# A Unified Theory for 3D Gear and Thread Metrology
## 2 State of the Art in Gear and Thread Metrology
### Classical Involute Gear Metrology  
Classical gear metrology is based on the mechanical measuring devices established prior to the advent of coordinate measuring technology (CMT) in the 1970s, a technology that kinematically imitated the gear's rolling process. Although CMMs have been established in gear metrology for several decades now, the principles behind the specific measurement and evaluation strategies do not follow the fundamental ideas pursued in other CMT applications. The major issue in state-of-the-art gear metrology is that each single measurement line is evaluated in an individual reference coordinate system. As a consequence, correlations between the different measurands cannot be detected and errors in the manufacturing process may not be identified properly.  
There is a strict separation of the three main gear inspection categories: profile deviations, helix deviations, and pitch deviations (Figure 1). This categorization can be considered a remnant from a past era when fully equipped gear inspection labs needed at least three separate mechanical machines: an involute (profile) checker, a lead (helix) checker, and a pitch checker.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Gear** & **Thread** \\ \hline Number of teeth \(z\) & Number of starts \(n\) \\ Reference diameter \(d\) & Pitch diameter \(d_{2}\) (\(D_{2}\)) \\ Pressure angle \(\alpha_{t}\) & Flank angles \(\beta_{1}\), \(\beta_{2}\) \\ Helix angle \(\beta\) & Lead angle \(\varphi\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Equivalents of the most relevant gear and thread determinants. The names and symbols are in accordance with ISO 21771:2007 and ISO 5408:2009, respectively [6; 24].  
The earliest mechanical generative involute measuring devices were designed to reproduce the base circle principle as the essential law in gear kinematics. They embodied a simple disc of the exact base circle diameter corresponding to the gear under test, and this disc was mounted to the spindle of the instrument. While the disc rotated around the spindle, a tangent ruler was pressed against it and slid along the gear profile. A measuring probe attached to this ruler would then directly record the deviations from the perfect involute, responding to either manufacturing errors or to misalignment of the gear against the disc. These deviations were written in real-time to a strip of paper moving in sync with the spindle's rotation.  
The mechanical lead checker followed the same approach. It used simple rotational and translational motions to rotate the gear and move the measuring probe in a way that emulated the exact lead of the part. Again, the probe directly measured the deviations from the nominal helix and wrote them onto a piece of moving chart paper.  
For pitch measurements, a dedicated indexing machine was needed.  
The resulting charts, which resembled seismographs or electrocardiograms, then needed to be read and interpreted. Since both the measurement and the output were completely analog, no numerical values were generated that could be used for calculations. Instead, the diagrams had to be evaluated by skilled technicians using rulers and triangles to determine slope and form deviations [26].  
When gear metrology made the move to CNC machines like universal CMMs and specialized gear measuring instruments in the late 1970s, computations and the evaluation of standardized parameters were significantly facilitated. Despite this, the separation of measurement routines and determinants into a line-based profile and helix inspection on the one hand and pointwise pitch evaluation on the other has been preserved until today. Although only one machine is now needed for all three inspection tasks, the analysis of the results still does not allow the holistic description of manufacturing errors in one common coordinate system.