# For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

I thought the answer to this question would just be when the two cylinders are tangent because otherwise they intersect in $3$D space, but I think I am missing a case. Are there any other cases?

• Are the cylinders filled in or are they just the skins of the cylinders? – Sean English Jan 11 '16 at 15:12
• I am not sure, but I don't think it matters. – user19405892 Jan 11 '16 at 15:15
• Sure it does, if they are just skins then consider a two circle Venn diagram. That arrangement would have them intersecting in two parallel lines, which is contained in a plane. – Sean English Jan 11 '16 at 15:19
• Yes, but they extend upwards as well so the intersection is not in just the $2$D plane. – user19405892 Jan 11 '16 at 15:21
• The intersection is a plane. For example if you have two circles that intersect in the xy plane at (0,1) and (0,-1), then we extend these to infinite cylinders in the z direction, they intersect at the lines of the form (0,1,z) and (0,-1,z) which are contained in the yz coordinate plane. – Sean English Jan 11 '16 at 15:30

HINT: Let $C,D$ be the two cylinders. If $C \cap D$ is contained in a plane $P$, then $$C \cap D = C \cap D \cap P \subset C \cap P$$ Now, you shold know that intersecting a cylinder with a plane gives you three possibilities: an ellipse, two parallel lines or just one line.
• Read my answer better. I am talking about the intersection of a cylinder $C$ with a plane $P$. You can get two parallel lines for example with $C= \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2-1=0\}$ and $P=\{ (x,y,z) \in \mathbb{R}^3 : y=0 \}$. – Crostul Jan 11 '16 at 15:22
• It is an infinite cylinder, though, so I see many parallel lines not just one. Oh so there are $2$ parallel lines I see. – user19405892 Jan 11 '16 at 15:26