This paper describes a novel construction of generalized barycentric coordinates of points on a sphere with respect to the vertices of a given spherical polygon that is contained in a common hemisphere. While in the standard approach such coordinates are derived from their classical planar counterparts (e.g. Wachspress, or mean value), we instead derive them from 3D barycentric coordinates of the origin and show that they are endowed with some useful properties such as edge linearity and Lagrange property. In addition, we show that spherical mean value coordinates of both approaches coincide while their corresponding spherical wachspress coordinates are in general different.