We present a simple approximation that permits us to obtain the axisymmetric two-dimensional supersonic solution of a rotating radiation-driven stellar wind from the Friend & Abbott one-dimensional model of the equatorial flow. Our solution predicts the formation of a dense equatorial disk if the rotation rate of the star is above a threshold value, which depends on the ratio of the terminal speed of the wind to the escape speed of the star. Along the upper main sequence (earlier than B2), both the observed and theoretical values for this ratio decrease monotonically toward later spectral types. For early O stars, the disk can only form if the rotation speed is in excess of 90% of the critical (breakup) speed. For B2 stars, the disk forms at rotation speeds above 50%-60% of the critical rotation speed, depending on the adopted terminal speed (observational vs. theoretical estimates). This corresponds to a rotation speed V(rot) > 230-300 km s-1. Later than B2, the theoretical terminal speed ratio increases, and at B9 the disk forms when the rotation speed exceeds 73% of the critical value. The change in the disk formation threshold as a function of spectral type qualitatively explains the frequency distribution of Be stars and indicates a maximum probability around B2. The disk is formed because the supersonic wind that leaves the stellar surface at high latitudes travels along trajectories that carry it down to the equatorial plane, where the material passes through a standing oblique shock on top of the disk. The ram pressure of the polar wind thus confines and compresses the disk. For Be stars, the disk is predicted to be quite thin (almost-equal-to 0.5-degrees opening angle) and has a density enhancement rho(eq)/rho(pole) 10(3). This compression is large enough to potentially explain the discrepancy between the inferred UV and IR mass-loss rates of Be stars. Adjacent to the disk, the standing shock heats the flow that enters the equatorial region to temperatures of 10(5)-10(6) K before the material finally mixes with the disk. This temperature is sufficient to produce superionization in the winds of Be stars, and the shock location explains observations indicating that C IV is concentrated toward the equator. In addition, the shock temperature indicates that Be stars will be EUV and soft X-ray emitters.
