# A Novel Measurement Standard for Surface Roughness on Involute Gears
## 2 Methods for the Analysis of Surface Roughness on Involutes
### Coordinate Systems Involved
#### 2.2.3 \(K_{\text{cart,profilometer}}\to K_{\text{cart,ideal}}\)  
It is assumed that the measurement on the profilometer has taken place sufficiently perpendicular to the gear axis. Therefore, both reference systems are located in the same 3D-plane. This reduces the transformation to a 2D-problem.  
The Cartesian coordinate systems \(K_{\text{cart,profilometer}}\) and \(K_{\text{cart,ideal}}\) only differ in position and orientation to one another. This means that the transformation between these coordinate systems can be performed by using one translation via a two-dimensional vector:  
\[v_{\text{trans}}=\left(\begin{array}{c}v_{x}\\ v_{y}\end{array}\right) \tag{5}\]  
and one rotation around an angle \(\varphi\) via the following operator.  
\[\hat{M}=\left(\begin{array}{cc}\cos(\varphi)&\sin(\varphi)\\ -\sin(\varphi)&\cos(\varphi)\end{array}\right). \tag{6}\]  
This, however, requires the knowledge of \(v_{\text{trans}}\) and \(\varphi\). In practice, the transformation can be implemented with different methods depending on the definition of \(v_{\text{trans}}\) and \(\varphi\). As visualised in Figure 3, one defines \(\varphi\) as the angle between the line through the first and last point of the measurement data \(g_{\text{profilometer}}\) and through the first and last point of the fitted data \(g_{\text{ideal}}\). The vector \(v_{\text{trans}}\) is defined as the vector from the first data point \(Z_{\text{profilometer,1}}\) in the \(K_{\text{cart,profilometer}}\)-system to the corresponding point \(Z_{\text{ideal,1}}\) in the \(K_{\text{cart,ideal}}\)-system. The transformation is then executed by rotating and shifting the data points accordingly. The measurement on a profilometer, however, does not provide any direct information about the position of the measured data relative to the ideal involute.  
Figure 3: Principle of the fitting process: The profile as measured by the profilometer (red) is fitted onto the ideal involute (black), yielding the measured profile in the coordinate system of the ideal involute (blue). This is performed by finding the shift-and-turn transformation that yields the minimal squared error of the transformed curve to the ideal involute via an optimisation algorithm.  
That is why an optimisation algorithm (fit) is implemented to find the optimal transformation parameters \(v_{\text{trans}}\) and \(\varphi\). The optimum is defined as the transformation that yields the transformed profile \(Z_{\text{ideal}}\) with the least mean square error to the ideal involute \(I_{\text{ideal},\downarrow}\). In other words, the profilometer profile \(Z_{\text{profilometer}}\) is shifted and turned until it matches the ideal involute best.  
In order to stabilize the optimisation procedure, choosing reasonable starting values for the transformation parameters will be helpful, e.g., choosing a vector that shifts the first point of \(Z_{\text{profilometer},1}\) to the first point of the ideal involute \(I_{\text{ideal},1}\).  
Alternative form fitting methods are available and are described in [15].