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

Let's recall two facts first:

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

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.