Helical gear reducers with parallel shafts have a wide range of applications.

　　From small clocks and precision measuring instruments (used for motion transmission) to large gears employed in marine propulsion systems (for power transmission), gears have an extremely wide range of applications and, alongside screws and bearings, rank among the most critical mechanical components.

　　There are many types of gears, but the simplest and most commonly used is the gear employed to transmit a specific speed ratio between two parallel shafts at a prescribed center distance. The gear shown, whose teeth are parallel to the axis, is known as a spur gear.

　　The simplest method for transmitting a specific angular velocity ratio between two parallel shafts is the rolling‑friction drive. This involves two cylinders whose diameters are inversely proportional to the speed ratio, rotating in contact without slipping—contact occurs on the outer side if the shafts rotate in opposite directions, and on the inner side if they rotate in the same direction. In other words, rotation is transmitted via the frictional force at the point of rolling contact. However, some slip is unavoidable, making reliable power transmission difficult to achieve. To transmit greater power, larger contact forces are required, which in turn lead to high bearing loads. For these reasons, this arrangement is unsuitable for handling substantial power. Consequently, an alternative approach was developed: machining appropriate tooth profiles at equal intervals along the cylindrical rolling surfaces, ensuring that at least one or more pairs of teeth remain in continuous contact. By engaging the driving shaft’s teeth with those of the driven shaft, robust power transmission can be assured. Such gears are called cylindrical gears, and the reference cylinder from which the teeth are cut is known as the pitch cylinder. Spur gears are a type of cylindrical gear.

　　When two shafts are neither parallel nor intersecting, a true rolling contact surface does not exist. Depending on the gear type, teeth are generated on a pair of reference mating rotating surfaces. In all cases, it is necessary to define the tooth profile so that the relative motion of the contacting tooth surfaces matches the relative motion of the teeth engaging on the reference surfaces.

　　When gears are treated as rigid bodies, to ensure that, at the point of tooth contact, they maintain a constant angular velocity ratio without colliding or separating, the common normal component of the relative velocity at the contact point must be equal for both gears. In other words, at the contact point, there is no relative motion along the common normal direction; relative motion occurs only tangentially to the contact surface. This tangential motion is known as sliding. Except at special points, gear surfaces invariably involve what is referred to as sliding-contact transmission.

　　To ensure that the tooth profile satisfies the aforementioned conditions, the desired tooth form can be obtained by employing the envelope method, which is a general-purpose approach.

　　Now, designate one side of gear A’s surface as the curved surface FA, and assign the two gears their specified relative rotation. Next, on the coordinate system attached to gear B, plot a set of successive positions of gear A’s surface FA. Consider the envelope of this set of curves and use it as the tooth surface FB of gear B. From the theory of enveloping surfaces, it is readily apparent that the two gear surfaces maintain constant line contact, and the two gears will exhibit the desired relative motion.

　　The tooth profile can also be derived as follows. In addition to a pair of gears A and B with specified relative motion, consider a third hypothetical gear C that meshes with both A and B, assigning it an arbitrary tooth surface FC—comprising only a curved surface without a solid tooth body—and an appropriate relative motion.