Replying to Allan Wechsler and Dan Asimov.

From: Allan Wechsler <acwacw@gmail.com> Date: 10/29/20, 10:54 PM

Maybe my visualization skills are tripping me up, but it seems to me that when you flatten the cone to a plane, the angle between the two edges must be 90 degrees.

Yes, it's 90 degrees. (*Not* 270.) I suppose this pie-slice angle should be theta. This raises another question. What's the angle you get if you slice the cone with a plane through its vertical axis-- call this phi--as a function of theta? Let's see, theta=0 -> phi = 0; theta=2pi -> phi = pi. So phi = theta/2, right?

From: Dan Asimov <dasimov@earthlink.net> Date: 10/30/20, 10:56 AM

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

Yes! That's what I meant by, "Maybe a nice diagram?" I suppose we can call it the fan diagram. There are two kinds of geodesics on a given cone: straight lines through the vertex, and scaled copies "that curve." The follow-up questions are, what are the relative distances- from-the-origin of the nearest point and any self-crossings, and how many self-crossings are there? --Steve