# Tooth surface reconstruction of logarithmic spiral bevel gear

Tooth surface reconstruction is the core part of the whole reverse engineering technology. The basic idea of logarithmic spiral bevel gear surface reconstruction is to construct a mathematical model by interpolation according to the given type value points in topological matrix array. The basic process is as follows: firstly, the data dot matrix PI, J, (I = 0,1,2…, R; J = 0,1,2…, s) of the topological rectangular array on the tooth surface of the real logarithmic spiral bevel gear is given by measurement, and the adjacent two points of each row of data points are connected by straight line segments to form a topological rectangular grid (as shown in Figure 1), The tooth surface of the gear is first fitted with the logarithmic curve of the tooth surface of the gear skeleton, and then the tooth surface is obtained by taking the logarithmic curve of the tooth surface as the tooth surface.

In the process of constructing interpolation surface, two parameter segmentation must be given first Δ U and Δ v. Make each data point establish one-to-one correspondence with the points in the parameter field on the U V parameter plane. According to the specific situation of the real tooth surface of the constructed logarithmic spiral bevel gear, the method and principle of tooth surface parameterization and boundary condition selection of logarithmic spiral bevel gear are determined through analysis. The existing tooth surface curve of logarithmic spiral bevel gear is selected to construct a k-curve × L-degree B-spline surface interpolates the given data points. The process of finding the tooth surface of real logarithmic spiral bevel gear is the same as that of finding the tooth surface curve. It also needs to solve a linear equation group of unknown control vertices first, but this linear equation group is too complex to solve by computer. Through analysis, the problem of finding control points on the tooth surface of logarithmic spiral bevel gear can be expressed as a two-stage curve inverse calculation problem, so the B-spline interpolation surface equation to be solved can be written as follows:

If a parameter value V is fixed, M + k points on these control curves are given. These points are also used as control vertices to define the isoparametric lines with u as the parameter on the tooth surface of logarithmic spiral bevel gear. When the parameter V value sweeps through its entire definition domain, an infinite number of isoparametric lines describe the whole tooth surface of logarithmic spiral bevel gear (as shown in Figure 2).

According to the characteristics and data processing of the tooth surface of logarithmic spiral bevel gear, a new profile control vertex close to the surface shape is obtained. Using the powerful surface construction function of Imageware software, read the data blocks after data processing, and generate multiple NURBS section curves interpolated at the type value points (u-direction and v-direction section lines are generated respectively). In order to ensure the continuity and closure of the generated surface, a closed NURBS boundary curve is constructed, which is interpolated at the boundary control points in U direction and V direction. As shown in the curve of u-direction and the curve of v-direction on the basis of NURBS, the curve of u-direction and the curve of v-direction are constructed on the basis of NURBS.

Comparing the reconstructed logarithmic spiral bevel gear tooth surface with the original point cloud data, the error result of the reconstructed logarithmic spiral bevel gear tooth surface can be obtained, as shown in Figure 4.

In order to ensure the minimum error between the tooth surface of the constructed logarithmic spiral bevel gear and the point cloud, the system parameters need to be continuously adjusted in the construction process, so as to maximize the surface fitting quality. In order to optimize the calculation time, only the following points contained in the surface are used for calculation: points in the maximum positive direction, points in the minimum positive direction, points in the maximum negative direction, points in the minimum negative direction and average distance points. The results detected by the software are shown in Figure 4, and the error analysis results are shown in Figure 4 Txt file. Through the analysis results, it can be known that the maximum geometric error of the reconstructed logarithmic spiral bevel gear tooth surface is 0.1533mm, which is less than the standard value of 0.5mm, the average error is 0.1mm, there is no transverse error, all within the error range, and meet the accuracy requirements of general curved surface.

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