The tooth surface of helical offset non orthogonal gear is formed by the tooth surface of helical gear shaper cutter. In Figure, SS rotates clockwise around axis ZS φ S gets SG, SG translates along the negative direction of axis ZG, l0 gets SM, SM translates along the positive direction of axis XM, e gets Sn, Sn rotates clockwise around axis xn γ M gets SP, which rotates counterclockwise around the axis ZP φ 2 get S2.
From the above coordinate transformation, the coordinate transformation matrix M2S from the spiral gear shaper coordinate system Ss to the face gear coordinate system S2 can be obtained, as shown in the formula:
According to the transformation relationship of the coordinate system, in the face gear coordinate system S2, the tooth surface equation and normal vector N2 of the face gear can be expressed by the formula as follows:
