In order to think about this, I find it useful to imagine first cutting open the cone by a cut from its vertex — making it into a sector of the plane — and then sewing together infinitely many copies of this sector, edge to edge, to that the whole thing has the topology of an infinite parking ramp (or a log surface in the complex plane) and laying it flat in the plane. Now any piece of a geodesic on the original cone can be transferred to this surface and despite its being flat, you can extend the geodesic indefinitely and it will always be in the correct corresponding location on one of the (infinitely many) sectors, each corresponding to the original cone. (We've put the vertex of the cone at the origin.) Of course the geodesic once it's on the plane will just be a straight line if thought of in the plane. The generators of the cone (straight lines through its vertex) correspond to rays emanating from the origin in the plane.
From this it's possible to see that (assuming the geodesic doesn't cross the cone's vertex) that it will reach a closest point and then veer away, and that it cannot circle the cone infinitely many times. (But by choosing a cone with arbitrarily small cone angle, one can find geodesic that do circle the cone arbitrarily many times.)
—Dan Steve Witham wrote: ----- Questions: Imagine you took apart the lampshade, cut straight through the vertical joint (so there was no overlap), and laid the cone material out flat. Maybe fill in the center part where the top hole was cut out. The material would form some pie-chart fraction. Measuring from the center/tip, an arbitrary unfolded cone could cover <180, 180 or >180 degrees of the pie. What is that angle for *my* lampshade? Considering whole cones again, how many times does a geodesic of a cone cross itself? Can that number be infinite? Can it be finite? If finite, what does the number of crossings depend on and how? What are the relative distances of the crossings from the tip of the cone? Are the even and odd crossings spaced differently? What about the non- self-crossing point where the curve is closest to the tip? Maybe a nice diagram? Does the 3D curve have a name? If you draw verticals from the curve down to a horizontal plane, what is that curve? If you project horizontally to a vertical plane what kinds of curves do you get? Can the threads be arranged so that they consistently follow the over-under weave pattern all the way around? If so, show how. Hint: imagine making the cone out of graph paper. Can you imagine a machine to weave it, or even a jig to help weave it by hand? If the consistent weave is impossible, prove it. -----