# A Numerical Gear Rolling Test Method for Face-Hobbed Hypoid Gears
## 3 Actual Tooth Surface Fitting  
The non-uniform rational B-spline (NURBS) method was adapted to fit the actual tooth surface. The mathematical equations of the NURBS surface are [17]  
\[P(u,v)=\frac{R(u,v)}{W(u,v)} \tag{1}\]  
\[R(u,v)=\sum_{i=0}^{m}\sum_{j=0}^{n}w_{i,j}d_{i,j}N_{i,k}(u)N_{j,l}(v) \tag{2}\]  
\[W(u,v)=\sum_{i=0}^{m}\sum_{j=0}^{n}w_{i,j}N_{i,k}(u)N_{j,l}(v) \tag{3}\]  
where \(u\) and \(v\) are surface parameters, \(w_{i,j}\) is the weight factor, \(d_{i,j}\) is the control vertex, \(N_{i,k}(u)\) is the canonical B-spline basis function of \(k\) times in the u direction, and \(N_{j,l}(v)\) is the canonical B-spline basis function of 1 times in the \(v\) direction.  
The fitting process is as follows.  
First, the curve fits along the \(u\) direction. The 17 \(\times\) 15 control vertex d and weighting factor \(w\) are calculated back in the \(u\) direction [18]. Then, the curve fits along the \(v\) direction. The 17 \(\times\) 15 control vertex \(d\) and weight factor \(w\) obtained in the previous step are taken as new measuring points to calculate the 17 \(\times\) 17 control vertex \(d\) and weight factor \(w\) in the \(v\) direction in the same method. Finally, the obtained 17 \(\times\) 17 control vertex \(d\) and weight factor \(w\) are introduced into Equation (1), and the NURBS surface equation of the actual tooth surface can be obtained.  
Based on the above method, a hypoid gear was taken as an example (see reference [19] for detailed parameters) to fit the 15 \(\times\) 15 data points on the theoretical tooth surface, the concave and convex of both pinion and gear can be obtained, as shown in Figures 2-5.  
The fitting tooth surface obtained by NURBS coincides with the actual tooth surface only at the measuring point, and the positions that are not at the measuring points are obtained by fitting algorithms. Therefore, it is necessary to analyze whether these points can represent the points on the actual tooth surface. That is, the fitting accuracy of these areas needs to be verified. The forward algorithm of NURBS surface fitting was adopted in this work to calculate the 14 \(\times\) 14 coordinates (\(x,y,z\)) of the center point [18], the verification points shown in Figure 6.  
Figure 2: Pinion concave: NURBS surface.  
Figure 4: Pinion convex: NURBS surface.  
Figure 5: Gear concave: NURBS surface.  
Figure 3: Gear convex: NURBS surface.  
The X and Y of the center point on the theoretical tooth surface are equal to the X and Y of the point on the NURBS surface, and the Z can be obtained on the theoretical tooth surface. The distance between the two center points was calculated, and the maximum distance between the center points of each column was used to draw the curve shown in Figure 7. These errors are the Z direction distances between the center of each surface piece of the original tooth surface and that of the fitting tooth surface.  
It can be seen from Figure 7 that the fitting errors of the pinion and gear do not exceed 0.1 \(\mu\)m, which can meet the engineering requirements [20].  
Figure 6: Grid center point obtained using forward algorithm.  
Figure 7: Curve of fitting errors.