In the process of transmission, the contact line of the line gear is formed by the motion track of the meshing point of the line gear, and the contact line of the line gear is a spatial curve. A section of curve on the normal plane of any meshing point on the contact line is taken as the generatrix, and then the generatrix is scanned along the contact line to generate the line tooth surface of the line gear. The contact line of a cylindrical gear is a cylindrical helix. A section of arc on the normal plane at any meshing point of the cylindrical helix is taken as the generatrix, and then the line tooth surface of the cylindrical gear can be obtained by scanning the arc generatrix along the cylindrical helix.

Establish the tooth surface model of cylindrical gear, as shown in Figure 1. In Figure 1: OM is the initial meshing point of the contact line; I → is the unit tangent of the contact line at Om; J → is the unit principal normal of the contact line at Om; K → is the unit sub normal of the contact line at OM. On the normal plane at the point OM, the arc generatrix is constructed with the point o as the center and R = oom as the radius; the angle between the vector oom → and K → is the modification angle of the linear gear tooth profile. The circular arc generatrix can be scanned along the contact line to get the single side tooth surface of cylindrical gear.

Establish cylindrical gear coordinate system, as shown in Figure 2. In Figure 2, the coordinate system o1-x1y1z1 is fixedly connected with the cylindrical gear, with the axis of the cylindrical gear as the Z1 axis and the lower end face of the cylindrical gear as the x1o1y1 plane. In the coordinate system o1-x1y1z1, the contact line of cylindrical gear is a cylindrical helix

Where: m is the helix radius of the contact line; n is the pitch of the contact line; t is the meshing point parameter of the contact line, TS ≤ t ≤ te, TS is the meshing point parameter t, Te is the meshing point parameter t; Xc, YC and ZC are the coordinate values of the contact line in the coordinate system o1-x1y1z1.

The space rectangular coordinate system om xmymzm is established by taking the meshing point om of the contact line when the meshing point parameter t = 0 in the formula as the coordinate origin (see Figure 2). Where: XM axis is the negative direction of the unit principal normal vector J → at the contact line at the point om; YM axis is the positive direction of the unit tangent vector I → at the contact line at the point om; ZM axis is the positive direction of the unit pair normal vector k → at the contact line at the point om; plane xmomzm is the normal plane of the contact line at the meshing point OM. Taking the point om as the coordinate origin, the normal plane coordinate system of cylindrical gear is established, as shown in Figure 3. In Figure 3: point O is the center of the arc generatrix; point m is any point on the arc generatrix; the vector om → rotates counterclockwise around point O is the positive direction; u is the angle between the vector om → and the positive direction of XM axis.

In the coordinate system om xmymzm shown in Fig. 2, the parameter equation of circular arc generatrix on the normal plane of cylindrical gear tooth at point OM is as follows:

Where: R is the radius of the arc generatrix on the normal plane of the meshing point; φ is the tooth profile correction angle of the linear gear; u is the angle between the vector om → and the positive direction of the XM axis, UMIN ≤ u ≤ Umax; XM, YM and ZM are the coordinate values of the arc generatrix in the coordinate system om xmymzm.

Through the formula and the coordinate transformation matrix m1m from the coordinate system om xmymzm to the coordinate system o1-x1y1z1, the parameter equation of the circular arc generatrix on the normal plane at the point OM in the coordinate system o1-x1y1z1 can be obtained

Where: arctan (M / N) is the helix angle of contact line of cylindrical gear; XM ‘, YM’ and ZM ‘are the coordinate values of arc generatrix in the coordinate system o1-x1y1z1.

The tooth surface can be obtained by scanning the arc generatrix on the normal plane of the meshing point along the contact line

Where: x1, Y1 and Z1 are the coordinate values of cylindrical gear tooth surface in the coordinate system o1-x1y1z1.