A novel and efficient cutter path planning method for machining intricately shaped curved surfaces, called the steepest directed tree method, is presented. The curved surface is defined by triangular facets, the density and structure of which are determined by the intricacy and form accuracy of the surface. Geometrical form definition and recognition of the topological features are used to connect the nodes of the triangulated surface meshes for the successive and interconnected steepest pathways, which makes good use of end milling characteristics. The planetary cutter centers are determined to locate along smoothly changing paths and then the height values of the cutter are adjusted to avoid surface interference. Several machined examples of intersecting and intricate surfaces are presented to illustrate the benefits of the new approach. It is shown that due to more consistent geometry matching between cutter and surface(in comparison with the current CC Cartesian method) surface finish can be typically improved. Moreover, the material in concave fillets which is difficult to be removed by ball mills can be removed efficiently. The built-in positioning of cutter to avoid interference runs minutely in the sharp and discontinuous regions. The steepest upward movement of the cutter gives a stable dynamic cutting state and allows increase in the feedrate and spindle speed while remaining the stable cutting state.