radius r. And I "cut it" (a cross-section) in the middle. If I cut it perpendicular to the axis, simple enough I will have a circle of radius r.

Now, what happens when I cut it non-perpendicularly at an angle $\alpha$ with respect to the axis?

I get an ellipse! Fair enough! And an ellipse has two focal points (foci?), F1 and F2. All right, but what I would like to know is how to calculate data about these foci. i.e, the distance between those and the distance between any point in the ellipse to these foci

Any advice greatly appreciated

• Nice question! However I am not sure whether it will actually be an ellipse, because cutting a cone gives an ellipse; intuitively a cylinder should be different. – shardulc Jun 26 '17 at 2:04
• Relevant, with images: blog.zacharyabel.com/2012/10/what-makes-ellipses-ellipses – Chappers Jun 26 '17 at 2:07
• Yes, I was just reading that site before posting this :) Wondering how to calculate F1 and F2 and distances based on radius and angle... – KansaiRobot Jun 26 '17 at 2:39
• See Dandelin spheres. – hypergeometric Jun 26 '17 at 13:08

If you cut the cylinder at an angle $\theta$ to the axis of symmetry, you will create an ellipse with minor axis $r$ and major axis $r \sec \theta$
$$d=r\sqrt{\sec^2 \theta - 1 } =r \tan \theta$$
• $\theta$ is not the angle between plane and axis of symmetry, but rather its complementary. – Aretino Jun 26 '17 at 21:33