To address a major technical challenge in simulating geometric models of machined sculptured surfaces in three-axis virtual machining, this paper presents an efficient, accurate approach to representing the 3D envelopes of a cutter sweeping sequentially through cutter locations; these envelopes embody the furrow patches of the machined surfaces. In our research, the basic mechanism of removing stock material in three-axis computer numerically controlled (CNC) milling of sculptured surfaces is investigated, and, consequently, an effective model is proposed to represent the 3D envelopes (or furrow patches). Our main contribution is that a new directrix (or swept profile) of the furrow patches (mathematically, ruled surfaces) is identified as a simple 2D envelope of cutting circles and is formulated with a closed-form equation. Therefore, the 3D cutter-swept envelopes can be represented more accurately and quickly than the existing swept-volume methods. With this innovative approach, a method of accurate prediction of the machining errors along tool paths in three-axis finish machining is provided, which is then applied to the optimization of tool-path discretization in two examples. Their results demonstrate the advantages of our approach and verify that the current machining-error-prediction methods can cause gouging in three-axis sculptured surface milling.