As we all know, in a plane, there is only one straight line through two points. But on a cylinder, there are countless helixes passing through two points. Why is that?

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    $\begingroup$ What kind of a "why" answer are you looking for? $\endgroup$ – user856 Nov 22 '19 at 15:02
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    $\begingroup$ There's no reason to expect geodesics on different surfaces to behave in the same way. I'll note that you can impose a "two points determine a unique line" property upon the cylinder by taking "line" to mean "shortest geodesic". $\endgroup$ – Blue Nov 22 '19 at 15:06
  • $\begingroup$ Why countless? There are only countably many. $\endgroup$ – Ivan Neretin Nov 22 '19 at 15:43
  • $\begingroup$ @kimchilover I'm sorry. $\endgroup$ – z.qmpx Nov 23 '19 at 13:29
  • $\begingroup$ @Blue The other helixes are not straight lines? $\endgroup$ – z.qmpx Nov 23 '19 at 13:33

Roughly speaking:

The "straightness" of a curve is a local property, determined by what happens near each point. A curve is straight if it's straight at each of its points. There "straightness" is determined by minimizing length. At each point there's a straight line that starts out in any direction.

Asking about straight lines that join two points is a global question. The answer may depend on the global topology of the surface and the location of the particular points. For example, on the sphere there are two "straight lines" joining any pair of nonantipodal points. There are infinitely many joining the North and South poles.

Among the straight lines joining two points you can search for a geodesic: a line of minimal length. It may or may not be unique. For nonantipodal points on the sphere it is, for a pair of poles it's not.


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