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It is known that 4 non-collinear, non-coplanar 3D points define a 3D sphere.

Is there an equivalent property/theorem for cylinder?

101010
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  • Essentially [cross-posted at Math SE](http://math.stackexchange.com/q/985539/35416). Please [don't do that](http://meta.stackexchange.com/q/64068/188688) without giving the first post some time to get answered. And please always include links between related posts. – MvG Oct 22 '14 at 08:56
  • @MvG The one posted in mathematics is relevant but completely another question. – 101010 Oct 22 '14 at 09:11
  • Do you mean a cylinder of revolution ? –  Oct 23 '14 at 12:19
  • Is it a finite or infinite cylinder? – John Alexiou Oct 25 '14 at 13:48

5 Answers5

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For cylinder you need 5 points. But I am not EXACTLY sure if 5 points uniquely defines a cylinder.

Following references justifies this:

http://library.wolfram.com/infocenter/Conferences/7521/cylinder_5_points_computation.pdf

Tarek
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  • That article suggests six distinct solutions in general, as discussed in the section “The size of the solution set”. It refers an example were most of these solutions contained complex numbers, but two of them were real-valued, so we can't hope for a unique solution in general. – MvG Oct 22 '14 at 08:59
  • Well, as I said already, I am not sure if 5 points uniquely defines a cylinder. 5 real points are generally required to form an cylinder. – Tarek Oct 22 '14 at 09:05
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A cylinder has 5 degrees of freedom: 4 for the axis (a line in 3D space), 1 for the radius, so in principle 5 points are required and enough.

But there can be several solutions: taking five point that form a regular bipyramid (two tetrahedra with a common base), there are 6 solutions, by symmetry.

  • What about where along the 3D line the base and top are. You will need 2 more parameters. 4 for the line, 1 for the radius and 2 for the ends = 7. – John Alexiou Oct 25 '14 at 13:38
  • The OP isn't explicit about this. Adding the basis, there are two extra DOFs and more ambiguities. –  Oct 25 '14 at 13:59
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This question is much more interesting than it looks like at a first look. It is relatively easy to see how 5 points define a cylinder but not uniquely: you can pick 3 of such points to define a circular cross section and let the other two define the bases. However it is not difficult to see that the choice of the three first points is not unique. It also depends on whether "define" means that the points have to lie on the surface (in which case the two last point have to lie inside the unbounded cylinder defined by the previous three) or not.

I think there is no simple elegant statement like in the case of the sphere.

Emerald Weapon
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  • You are referring to a cylinder of finite extent here, with points on its end caps. The question doesn't state which kind of cylinder to consider. But I'd say that for this finite case you'd need six points on the side and one on each cap to uniquely define the cylinder. Those six points on the side correspond exactly to the infinite case, so that's a subproblem of the finite case. – MvG Oct 22 '14 at 09:14
  • @MvG Why six points on the side? Do you mind expanding a bit? And yes, I was assuming a finite cylinder. – Emerald Weapon Oct 22 '14 at 09:18
  • See [my answer on Math SE](http://math.stackexchange.com/a/985569/35416). Five points lead to a finite number of solutions, a sixth point can be used to choose from these. – MvG Oct 22 '14 at 10:34
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For a finite cylinder you need a total of 7 parameters.

A 3D line needs 4 parameters (minimum distance from origin, and 3 for orientation). Then from the point closest to the origin you need 2 distances defining the beginning and end of the cylinder. One more parameter is needed for the radius, and voila, you have a 3D cylinder in space defined.

You can also use two 3D points plus a radius which also needs 7 parameters.

For in infinite cylinder you need 5 parameters. 4 for the line and 1 for the radius.

John Alexiou
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Sticking to the exact vocabulary of the question, you only need two points (really one point and a scalar for the radius) for a sphere.

A cylinder needs no more that 3 points. Two to define the axis and end points, plus a 3rd (really, 2 points and a scalar) to get the radius.

burhop
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