In previous work, we and others have developed visibility-based tool path generation schemes. Almost all previous research implicitly assumes that all visible parts are machinable. Though usually true practice, this assumption hides several subtleties inherent to the geometry of the machining process. Here, we define machinability in a stricter sense, as a generalization of the robotic path planning problem. Then, we define various “tight” necessary conditions for strict machinability, and show the connections between these conditions. After demonstrating the richness of the information contained in visibility, we show how to compute visibility effectively. Visible directions constitute an approximate feasible configuration space of a cutting tool. We also address questions pertaining to the topological connectivity of the feasible space. The theoretical results of this paper lay down a firmer foundation of machining path planning.