Optimal workpiece orientation for multi-axis sculptured part surface machining is generally defined as orientation of the workpiece so as to minimize the number of setups in 4-, 5- or more axis Numerical Control (NC) machining, or to allow the maximal number of surfaces to be machined in a single setup on a three-, four-, or five-axis NC machine. This paper presents a method for computing such an optimal workpiece orientation based on the geometry of the part surface to be machined, of the machining surface of the tool, and of the degrees of freedom available on the multi-axis NC machine. However, for cases in which some freedom of orientation remains after conditions for machining in a single setup are satisfied, a second sort of optimality can also be considered: finding an orientation such that the cutting condition (relative orientation of the tool axis and the normal to the desired part surface) remains as constant, at some optimal angle, as possible. This second form of optimality is obtained by choosing an orientation (within the bounds of those allowing a single setup) in which the angle between the neutral axis of the milling tool and the area-weighted mean normal to the part surface, at a “central” point with a normal in that mean direction, is zero, or as small as possible. To find this solution, Gaussian maps (GMap) of the part surfaces to be machined and the machining surface of the tool are applied. To our knowledge, we are the first [1] who have picked up this Gauss’ idea to sculptured part surface orientation problem and who have developed the general approach to solve this important engineering problem [2]. Later a similar approach was claimed by Gan [3]. By means of GMaps of these surfaces, the problem of optimal workpiece orientation can be formulated as a geometric problem on a sphere. The GMap on a unit sphere finds wide application for orientation of workpiece for NC machining, for probing on coordinate measuring machines, etc. GMaps are useful for selecting the type of cutting tool, its path, workpiece fixturing, and the type of NC machine (its kinematic capabilities). The primary process application addressed is 3- and 4-axis NC milling, although the techniques presented may be applied to machines with more general articulation. The influence of tool geometry is also discussed and incorporated within a constrained orientation algorithm. This paper covers the following topics: a) the derivation of the equations of the GMap of the part surface to be machined and the machining surface of the tool; b) calculation of the parameters of the weighted normal to the part surface; c) optimal part orientation on the table of a multi-axis NC machine; d) introduction of a new type of GMap for a sculptured part surface—its expandedGMapE; and e) introduction of a new type of indicatrix of a sculptured part surface and a particular cutting tool–the indicatrix of machinability.