## 3 Additional Observation  
_Module as a Random Variable in the Population_  
During the research, a puzzling fact for the authors was whether the reproduced value of the module should be treated as a random variable. Is it not the case that the tested modules are a sample from the population that can reproduce the values of the nominal modules? The authors tried to answer the question of the relationship between the original gear wheel with a non-standard module and the reconstructed gear wheel with a different, but still non-standard, module, both very similar in value. It is important that we are talking about nominal geometries here without taking into account real and observed dimensions. We are only talking about the tolerable dimensions according to ISO and the generally tolerable dimensions included in the technical drawing of the original gear wheel and the recreated gear wheel. The authors wanted to answer whether it is necessary at all (and if so, to what extent) to get as close as possible to the original value of the nominal gear module when reconstructing the geometry of this gear. It was found that for two wheels to be the same, they must have identical tooth thickness at any diameter. The base diameter of both wheels must be the same. Taking the above into account, after several transformations, an equation with two unknowns was finally obtained, allowing the selection of a pair of parameters, _module-shift factor_, which can be used as an alternative to the originally assumed/calculated ones. The diameter of the base is treated here as a known value. First, the formula \(s=\left(\frac{\pi}{2}+2\cdot x\cdot\tan\alpha\right)\cdot m\) was inserted into the expression \(s_{x}=d_{x}\cdot\left(\frac{\pi}{d}+\text{inv}\alpha-\text{inv}\alpha_{x}\right)\):  
\[s_{x}=d_{x}\cdot\left[\frac{\left(\frac{\pi}{2}+2\cdot x\cdot\tan\alpha\right) \cdot m}{z\cdot m}+\text{inv}\ \alpha-\text{inv}\ \alpha_{x}\right] \tag{16}\]  
Then, two steps were performed: The following were inserted into the above: \(\text{inv}\alpha=\tan\alpha-\overline{\alpha}\) and \(\text{inv}\alpha_{x}=\tan\alpha_{x}-\overline{\alpha}_{x}\) and then \(\alpha=\cos^{-1}\left(\frac{d_{b}}{z\cdot m}\right)\) and \(\alpha_{x}=\cos^{-1}\left(\frac{d_{b}}{dz}\right)\):  
\[s_{x}=d_{x}\cdot\left\{\frac{\frac{\pi}{2}+2\cdot x\cdot\tan\left[\cos^{-1} \left(\frac{d_{b}}{z\cdot m}\right)\right]}{z}+\left\{\tan\left[\cos^{-1} \left(\frac{d_{b}}{z\cdot m}\right)\right]-\cos^{-1}\left(\frac{d_{b}}{z\cdot m }\right)\right\}-\left\{\tan\left[\cos^{-1}\left(\frac{d_{b}}{dz}\right) \right]-\cos^{-1}\left(\frac{d_{b}}{dz}\right)\right\}\right\} \tag{17}\]  
After transformation and simplification, the following equation was obtained, called the _Equation of Identical Gears_ (EIG) [20; 21]:  
\[x=z\cdot\frac{\frac{\pi_{b}}{d_{x}}-\frac{\pi}{2\cdot z}-\left\{\tan\left[\cos ^{-1}\left(\frac{d_{b}}{z\cdot m}\right)\right]-\cos^{-1}\left(\frac{d_{b}}{z \cdot m}\right)\right\}+\left\{\tan\left[\cos^{-1}\left(\frac{d_{b}}{dz} \right)\right]-\cos^{-1}\left(\frac{d_{b}}{dz}\right)\right\}}{2\cdot\tan\left[ \cos^{-1}\left(\frac{d_{b}}{z\cdot m}\right)\right]} \tag{18}\]This proves that you should not worry about the fact that the recreated value of the module may most likely be completely different than the original value, because, in the context of the correctness of the geometry reconstruction process, it has no mathematical significance. Therefore, the module cannot be used as a parameter to compare the original wheel with the recreated wheel. An explanation of this aspect is presented in Table 2, which shows that the use of alternative parameters does not change either the base pitch or the tooth thickness measured along the arc of any chosen circle. An interesting issue worth paying attention to in future research is the fact that, depending on the set (Table 2), the same gear is shifted negatively (\(x<0\)), and sometimes positively (\(x>0\)).