For the non-circular gear, assuming that the radius of the imaginary flat top gear is equal to the spherical radius of the non-circular gear, its center position o is coincident with the bevel gear and is also fixed. The position OC of the disc milling cutter will rotate with the rotation of the imaginary flat top gear, as shown in figure 1, and the instantaneous position can be obtained as shown in the formula.

Where, (x, y, z) is the position of the imaginary flat top gear center (wheel blank Center), and M is the coordinate transformation from the imaginary flat top gear center O to the disc milling cutter OC. The starting position is shown in the formula, and the parameters are shown in the figure

Where V, h and T are the distances of OC relative to o point in three directions; RC is the maximum radius of cutter head; β A is the large end spiral angle. L = R0, R0 is the spherical radius of the large end of the bevel gear; δ is the pitch cone angle; γ is the root cone angle; and θ I is the rotation angle around the shaft at any position I of the cut gear. When cutting a tooth, from the big end to the small end, the tooth must be in the horizontal line position. Starting from the starting position of the wheel blank, with the wheel blank turning the tiny angle Δ φ (the maximum angle of the jth tooth is φ J), the imaginary flat top gear also turns the corresponding micro angle Δ Φ (the maximum angle of the jth tooth is Φ J), and φ and Ф are obtained by the formula.

The distance between the initial position of the milling cutter head and the center of the imaginary flat top gear is expressed by s, and the angle is represented by Ф 0

If the wheel blank rotates at a constant speed and the rotation angle is φ, the corresponding flat top gear angle is Ф. The relationship between the two is shown in the formula. The subscripts 1 and 2 represent two gears (the cut gear and the imaginary flat top gear).

The instantaneous position of milling cutter head is shown in the formula. It can be seen that according to the curve of large end section, the position coordinates of milling cutter head can be obtained.