  Let $G = \langle A,B \rangle$ be a non-elementary two generator subgroup of
the isometry group of $\mathbb{H}^2$, the hyperbolic plane. If $G$ is discrete
and free and geometrically finite, its quotient is a pair of pants and in prior
work we produced a formula for the number of essential self intersections
(ESIs) of any primitive geodesic on the quotient. An ESI is a point where the
geodesic has a self-intersection on a seam. Self-intersections of geodesics on
arbitrary hyperbolic surfaces have recently been studied by Basmajian and Chas.
Here we extend our results to two generator subgroups $G$ of isometries of
$\mathbb{H}^3$, hyperbolic three-space, which are discrete, free and
geometrically finite. We generalize our definition of ESIs and give a geometric
interpretation of them in the quotient manifold. We show that they satisfy the
same formulas.
