  In this paper, we first extend the finite distortion problem from the bounded
domains in $\mathbb{R}^2$ to the closed genus-zero surfaces in $\mathbb{R}^3$
by the stereographic projection. Then we derive a theoretical foundation for
spherical equiareal parameterizations between a simply connected closed surface
$\mathcal{M}$ and a unit sphere $\mathbb{S}^2$ via minimizing the total area
distortion energy on $\overline{\mathbb{C}}$. Provided we determine the
minimizer of the total area distortion energy, the minimizer composed with the
initial conformal map determines the equiareal map between the extended planes.
Taking the inverse stereographic projection, we can derive the equiareal map
between $\mathcal{M}$ and $\mathbb{S}^2$. The total area distortion energy can
be rewritten into the sum of Dirichlet energies associated with the southern
and northern hemispheres, respectively, and can be decreased by alternatingly
solving the corresponding Laplacian equations. Based on this foundational
theory, we develop a modified stretch energy minimization for the computation
of the spherical equiareal parameterization between $\mathcal{M}$ and
$\mathbb{S}^2$. In addition, under some mild conditions, we verify that our
proposed algorithm has asymptotically R-linear convergence or forms a
quasi-periodic solution. Numerical experiments on various benchmarks validate
the assumptions for convergence always hold and indicate the efficiency,
reliability and robustness of the developed modified stretch energy
minimization.
