What name and formula describes the curve of where a plane intersects a cylinder?

Question

Given a cylinder of infinite height and a plane that intersects the cylinder where the plane may be inclined at an arbitrary angle relative to the plane whose intersection produces a perfect circle, what is the class of curves one can produce and what is their formula?

I'm thinking that since you get a circle at zero degrees it is likely the set of curves are all ellipses, but I'm not sure how to demonstrate that. If it they are ellipses, how do you compute the foci and formulas as a function of the angle?

Answer 1

They are ellipses. You build two spheres tangent to the plane inside the cylinder. Then you consider the point of intersection of the sphere with the plane as foci $F_1$ and $F_2$ of the ellipse, then you take an arbitrary point $P$ on the intersection and show that the sum of $PF_1+PF_2$ is a constant, more exactly is the distance between the two circumferences where the spheres are tangent to the cylinder.

Hope this helps

Answer 2

Forget about the locus definition of ellipses. In the first place an ellipse is the image of a circle under parallel projection, or the curve you are talking about in your question, or the curve obtained by linearly scaling a circle in one of two plane directions.

It so happens that ellipses have some interesting geometric properties, e.g., this thing about foci, reflections of rays through the foci at the boundary, etcetera.