Am seeking help towards a problem that has occurred to me but is beyond my current skill-set.

On the Euclidean plane, the ratio of a circle's diameter to its circumference is always pi. But on a curved surface of course this is not always so. On a sphere, for instance, the very largest circle you can draw has a circumference which is twice the length of the diameter. It is easiest for me to visualize this by imagining a globe; the equator is a "great circle," and the diameter, running through the north (or south) pole, is exactly half as long as the equator.

As the diameter of the circle shrinks (say, as you get closer to the north pole), the ratio gets bigger, approximating Euclidean conditions, until you get to the pole itself, and you get pi. Of course this is a paradoxical situation, since this polar 'circle' doesn't strictly exist on the sphere's surface, as it is infinit'simal.

But my question is not about the paradox. It just occurred to me that since the range of possible values for this ratio C/d lies between 2 and pi, there is a particular distance from the 'pole' at which the ratio is precisely 3. I.e., at what "latitude" is the circle I can draw around the pole precisely 3x its diameter, or 6x its radius (its distance from the pole?)

I have thought quite a bit about how to figure this, but I keep bottoming out. My thought is that I need to figure the angle between the center of the sphere and the line and center of the circle. A 'great circle' that has the ratio C/d=2 will have a right angle between the axis of the sphere (the line running through the center of the sphere and the center of the circle) and a line from the sphere's center intersecting the circle. As this angle gets smaller, the ratio C/d approaches pi. (or does it approach infinity? That doesn't seem to make sense, and it verges on the aforementioned paradox I said didn't really interest me. I guess I have to admit a small degree of interest). So can I use this to reverse-engineer an angle that gives me a ratio of 3 for C/d? Or am I on the wrong track altogether?

Many thanks in advance,
~~Bryan,
Seattle WA

Let α = ∠AON and assume the radius of the sphere is 1, so that OA = OB = ON = 1.

Then

U(α) = arc ANC = 2π × 2α / 2π = 2α.

From ΔAOB, AB = sin(α) and the circumference of the circle AC is

V(α) = 2π sin(α).

We are interested in the ratio, say,

f(α) = V(α) / U(α) = π sin(α)/α.

Let's check:

• f(π/2) = π × 1 / (π/2) = 2. That's OK.

• f(0) = π lim_α→∞sin(α)/α = π. Also all right.
  

You ask about α for which

π sin(α) / α = 2.

Well, this is a transcedental equation that could only be solved approximately. It definitely has a solution between 0 and π/2.