Abstract When a ball-end milling tool cuts a given path on a smooth surface, it is desirable to maintain a fixed angle ψ between the tool axis a and the local surface normal n at each point, to ensure a constant speed of the tool cutting edge against the surface. This means that the tool axis a must lie on a cone of angle ψ about the surface normal n at each point, but its azimuthal position on this cone remains indeterminate. To resolve this indeterminacy, while minimizing actuation of the rotary axes that orient the workpiece relative to the tool, the component of a in the surface tangent plane is specified through the parallel transport of a given initial state along the path. This amounts to the integration of coupled first-order differential equations that involve the Christoffel symbols for the given surface. Alternatively, the tangent plane component of the tool axis a is shown to be rotation-minimizing with respect to the surface normal n, and its orientation relative to the Darboux frame along the tool path can be determined by integrating the geodesic curvature along that path. The method is illustrated by closed-form solutions for simple analytic surfaces, and numerical integration using an object-oriented software implementation for free-form surfaces. The real-time implementation of such rotation-minimizing 5-axis tool motions for free-form surfaces is well within the scope of modern CNC systems.