The paper describes a new theory and methods for designing hyperbola, degenerate circles, involute and roulette curves which have been applied in grinding convex involute gears, cutting special profiles on a gear shaper, grinding the race-way of ball thrust bearings, form grinding and grinding cyclo-cycloidal gears with an abrasive belt. All these applications have been used in industruy with considerable benefits. The curves are produced by different combinations of straight line and circle motions, Given the curve parameters of the workpieces to be machined, the best approximate curve can be obtained from a computer program designed for this purpose.