# How many curves can fit on a sphere without intersecting?

(This is more or less a generalized and more abstract version of this m.SE question.)

Let's recall two facts first:

1. Any (arc-length parametrized) space curve is uniquely determined (up to rigid motions) by its curvature $\kappa(s)$ and its torsion $\tau(s)$.

2. The curvature and torsion of a unit-spherical curve satisfies the equation $$\left(\tau\frac{\mathrm d}{\mathrm ds}\frac1{\kappa}\right)^2+\frac1{\kappa^2}=1$$

In my m.SE question, the unique properties of the loxodrome and the existence of the Mercator projection allowed the determination of when $n$ loxodromes can be placed on a sphere without intersecting. I am now considering the following question:

What conditions should the curvature and torsion of a spherical curve satisfy so that $n$ copies of it can be placed on a sphere without intersecting?

Alternatively, any spherical curve can also be defined as parametric equations of the longitude $\theta=\theta(t)$ and co-latitude $\varphi=\varphi(t)$, or in Cartesian form (for a unit sphere):

\begin{align*}x&=\cos\,\theta(t)\sin\,\varphi(t)\\y&=\sin\,\theta(t)\sin\,\varphi(t)\\z&=\cos\,\varphi(t)\end{align*}

so we can also ask

What conditions should $\theta(t)$ and $\varphi(t)$ satisfy so that $n$ copies of it can be placed on a sphere without intersecting?

If there are better ways to characterize the conditions than these, I'd be interested in hearing about them.

For convenience, let's consider only the unit sphere.

One theorem(?) I have along these lines is

If a spherical curve fits within a hemisphere without touching a great circle, then two copies of the curve can be fit on the sphere without intersecting.

but I haven't figured out how to generalize this.

-
Short note: I got no bites weeks after first asking this on m.SE, so I post it here. – J. M. Aug 17 '11 at 3:15
I'm a bit worried. I can foliate the sphere by a family of circles centred about two antipodal points, and any discrete subset of this family does not have any intersection points and arbitrarily large curvature. Have I missed the point? – Glen Wheeler Aug 17 '11 at 9:00
Glen: I think the idea is to look at curves with small curvature. For example, the number of closed curves of 0 curvature that fit on the sphere without intersecting is 1. – Ben McKay Aug 17 '11 at 12:10
@Glen: I think the curves are supposed to be isometric to each other, so circles of different sizes wouldn't count. – Andreas Blass Aug 17 '11 at 16:57
I guess the question should include these assumptions, if appropriate. – Glen Wheeler Aug 19 '11 at 11:06