  We give a geometric account of the relative motion or the shape dynamics of
$N$ point vortices on the sphere exploiting the $\mathsf{SO}(3)$-symmetry of
the system. The main idea is to bypass the technical difficulty of the
$\mathsf{SO}(3)$-reduction by first lifting the dynamics from $\mathbb{S}^{2}$
to $\mathbb{C}^{2}$. We then perform the $\mathsf{U}(2)$-reduction using a dual
pair to obtain a Lie--Poisson dynamics for the shape dynamics. This
Lie--Poisson structure helps us find a family of Casimirs for the shape
dynamics. We further reduce the system by $\mathbb{T}^{N-1}$-symmetry to obtain
a Poisson structure for the shape dynamics involving fewer shape variables than
those of the previous work by Borisov and Pavlov. As an application of the
shape dynamics, we prove that the tetrahedron relative equilibria are stable
when all of their circulations have the same sign, generalizing some existing
results on tetrahedron relative equilibria of identical vortices.
