A realistic model of a point-cathode electron source is analyzed in this paper, and its electron optical properties are discussed. Both cathode and anode are modeled as equipotential surfaces of a sphere on orthogonal cone (SOC). The model predicts that the radius of the apparent electron source produced by this diode may be as small as 2-3 nm for field emitters and as small as 4-6 nm for strong-field Schottky emitters. This source is found to be a virtual image of the emitting area of the cathode. The image position and magnification are determined for different representative cathode shapes, and quantitative results are also presented on geometric aberrations, effects arising from the initial energies of emission, and on source current and current density. An optimum cathode apex radius for smallest source sizes is predicted to be about 200 nm for field emission and about 700 nm for Schottky emission. These results are compared with those from the often used, but less accurate, sphere model.