Smooth continuous spiral tool paths are preferable for computer numerical control (CNC) machining due to their good kinematic and dynamic characteristics. This paper presents a new method to generate spiral tool paths for the direct three-axis CNC machining of the measured cloud of point. In the proposed method, inspired by the Archimedean spiral passing through the radial lines in a circle, 3D radial curves on the cloud of point are introduced, and how to construct the radial curves on the complex cloud of point is discussed in detail and then a practical and effective radial curve construction method of integrating boundary extraction, region triangulation, mesh mapping, and point projection is proposed. On the basis of the radial curves, the spiral tool path can be generated nicely by interpolating the radial curves using a spiral curve. Besides, the method of identifying and eliminating the overcuts and undercuts in the spiral tool path resulting from the interpolation error is also presented for good surface quality. Finally, several examples are given to validate the proposed method and to show its potential in practical applications when quality parametric models and mesh models are not available.