Spherical Functions (SF) play a pivotal role in Diffusion MRI (dMRI) in representing sub-voxel-resolution micro- architectural information of the underlying tissue. This in- formation is encoded in the geometric shape of the SF. In this paper we use a polynomial approach to extract geometric characteristics from SFs in dMRI such as the maxima, min- ima and saddle-points. We then use differential geometric tools to quantify further details such as principal curvatures at the extrema. Finally we propose new scalar measures like the Peak Fractional Anisotropy (PFA) and Total-PFA, to represent this rich source of information for characteriz- ing white-matter (WM) fibers. As an example we illustrate our method on the Orientation Distribution Function (ODF) estimated from real data.