A mathematical model for sputtering a shape or cavity with an arbitrary cross-sectional profile has been developed for focused ion beam milling. The ion beam is assumed to have a Gaussian intensity distribution and a submicron width. The model solves for ion beam dwell times on a pixel grid which yields the desired feature depth as a function of the pixel (x,y) coordinate. The solution is unique and accounts for the ion beam flux contribution at any point from all other pixels in the address matrix. A semiempirical sputter yield treatment allows for a very wide range of ion beam/solid combinations and for yield variations with ion energy and angle of incidence. Solutions have been obtained for parabolic surfaces of revolution, a parabolic trench (with a plane of symmetry) and a hemispherical pit. Either a square or a circular pixel matrix was used for the parabolic shapes. Correspondence between the predictions of the model and experimental 20 keV Ga+ sputtering of a parabolic cross-section trench in Si(100) was within the limits of the accuracy of the experimental control.