（1） Undercutting theory of helical gears
When the face gear engages with the tool, the speed of the former at the contact point is zero. At this time, undercutting phenomenon will occur on its tooth surface, and undercutting boundary line will appear on the tool tooth surface. In order to find out the minimum inner radius of helical gear without undercutting at the tooth root, the intersection of undercutting limit on the tooth surface of the tool and the top circle of the tool is used θ S value. Then there are:
The conditions under which undercutting does not occur on the tooth surface of helical gear during generation are as follows:
Expand the above formula to obtain the parameters ϕ s 、 θ S and ξ Relationship between S:
The nonlinear equations are solved by successive approximation method, and the parameter s is solved ϕ And S ξ Value of. The value is substituted into the tooth surface equation of face gear
According to the corresponding coordinates x * 2 and y * 2, the minimum inner radius of the face gear without undercutting during machining is obtained:
（2） Theory of tooth tip sharpening of helical gear
The intersection of the tooth surfaces on both sides of the helical gear tooth and the thickness of the tooth top is zero, which is an important geometric feature of the sharpening of the tooth top of the helical gear. Now suppose that the cusp of the helical gear is located on the two tooth surfaces of a tooth and on the X2 axis, and the distance from this point to the coordinate origin o is L2, as shown in the figure.
By substituting the assumptions of the above formula into the surface gear equation, the maximum outer radius of the helical gear with constant tip can be obtained:
（3） Formation of transition surface of helical gear
The tooth root transition surface of helical gear is formed by cutting the tooth top line of the tool, that is, the tooth top line of the tool is transformed into the coordinate system of helical gear through the method of coordinate transformation, which is a part of the curve obtained. In the coordinate system S2 of helical gear, the basic equation of the transition surface of helical gear is obtained as follows:
By substituting the formula and meshing equation into the basic equation of the transition surface, the component expression of the transition surface can be obtained.