Spherical geometries arise in many scientific areas. In this synthesis, we discuss the main ideas about the double Fourier sphere (DFS) method. We apply the DFS method to derive barycentric interpolation formulas for a sphere from one dimension trigonometric barycentric formulas. The formulas were derived for three commonly utilized grid types in numerical weather prediction: equally spaced, shifted equally spaced, and Gauss-Legendre grids. Numerical results show that barycentric interpolation on a sphere converges exponentially for smooth functions. Numerical results also show that we have ill-conditioning in the computation of interpolation weights for the latitude direction when interpolating using a HEALPix grid, making interpolants unstable for considerably large grid values. We demonstrate that a Floater-Hormann type of barycentric interpolation can be used to construct the variation of the barycentric interpolant for the sphere that is stable for HEALPix grids.