Toroidal area-preserving parameterizations of genus-one closed surfaces

We consider the problem of computing toroidal area-preserving parameterizations of genus-one closed surfaces. We propose four algorithms based on Riemannian geometry: the projected gradient descent method, the projected conjugate gradient method, the Riemannian gradient method, and the Riemannian conjugate gradient method. Our objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of ring tori embedded in three-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the effectiveness of the proposed framework. Finally, we show how to use the proposed algorithms in the context of surface registration and texture mapping applications.