Parameter estimation for exponential sums is a classical problem in signal processing. Recently, a new concept for estimating parameters of bivariate exponential sums has been proposed. The resulting method relies on parameter estimations for univariate exponential sums along several lines in the plane. These (univariate) parameter estimations are being used to first compute the projections of the unknown bivariate frequency vectors onto these lines, before they are combined to obtain estimations for the sought frequency vectors of the bivariate exponential sum. In this paper, we address theoretical questions concerning this new concept, namely (a) how many lines are needed for exact reconstruction, and (b) how to recover linear combinations of shifted positive definite functions.