Data from the polar-orbiting satellite DE 2 are used to calculate one-dimensional electrostatic potential distributions across the polar cap region. Using passes that lie within ¿3 hours MLT of the dawn-dusk line, various parameters of the polar potential distribution (location and magnitude of the maxima and minima, location of the zero potential point, etc.) are analyzed in relation to each other and to the interplanetary magnetic field (IMF). The resulting dependences are used to derive a two-dimensional model of the distribution of the electrostatic potential in the high-latitude ionosphere during times of southward IMF. This model can be generated using as inputs either the ionospheric potential parameters or, based on the relationships analyzed here, the IMF conditions. The capabilities of the resulting mathematical model are illustrated, and the importance of retaining a flexibility in the model to accommodate individual observations is emphasized.