This English translation, with revisions, of the well-known Chinese edition presents systematically the geometric theory of conjugate tooth surfaces in a more or less rigorous form. The concepts of the two kinds of limit points and limit curves are explained in some detail and a general formula for induced normal curvature is derived, of which the formula of Euler-Savary appears as a direct consequence. The idea of relative differentiation, initiated by Zhida Yan, simplifies the presentation considerably. The phenomenon of secondary contact, closely related to the limit curve of the second kind, is treated in full and its applications to direct and indirect generation are explained; concrete formulas for secondary plane envelope are derived.
Notes to surface theory; motion and relative motion; conjugate surfaces; induced normal curvature; derivative of induced normal curvature; secondary contact and direct generation; secondary plane envelope (direct generation); secondary contact and indirect generation.