Computed swept-frequency traces of minimum group-path in backscatter-radar ionograms are presented for ionospheric electron distributions in the polar region. Some of the traces for ground backscatter contain cusps that are shown to result when a small fraction of rays from the radar encounter a region in which the vertical electron-density gradient decreases sharply with distance after the rays passed through apogees. An accurate geomagnetic field model is used in obtaining traces for direct backscatter from field-aligned ionization that causes radar auroral clutter. A method is described for representing three-dimensional electron distributions. The method is sufficiently flexible for iterative ray computations, which appear as a reasonable and potentially reliable approach to the problem of converting backscatter-radar ionograms to electron distributions.