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The solution is like this:
Open the cone and make it flat, and connect two points P and Q with a straight line.

But I cannot understand why it is possible to cut a cone and make it flat.

Is it obvious or not?

By the way, it is possible to cut a cylinder and make it flat and it is impossible to cut a sphere and make it flat.

Parcly Taxel
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tchappy ha
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  • Yes, this is completely obvious. https://frogprincepaperie.com/wp-content/uploads/2013/12/christmas-paper-cone-tree-5.jpg –  Sep 27 '17 at 08:04
  • But I do not think it is obvious. What should I do? What is mathematical formulation and proof? – tchappy ha Sep 27 '17 at 08:07
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    You should get a piece of paper, make it into a cone as shown on that picture, tape it togehter, then cut it open and lay it flat to see that it can be done. Alternatively, cut out a circle of paper, remove a wedge from that circle (make a "pac-man"), join the two edges of the cut and see that you get a cone from a perfectly flat piece of paper. – Arthur Sep 27 '17 at 08:08
  • You were asking "why it is possible". –  Sep 27 '17 at 08:09
  • A link for the question in title https://math.stackexchange.com/questions/140636/what-is-the-shortest-path-equation-between-2-points-on-a-cone/140886#140886 – zwim Sep 27 '17 at 08:23

2 Answers2

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It all depends on the Gaussian curvature of the surface in question. A surface can be cut and made flat if and only if its Gaussian curvature is equal to that of the plane, i.e. zero.

The cone and cylinder have zero Gaussian curvature, so can be flattened while preserving distances, which explains the validity of the solution to the shortest-path-on-a-cone problem. The sphere has positive Gaussian curvature and cannot be flattened in a similar manner, which has long been a source of frustration among cartographers.

Parcly Taxel
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Surfaces that can be made flat are called developable. They need to be ruled, i.e. made of straight lines, like cylindres and cones, but this is not enough.

https://en.wikipedia.org/wiki/Developable_surface