# Determining Optimal Parameters of Regular Microrelief Formed on the End Surfaces of Rotary Bodies
## 3 Results
The formula determines angle \(\alpha_{\rho}\):  
\[\alpha_{\rho}=2\cdot\arccos\left(\frac{S_{o}-A_{k1}-A_{k2}}{2\cdot\rho_{k}}\right) \tag{10}\]  
Taking into account (10), Formula (8) takes the following form:  
\[S_{ov}=\rho_{k}^{2}\cdot\left[\frac{2\cdot\pi\cdot\arccos\left(\frac{S_{o}-A_{ k1}-A_{k2}}{2\cdot\rho_{k}}\right)}{180^{\circ}}-\sin\left(2\cdot\arccos\left( \frac{S_{o}-A_{k1}-A_{k2}}{2\cdot\rho_{k}}\right)\right)\right] \tag{11}\]  
The area of existence for this formula was determined by the limit value of the axial pitch of the PRMR, which was determined from the following dependence:  
\[\left\{ \begin{array}{c}A_{k1}+A_{k2}+2\cdot\rho_{k}>S_{o}\geq A_{k1}+A_{k2}+ 2\cdot\rho_{k}\cdot\sin\left(\frac{\alpha_{k2}}{2}\right)\\ \alpha_{k2}>\alpha_{k1}\end{array} \right. \tag{12}\]  
Stage II. Consider the case when the groove vertices intersect within the circular segment with radius \(\rho_{k}\) and chord height \(\Delta_{\rho_{1}}\) and partially rectilinear sections (Figure 4). Figure 3: Scheme of overlap of V-shaped grooves of PRMR (stage I). We calculated the overlap area of the grooves for this case. In this case, the overlap area will be defined as the sum of the areas of two circular segments with radius \(\rho_{k}\) and the areas of two trapezoids (\(ACDB\) and \(AELB\)) (Figure 4). \[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]  
where \(S_{s1}\) is the segment area of groove 1, mm\({}^{2}\); \(S_{t1}\) is the trapezoid area of groove 1, mm\({}^{2}\); \(S_{s2}\) is the segment area of groove 2, mm\({}^{2}\); \(S_{t2}\) is the trapezoid area of groove 2, mm\({}^{2}\). To determine the overlap area according to the scheme (Figure 4), it was advisable to determine the coordinates of point \(B\) pertaining to the intersection of groove elements. To do this, we wrote down the equation of the axial lines of grooves 1 and 2 in the coordinate system \(xy\). The equation of centerline \(a\)-\(a\) (groove 1) is written in the following form:  
\[y_{i(a-a)}=tg\Big{(}90-\frac{\alpha_{k1}}{2}\Big{)}\cdot x_{i}+\frac{S_{o}-A_{ k1}-A_{k2}}{2} \tag{14}\]  
The equation of centerline \(b\)-\(b\) (groove 2) is written in the following form:  
\[y_{i(b-b)}=-tg\Big{(}90-\frac{\alpha_{k2}}{2}\Big{)}\cdot x_{i}-\frac{S_{o}-A_ {k1}-A_{k2}}{2} \tag{15}\]  
The equation of the imprint of side surface \(a^{\prime}\)-\(a^{\prime}\) (groove 1) is written in the following form:  
\[y_{i(a^{\prime}-a^{\prime})}=tg\Big{(}90-\frac{\alpha_{k1}}{2}\Big{)}\cdot x_{ i}+\frac{S_{o}-A_{k1}-A_{k2}}{2}-\frac{\rho_{k}}{\sin\big{(}\frac{\alpha_{k1}}{2} \big{)}} \tag{16}\]  
The equation of the imprint of side surface \(b^{\prime}\)-\(b^{\prime}\) (groove 2) is written in the following form:  
\[y_{i(b^{\prime}-b^{\prime})}=-tg\Big{(}90-\frac{\alpha_{k2}}{2}\Big{)}\cdot x_ {i}-\frac{S_{o}-A_{k1}-A_{k2}}{2}+\frac{\rho_{k}}{\sin\big{(}\frac{\alpha_{k2 }}{2}\big{)}} \tag{17}\]  
We equate expressions (16) and (17) to find value \(x_{i}=\Delta_{x}\), at which values \(y_{i(a^{\prime}-a^{\prime})}\) and \(y_{i(b^{\prime}-b^{\prime})}\) will be the same. That is, we find coordinate \(x\) of point \(B\). \[\Delta_{x}=\frac{\rho_{k}\cdot\left(\frac{1}{\sin\big{(}\frac{k2}{2}\big{)}}+ \frac{1}{\sin\big{(}\frac{k1}{2}\big{)}}\right)-\left(S_{o}-A_{k1}-A_{k2} \right)}{tg\big{(}90-\frac{\alpha_{k1}}{2}\big{)}+tg\big{(}90-\frac{\alpha_{ k2}}{2}\big{)}} \tag{18}\]  
Figure 4: The scheme of overlapping of V-shaped grooves of PRMR (stage II). Substituting the obtained value into Formula (16), we obtain value \(\Delta_{y}\).