How to identify the surface of a cylindrical patch? Consider that I have a 3D facet/patch that lies on the surface of a sphere. Taking four non-collinear, non-coplanar points that lie on the facet/patch I can find the patch's underlying sphere/surface.
Is there an equivalent/similar way to identify the underlying surface of a patch/facet that lies on a cylinder (i.e., to identify the cylinder that the cylindrical patch lies on)?
 A: A cylinder (of infinite height) has five real degrees of freedom: two for the direction of its axis, two for the location of that axis and one for its radius. (A cylinder of finite height would have two more parameters indicating the positions of its end caps, but I'll assume you are referring to the infinite case.) A point on the surface determines one degree of freedom, since it is in three-dimensional space but you can move it in two directions along the surface of the cylinder without modifying the cylinder. So five points should determine a cylinder, based on the number of degrees of freedom.
It might however be the case that a finite number of different solutions can exist for any combination of five possible points. Cylinders Through Five Points: Computational Algebra and Geometry by Daniel Lichtblau, found by user Tarek in this answer on Stack Overflow, suggests that we may in general expect six distinct sets of parameters for every combination of five input points in general position. Some of them may be complex, but there was a sufficiently general example where two of the solutions were real. Therefore even if you have five points, the cylinder is still not uniquely defined. If you add a sixth point, you will most likely be able to choose between the different cylinders for the first five points.
Actually finding a cylinder from five points is certainly a lot harder than finding the sphere through four given points. Geometrically, in the case of the sphere you'd construct the perpendicular bisector plane between every pair of points, and these intersect in the center of the sphere. That is because you can say that each such bisecting plane is the locus of points which have equal distance from both the given points. The analogous situation for the cylinder is harder, because you'd have to talk about a locus of lines not points, and these lines would not lie within a common plane so the problem appears to be more non-linear.
If you accept the problem as being non-linear (and as discussed above essentially of degree six) in nature, you can of course formulate an implicit (i.e. not parametric) representation of a cylinder, using five variables (or more than five and a sufficient number of additional constraints to bring the number of degrees of freedom back to five). Then you could plug each of the points into that representation, and you'd end up with a system of five (non-linear) equations. Or an over-constrained system of six equations if you add a sixth point.
If I find the time, I might do that later on and edit my post to include my result. I'd also run some example tests, see whether the solutions are unique or not. If someone else beats me to it, feel free to edit my post of write a post of your own. Until then, this answer will remain incomplete.
