All fundamental problems in electric power system engineering involve a set of equations relating active and reactive power injections and branch flows in a network as a function of node voltages. They are traditionally expressed either in terms of polar coordinates for the complex voltages using trigonometric functions, or in rectangular coordinates using coupled multivariable polynomials. Solving these equations can be challenging. The equations admit multiple solutions and there is no practical and scalable method to provably find them all. In this presentation we derive a representation of the power flow equations as high dimensional ellipses, and discuss their use for finding all the real-valued power flow solutions. We also discuss extensions of this work for application to optimal power flow. For example, this particular representation may aid in identification and elimination of redundant constraints used in current OPF formulations, and algorithms may be developed to exploit the elliptical form.