1680e9fa7b4dd5d62ece800239bb53bd-Supplemental.pdf

We analyze here briefly some basic notions of the geometry of the sphere that we use in our algorithm and convergence analysis. We refer the reader to [1, p. 73-76] for a more comprehensive presentation. Tangent Space: The tangent space of the r-dimensional sphere Sr at a point p is an r-dimensional vector space, which generalizes the notion of tangent plane in two dimensions. We denote it by TpSr and a vector v belongs in it, if and only if, it can be written as α(0), where α: ( ε,ε) Sr (for some ε > 0) is a smooth curve with α(0) = p. The tangent space at pcan be given also in an explicit way, as the set of all vectors in Rr+1 orthogonal to p with respect to the usual inner product.