# Reconstruction of tooth surface curve of logarithmic spiral bevel gear

Curve is the basis of building surface. In reverse engineering, the commonly used curve reconstruction method is to interpolate the data points into spline curve, and then use modeling tools to complete the modeling and editing of surface.

In the tooth surface curve reconstruction of logarithmic spiral bevel gear, the collected data points are preprocessed first. The mathematical model of logarithmic spiral bevel gear tooth surface established by these data points can truly restore its existing surface shape, that is, to determine the parameter equation of the surface, the parameter change law of the surface must be determined by these data points, that is, parameterization. The basis function equation of the curve is brought into by the nodes calculated by the parameter method, and the control points of the curve are calculated by the programming software, and finally the interpolation process of the curve is completed. When interpolating a batch of original data points, the adjacent two points of each row of data points are connected with straight lines to form a rectangular grid in the sense of topology.

Select a tooth surface of logarithmic spiral bevel gear as the research object, and obtain the existing data point Pi of logarithmic spiral bevel gear tooth surface in Imageware software, (I = 0,1,… N). When constructing cubic B-spline interpolation curve, the first and last data points shall be consistent with the first and last endpoints of the spline curve, then the cubic B-spline interpolation curve of the data points on the tooth surface of logarithmic spiral bevel gear shall have n + 3 control vertices, and the node vector shall be u = [U0, U1,… UN + 6], The process of determining the curve definition domain as u ∈ [U3, UN + 3] nodes is actually the process of parameterizing data points. The cubic B-spline interpolation curve equation can be written as:

Replace the node values in the curve definition domain u ∈ [U3, UN + 3] into the curve equation in turn, which shall meet the interpolation conditions, namely:

According to the above equation, the linear equations of N control points of the tooth surface curve of logarithmic spiral bevel gear are as follows:

In order to make the tooth surface curve of logarithmic spiral bevel gear more smooth at the end, the tangent vector boundary condition is used to limit the curve. Since the first and last data points of the curve coincide with the first and last control points, that is, d0 = P0, DN + 2 = PN, and the coincidence degree r = 3, there is an additional equation:

After the above boundary condition processing and algorithm processing of the above linear equations, the following linear equations can be obtained:

In the process of reconstructing the tooth surface curve of logarithmic spiral bevel gear, the curves in the transverse and longitudinal directions should be interpolated respectively to ensure that the curve passes through the type value point. In the process of interpolation, when the type value point of the transverse section interpolation curve approaches or passes through the corresponding longitudinal section interpolation curve, the type value point is effective. If the error is large, Repair this type of value point. By judging in turn, we can get all the shape value points of the new construction surface, and also get the shape value vertices of the new surface shape. The curve obtained by interpolation of data points on the tooth surface of logarithmic spiral bevel gear is shown in the figure.

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