# A Unified Theory for 3D Gear and Thread Metrology
## 3 3D Model of Helical Machine Elements
Especially in gear metrology, where the transverse section of the flanks is often considered, the coordinate \(z\) is used rather than \(\psi\). Since  
\[\psi=\mathtt{hand}\cdot c\bigg{(}z+\mathtt{flank}\cdot m\sqrt{r^{2}-r_{b}^{2}} \bigg{)} \tag{8}\]  
the flanks are then given by  
\[S_{\mathcal{P}}(r,z)= \begin{pmatrix}r\cdot\cos(\varphi_{b}+\mathtt{hand}\cdot cz +\mathtt{flank}\cdot\hat{\varphi}_{\mathcal{P}}(r))\\ r\cdot\sin(\varphi_{b}+\mathtt{hand}\cdot cz+\mathtt{flank}\cdot\hat{\varphi}_{ \mathcal{P}}(r))\\ z\end{pmatrix} , \tag{9}\]  
where we introduced the transverse profile function  
\[\hat{\varphi}_{\mathcal{P}}(r)=\mathtt{hand}\cdot cm\sqrt{r^{2}-r_{b}^{2}}- \arccos\Big{(}\frac{r_{b}}{r}\Big{)}. \tag{10}\]  
Equation (9) describes the surface of a flank as the screw motion of a curve on the \(xy\)-plane. While for generalized ruled helicoids the function \(\hat{\varphi}_{\mathcal{P}}\) is of the form given in Equation (10), one might of course consider more general functions \(\vartheta\) leading to other surfaces. Objects like gears or threads are constructed of left and right flanks for several teeth (or starts). In the case of gears, the definition  
\[\mathtt{flank}= \begin{cases}-1&:\mathtt{left}\ \mathtt{flank}\\ +1&:\mathtt{right}\ \mathtt{flank}\end{cases} \]  
leads to results for the flanks which are consistent with the definitions in [20]. The same convention will be used in this paper in the case of screw threads. The construction of a thread from the two flanks is depicted in Figure 4. Note, however, that for other objects this definition of flank might not be suitable. If \(N\) is the number of teeth, we denote by  
\[\mathcal{P}_{i}^{\text{ri}}=(\mathtt{hand}_{i}^{\text{ri}},1,r_{b,i}^{\text{ri}},m _{i}^{\text{ri}},c_{i}^{\text{ri}},\varphi_{b,i}^{\text{ie}})\qquad(i=1,\dots,N) \tag{11}\]  
the parameters of the right flanks, and by  
\[\mathcal{P}_{i}^{\text{le}}=(\mathtt{hand}_{i}^{\text{le}},-1,r_{b,i}^{\text{ le}},m_{i}^{\text{le}},c_{i}^{\text{le}},\varphi_{b,i}^{\text{le}})\qquad(i=1,\dots,N) \tag{12}\]  
those of the left flanks. The corresponding flanks are denoted by \(S_{i}^{\text{ri}}\) and \(S_{i}^{\text{le}}\), respectively. For the nominal object, the values of \(\mathtt{hand}\) and \(r_{b}\) are typically the same for the left and right flanks and for all teeth. While there might be different values of \(m^{\text{ri}}\), \(c^{\text{ri}}\) and \(m^{\text{le}}\), \(c^{\text{le}}\) for the left and right flanks, these values are still the same for all teeth. The nominal values for \(\varphi_{b}\) satisfy  
\[\varphi_{b,i}^{\text{ri}}=\varphi_{b,1}^{\text{ri}}-\frac{2\pi}{N}(i-1)\ \text{mod}\ 2\pi\qquad(i=1,\dots,N), \tag{13}\]  
and accordingly for the left flanks. The values \(\varphi_{b,i}^{\text{le}}-\varphi_{b,i}^{\text{ri}}\) are specified by the desired position of the right and left flanks relative to each other. This can be defined via a reference circle in the \(xy\)-plane with a suitable radius \(r_{0}>r_{b}\) with the condition that the arc length of the intersection of the reference circle with a tooth is \(\frac{\pi}{N}r_{0}\). From Equation (10) it follows that this implies  
\[\varphi_{b,i}^{\text{le}}-\varphi_{b,i}^{\text{ri}}=\frac{\pi}{N}+\mathtt{ hand}\cdot(c^{\text{le}}m^{\text{le}}+c^{\text{ri}}m^{\text{ri}})\sqrt{r_{0}^{2}-r_{b} ^{2}}-2\cdot\arccos\left(\frac{r_{b}}{r_{0}}\right)\ \text{mod}\ 2\pi. \tag{14}\]  
Figure 4: Screw thread definitions.