# A Unified Theory for 3D Gear and Thread Metrology
## 4 Relations to Standardized Measurands
### Thread Measurements  
For threads, the parameter \(r_{b}\) is fixed by \(r_{b}=0\), so this parameter is not related to a measurand. The flank angles \(\gamma_{i}^{\text{ri}}\) and \(\gamma_{i}^{\text{le}}\) for tooth \(i\) can be obtained from \(m_{i}^{\text{ri}}\) and \(m_{i}^{\text{le}}\) by  
\[\gamma_{i}^{\text{ri}}=-\arctan(m_{i}^{\text{ri}})\quad\text{and}\quad\gamma_ {i}^{\text{le}}=\arctan(m_{i}^{\text{le}}). \tag{30}\]  
The flank angles \(\beta_{1,i}\) and \(\beta_{2,i}\) for gap \(i\) as defined in ISO 5408:2009 [6] are then obtained according to the following table: \begin{tabular}{l c c} \hline \hline hand & \(1\) & \(-1\) \\ \hline \(\beta_{1,i}\) & \(|\gamma_{i}^{\text{le}}|\) & \(|\gamma_{i-1}^{\text{ri}}|\) \\ \(\beta_{2,i}\) & \(|\gamma_{i-1}^{\text{ri}}|\) & \(|\gamma_{i}^{\text{le}}|\) \\ \hline \end{tabular}   
Here and in the following formulas in this section, a flank index value of \(0\) is understood as \(N\). The thread angle for gap \(i\) is calculated with  
\[\alpha_{i}=\texttt{hand}\cdot(\gamma_{i}^{\text{le}}-\gamma_{i-1}^{\text{ri}}). \tag{31}\]  
The lead of a single right or left flank is further given by  
\[Ph_{i}^{\text{ri},\text{le}}=\frac{2\pi}{c_{i}^{\text{ri},\text{le}}}. \tag{32}\]  
The pitch diameter for gap \(i\) in the transverse section at position \(z\) is the diameter where the transverse gap width of gap \(i\) takes the value \(d_{0,i}(z)\frac{\pi}{2N}\). It is given by  
\[d_{2,i}(z)=d_{0,i}(z)=2\cdot\frac{\varphi_{b,i}^{\text{le}}-\varphi_{b,i-1}^{ \text{ri}}+2\pi k_{i}+\texttt{hand}\cdot(c_{i}^{\text{le}}-c_{i-1}^{\text{ri} })z+\pi+\frac{\pi}{N}}{\texttt{hand}\cdot(c_{i}^{\text{le}}m_{i}^{\text{le}}+c _{i-1}^{\text{ri}}m_{i-1}^{\text{ri}})}, \tag{33}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Measurement Category** & **Free Parameters** & **Parameters Set to Nominal Values** \\ \hline Profile evaluation & \(r_{b},\varphi_{b}\) & \(\texttt{hand},\texttt{flank},c\) \\ Helix evaluation & \(\texttt{hand},c,\varphi_{b}\) & \(\texttt{flank},r_{b}\), \\ Pitch evaluation & \(\varphi_{b}\) & \(\texttt{hand},\texttt{flank},r_{b},c\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: As classical gear metrology treats the three measurement categories (Figure 1) in different reference systems, the number of free geometry parameters should be reduced when the emphasis is on comparability. where \(k_{i}\in\mathbb{Z}\) must be chosen such that \(d_{2,i}(0)\) is in a reasonable range with respect to the nominal value. These \(z\)-dependent diameters form a cone with taper angle  
\[\omega_{i}=\arctan\left(\frac{c_{i-1}^{\text{ri}}-c_{i}^{\text{le}}}{c_{i}^{ \text{le}}m_{i}^{\text{le}}+c_{i-1}^{\text{ri}}m_{i-1}^{\text{ri}}}\right). \tag{34}\]  
Furthermore, the two-flank lead of gap \(i\) is given by  
\[Ph_{2,i}=2\pi\frac{c_{i}^{\text{le}}m_{i}^{\text{le}}+c_{i-1}^{\text{ri}}m_{i- 1}^{\text{ri}}}{c_{i}^{\text{le}}c_{i-1}^{\text{ri}}(m_{i}^{\text{le}}+m_{i-1} ^{\text{ri}})}. \tag{35}\]  
For cylindrical screw threads, the pitch \(P_{i}\) between the flanks \(i-1\) and \(i\) (right or left) measured at the nominal pitch radius \(r_{0,\text{nom}}\) is obtained with  
\[P_{i}(z)=2\cdot\frac{\zeta_{i}+\texttt{hand\cdot flank}(c_{i-1}m_{i-1}-c_{i}m _{i})r_{0,\text{nom}}+\texttt{hand}(c_{i-1}-c_{i})z}{c_{i}+c_{i-1}} \tag{36}\]  
with \(\zeta_{i}=\varphi_{b,i-1}-\varphi_{b,i}+2\pi k_{i}\), where \(k_{i}\in\mathbb{Z}\) must be chosen such that \(P_{i}(0)\) is in a reasonable range with respect to the nominal value. The position \(z\) is the \(z\)-coordinate of the center of the two radially shifted points on the cylinder with radius \(r_{0,\text{nom}}\) between which the pitch is measured. The general model derived in Section 3 constructs the screw thread geometry in the transverse section rather than in the axial section as is done in classical thread metrology.