# 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

• 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