Oh Give Me a Cone
By Bobby Neal Winters
The shortest distance between two points is a straight line. Remember that even when it is not true. And there are times when it is not true. The surface of the planet earth would be one example. The world, as has been known since the Greeks is a sphere. (There are a group of hyper-correcting sphincteroids out there who will said the earth isn’t a sphere, but an oblate spheroid, but quite frankly it is so close to a perfect sphere that you can’t tell without some pretty accurate measurements. But I digress.)
In any case, the world is a sphere, and there are no straight lines on it because spheres are round as round can be. I want you to think about two points on the surface of a sphere. The center of the sphere is a third point. These three points do not lie on a straight line, so they determine a plane. That plane will cut through the sphere in a circle and that circle is called a great circle. Are you with me so far? If not, go back and read this paragraph until you are. Have you done that? Okay, then, let us proceed.
Any two points will cut a circle into two arcs. The shorter of the two arcs is the shortest distance between two points. This is a nice example. The solution is elegant. Everything can be stated it simple language. We walk away from it loving life, mathematics, and geometry while we whistle zippity doo dah.
We now wish to apply our mathematical skills in other areas. This leads us to the concept of a surface. Surfaces are mathematical constructs that generalize the notion of the plane (where the shortest distance between two points is a plane) and the sphere where the shortest distance between two points is an arc of the great circle.
It is at this point in the discussion of surfaces that one typically runs of into a discussions of toruses (tori) of the one-holed, two-holed, and n-holed variety. That would be an interesting discussion, but I want to talk about something that seems, at first blush, to be simpler: the cone. I was drawn into this by the Facebook discussion of some intelligent amateurs (in the literal sense of that word) who got considerably more than they bargained for. The same has happened to professionals.
The question is: what is the shortest path between two points on the surface of a cone?
Before we go on to engage this question, I would like to lay down some information on the cone. The cone is known from antiquity. I suppose that anyone who had papyrus or vellum laying around wouldn’t have to use too much imagination to roll it into a cone. The ancient Greeks in the centuries before Christ did work with what we call the conic sections, discovering that if you intersect a cone with a plane at various angles you get ellipses, hyperbolas, and parabolas. These are very important curves and became even more important when Isaac Newton discovered that objects under the influence of gravity follow such curves.
So you take what you know about the great circle and shortest distance and you add that with what you know about how the plane interacts with the cone, and it is quite natural to form the believe that the shortest distance between two points on a cone will be the shortest arc of some conic section.
It is also quite wrong.
Let us now engage with the issue of the shortest path between two points on a cone. In some sense, this is an easy question. Mark the two points and then take your mathematical scissors out to cut from the base of the cone to the point of the cone in such a way as to not cut the shortest path. You can then roll the cone out flat on the plane. We can then see the shortest path is a line segment in the plane, just as we said in the first paragraph.
This is not something that can be done with the great circle on the sphere. The sphere has a property called constant positive curvature. The cone, even though it looks curved, has zero curvature, like the Euclidean plane. (The hyperbolic plane has constant negative curvature; this means it looks like a Pringles potato chip at every point, but a discussion of this would take us too far afield again.)
So when we have the cone rolled out flat on the plane, we see that all we have to worry about are lines, but it is still possible that when roll the cone up in space that straight line will yield a conic section of some sort. In order to show that our hopes and dreams of such an elegant solution are for naught all I have to do is to come up with one curve on the surface of a cone obtained from a line in the way described that is not an conic section.
Consider the cone obtained from folding the half plane in such a way that the origin becomes the cone point.
Now consider a vertical line that is parallel to the y-axis at a distance of one unit. In the cone, this curve looks like it might be a parabola or a branch of a hyperbola a first glance, but it can’t be. It has two ends and those two ends remain within two units of each other. In either a parabola or a hyperbola, the two ends become infinitely far apart.
When I first began playing with the problem of geodesics on a cone (geodesics are the technical name for the type of curve that gives the shortest path between points), I’d thought that the circles on the cone that are centered at the cone point would be geodesics. I was quickly disabused of this idea, however, when I observed that the chords of these circles (when they are laid flat on the plane) are shorter than the subtended arcs on those circles.
So this was a failed mathematical experiment, but it leads to a thought experiment. Any geodesic will be a line in the plane when the cone is unrolled. Consider now the family of circles centered at the cone point. Begin at the cone point, and let the circles increase in size from 0 radius until it makes first contact with the geodesic. That initial point of contact (thinking of the cone point being at the top) will be a maximum point of the geodesic. Let the circle increase in size until it is just past the maximum point of the geodesic.
When we lay this flat, it the geodesic will be a chord of the circle centered at the cone point. Take a perpendicular bisector of the chord through the cone point and cut along it. From this we see that any geodesic for the cone can be obtained from a perpendicular to a radius for the cone.
A little playing with this will show you that how steep the cone is will matter a lot regarding the type of geodesics you get. For example, if you make a cone from a quarter plane, the geodesics will cross themselves.