How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
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asked Dec 3, 2014 at 19:06
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$\begingroup$ Solid cylinders, or shells? $\endgroup$– The Masked AvengerCommented Dec 3, 2014 at 19:12
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$\begingroup$ @TheMaskedAvenger: points on the infinite cylinder surface (shells). $\endgroup$– David LampertCommented Dec 3, 2014 at 19:17
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$\begingroup$ As noted in a disappeared comment, there is no cylinder if one point is interior to the convex hull of the other 4. The question is what are the other possible numbers of cylinders, especially what is the maximum? $\endgroup$– David LampertCommented Dec 3, 2014 at 21:50
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$\begingroup$ Aside - this theoretical question arose years ago when I worked on an industry coordinate measuring machine application to compute and display cylinders (and other manufactured shapes) to best fit a number (greater than 5) of probe measured points. $\endgroup$– David LampertCommented Dec 3, 2014 at 22:09
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1$\begingroup$ The maximum number of cylinders is at least 3, since you can find 3 cylinders sharing the 8 vertices of a cube. But maybe you meant to rule this out when you asked about "general position". $\endgroup$– Gerry MyersonCommented Dec 3, 2014 at 22:35
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1 Answer
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If the 5 points are generic then I believe Lemma 7.1 in arXiv: 1306.2346 is relevant to your question.
The Lemma, for the case of cylinders, says if there are two cylinders that pass through the same 5 generic points and there is an isometry that maps one of the cylinders onto the other, then the two cylinders are the same.
(This is only an answer rather than a comment because I am not allowed to post comments.)
Edit: there's also a relevant paper called "On circular cylinders by 4 or 5 points in space" by Devillers, Mourrain, Preparata and Trebuchet.
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$\begingroup$ Thanks! I think the second paper answers the question: 6. $\endgroup$ Commented Dec 3, 2014 at 22:55
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$\begingroup$ library.wolfram.com/infocenter/Conferences/7521/… shows that there are 0,2,4,or 6 cylinders through 5 generic points. There is also discussion of the optimization problem with more than 5 points. $\endgroup$ Commented Dec 3, 2014 at 23:49
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$\begingroup$ They give 2 different configurations with 6 cylinders and conjecture that all configurations with 6 are perturbations of one of them. $\endgroup$ Commented Dec 3, 2014 at 23:55