# A Novel Measurement Standard for Surface Roughness on Involute Gears
## 2 Methods for the Analysis of Surface Roughness on Involutes
### Coordinate Systems Involved  
The modular design of PMS facilitates measurements on both gear measuring systems and classical profilometers. While the former directly uses gear coordinates and mainly _roll length \(l\)_ as their reference system, the latter detects the profile of a probe by using _Cartesian coordinates_, in this case on the surface of a tooth. Roughness parameters, however, are defined by assuming planar surfaces, thus using Cartesian coordinates on the surface of the workpiece. The surface is considered as a skin that is unwrapped from a geometrical element. In the case of a gear tooth, the involute shape is removed by delivering the positions along the _arc of the tooth_ in a two-dimensional Cartesian coordinate system. The lateral coordinate values lie on the arc of the fitted involute, and the vertical coordinate  
Figure 1: Left flank involute (blue) defined by the base circle radius \(r_{\text{b}}\) and parameterised by the roll angle \(\xi\) as defined in Equation (1). values represent the deviations of the real workpiece surface normal to the fitted involute. These point pairs represent the so-called primary profile. While a gear measuring machine delivers roll angles or roll lengths and the full geometry of the workpiece, a profilometer has no information on the geometry of the gear and, therefore, no access to the involute shape. The profilometrically obtained topography data do not provide information on the actual involute arc geometry and represent the profile points in a Cartesian system with its lateral axis lying parallel to some secant of the arc with its absolute position relative to the involute remaining unknown. This means that at least four different reference systems have to be considered when dealing with the different methods of measuring and analysing the profile of this gear-roughness standard. Those reference systems are illustrated in Figure 2 via an artificial texture as sinusoidal profile to illustrate the non-linearity of the transformation between the coordinate system of roll angles and chord positions; they are defined as follows:  
1. The Cartesian system \(K_{\text{cart,ideal}}\) (Figure 2a) in which the ideal involute is defined following Equation (1);
2. The Cartesian system \(K_{\text{cart,profilometer}}\) (Figure 2b) in which the profilometer measures the surface points;
3. The surface system \(K_{\text{arc}}\) (Figure 2c) using the arc length \(s\) (actual definition of roughness properties);
4. The gear system \(K_{\text{rolllength}}\) (Figure 2d) using roll length \(l\) (and the proportional roll angle \(\xi\)) (gear measurement systems). One deviation profile will appear differently when displayed (or measured) in different reference systems, as shown in Figure 2a,d, with an artificial profile. A profile that appears as a regular sine with respect to the arc length (Figure 2d, \(K_{\text{arc}}\)) also includes the shape of the involute, when displayed in Cartesian coordinates including the involute shape (Figure 2a, \(K_{\text{cart,ideal}}\) and Figure 2b, \(K_{\text{cart,profilometer}}\)). Using roll length \(l=r_{\text{b}}\cdot\xi\) as the reference length (\(K_{\text{rolllength}}\)) shows the same profile stretched and compressed in the \(x\)-axis, as shown in Figure 2d. Most measurements in gear metrology are described in reference to the roll length \(l\). Hence, the distinction between \(l\) and the arc length \(s\) marks the gap between the gear metrology sector and the classical roughness measurement sector. This means that it is of utmost importance to always ensure that measurements and evaluations are performed by using the adequate coordinate system. To be able to correctly evaluate and compare the measured data, one first needs to be able to transform the data between these reference systems. The relevant transformations are presented subsequently.