Let $\gamma=\gamma(L)$ be a
simple (non-self-intersecting) closed curve of length $L$
on the unit-radius sphere $S$.
So if $L=2\pi$, $\gamma$ could be a great circle.

I am seeking the most equitably distributed
$\gamma(L)$, distributed in the sense that
the length of $\gamma$ within any disk is minimized.
This is something like placing repelling electrons on a sphere,
but here the curve self-repels.
So there should be no "clots" of $\gamma$ anywhere on $S$.
I am especially interested in large $L$.
A possible $\gamma$ is shown below, surely not optimal for its length:

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Here is an attempt to capture more formally "equitably distributed."
I find this an awkward definition, and perhaps there is a more
natural definition.

Around a point $c \in S$, measure the $r$-density
of $\gamma$ as the total length within an $r$-disk:
$$d_\gamma(c,r) = | \gamma \cap D(c,r)|$$
where $D(c,r)$ is the disk of geodesic radius $r$
centered on $c$.
Then define $d_\gamma(r)$ as the maximum of $d_\gamma(c,r)$ over
all $c \in S$.

Finally, we can say that, for two curves $\gamma_1$ and
$\gamma_2$ of the same length $L$, that
$\gamma_1 \le \gamma_2$
if $d_{\gamma_1}(r) \le d_{\gamma_2}(r)$
for all $r \in (0,\pi)$, i.e.,
$\gamma_1$ is less concentrated than $\gamma_2$ for all $r$
up to a hemisphere.

This definition provides a partial order on curves of a given length $L$.
One version of my question is:


  Q. What do the minimal elements of this poset
  look like, especially as $L$ gets large?


These minimal curves are in some sense nowhere densely clotted.

Update. Acknowledging Gerhard Paseman's remark, I thought I would include
this attractive image of a space-filling curve on a sphere:

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(Image from this website).

But notice it is certainly not equidistributed in any sense, crowding near the northpole.


Question: Does reach let you threaten an opponent if there is another opponent between you?
Answer: As you have -- and per the PF SRD (http://www.d20pfsrd.com/gamemastering/combat#TOC-Threatened-Squares), "You threaten all squares into which you can make a melee attack".  Having an additional character between two opponents provides cover to both, but there's no mention in the rules that cover negates threatening, so flanking is still valid.  In the PF SRD there's an example of an ogre providing flanking despite not being adjacent to the target he threatens.