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?

  • $\begingroup$ Are the cylinders filled in or are they just the skins of the cylinders? $\endgroup$ – Sean English Jan 11 '16 at 15:12
  • $\begingroup$ I am not sure, but I don't think it matters. $\endgroup$ – user19405892 Jan 11 '16 at 15:15
  • $\begingroup$ 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. $\endgroup$ – Sean English Jan 11 '16 at 15:19
  • $\begingroup$ Yes, but they extend upwards as well so the intersection is not in just the $2$D plane. $\endgroup$ – user19405892 Jan 11 '16 at 15:21
  • $\begingroup$ 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. $\endgroup$ – 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.

Can you go on from here?

  • $\begingroup$ I don't see how there are other possibilities than them being just tangent. $\endgroup$ – user19405892 Jan 11 '16 at 15:08
  • $\begingroup$ Your intuition is correct: a proof of this fact can be found using my argument. Anyway, the two cylinders can also intersect just in one single point (if they are tangent but not parallel), which lies in a plane. $\endgroup$ – Crostul Jan 11 '16 at 15:12
  • $\begingroup$ How can you get two parallel lines? $\endgroup$ – user19405892 Jan 11 '16 at 15:17
  • $\begingroup$ 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 $P=\{ (x,y,z) \in \mathbb{R}^3 : y=0 \}$. $\endgroup$ – Crostul Jan 11 '16 at 15:22
  • $\begingroup$ It is an infinite cylinder, though, so I see many parallel lines not just one. Oh so there are $2$ parallel lines I see. $\endgroup$ – user19405892 Jan 11 '16 at 15:26

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