The accelerating field of a point-cathode electron source is divided into two regions, a diverging spherical-field region located in the immediate vicinity of the point cathode, and a converging-field region which occupies most of the cathode-anode space. Analysis of this model gives the Gaussian source size, axial position of the source, and spherical- and chromatic-aberration constants of the pointed-cathode electron gun. After including diffraction effects, the electron-optical parameters are optimized for minimum possible source size. The current density which appears to be emitted by the apparent source is introduced as a figure of merit for this type of electron gun, and the performance of field-emission and Schottky-emission pointed cathodes are compared. Source diameters below 100 Å are predicted for oriented-tungsten field-emission pointed cathodes, with apparent current densities exceeding 104 A/cm2; source diameter below 1000 Å are predicted for oriented-tungsten Schottky-emission pointed cathodes, with apparent current densities of approximately 100 A/cm2.