# Validation of a Manual Methodology for Measuring Roundness and Cylindicity Tolerances
## 2 Materials and Methods
### Theoretical and Analytical Fundamentals and Development
#### 2.1.2 Cylindricity Tolerance  
ISO 1101 standard [19] defines the cylindricity tolerance zone of a considered part as the volume bounded by two coaxial cylinders containing all points of the profile. The radial distance between these cylinders is the cylindricity tolerance, Figure 3. Cylindricity tolerance is closely related to the concepts of roundness tolerance and coaxiality. It goes without saying that the determination of the cylindricity tolerance involves the previous calculation of the roundness tolerance in various sections of the part. The more sections are evaluated, the more accurate this evaluation is, although it is much more time-consuming. Figure 3: Cylindricity tolerance zone definition according to UNE-EN ISO 1101 standard [19]. As in the previous study of roundness, the cylindricity tolerance can be obtained by considering four reference cylinders, as well as cylindricity deviations. So, in a similar way to roundness, the ISO standard suggests four methods: the minimum zone cylinders (MZCy), the least squares cylinder (LSCy), the maximum inscribed cylinder (MICy), and the minimum circumscribed cylinder (MCCy). Again, according to the cylindricity tolerance definition, only the MZC method leads to the minimum value of tolerance. Nevertheless, this method implies complex algorithms based on \"simplex approximations\" or iterative calculus [23; 24]. For that reason, it is more adequate for educational environments the use of LSCy, widely accepted in the industrial field, with tolerance values that are very close to those obtained by MZC [24]. LSCy fits the least squares cylinder to the measured profile of the whole part so that the sum of the squares of radial distances between the fitted cylinder and each point of the surface is minimized. It is well known that the difference between the largest and smallest distances from the points on the surface of the part to the axis of the least squares cylinder is the cylindricity tolerance. Typically, the determination of the least squares cylinder that fits a set of points on the workpiece is equivalent to obtaining the equation of the axis that best fits the centers of the least squares circles of each considered section of the cylinder, see Figure 4. Therefore, to evaluate the cylindricity tolerance of a part, it is necessary to assess the roundness corresponding to different sections in a previous step. And it can be checked that the more planes are measured, the greater the accuracy is obtained. Then, it is assumed that the centers of the least squares circles of each section of the part have been calculated according to the method described above, resulting in a set of points, \(C_{i}(x_{i}\),\(y_{i}\),\(z_{i})\). The mathematical problem consists of finding the equation of the straight line, r, that fits this series of points. The cylinder axis follows the general Equation (8), where \(P_{0}(x_{0}\),\(y_{0}\),\(z_{0}\)) and \(\stackrel{{\leftrightarrow}}{{V}}=(u,v,w)\) are a point and a vector of the axis, respectively. \[r:\ \frac{x-x_{o}}{u}=\frac{y-y_{o}}{v}=\frac{z-z_{o}}{w} \tag{8}\]  
Figure 4: (**a**) Representation of the least squares cylinder fitting several sections of the same part; (**b**) axis of the least squares cylinder by adjusting the centers of the LSC of each section. It can be mathematically demonstrated that the point \(\overline{X}(\overline{x},\overline{y},\overline{z})\), whose coordinates are the average values of those corresponding to all points to be fitted, is a point on the cylinder axis, see Appendix B. Thus, a director vector of the axis can be written as \(\overset{\rightarrow}{V}=(u,v,1)\), obtaining the Equation (9). \[r:\ \frac{x-\overline{x}}{u}=\frac{y-\overline{y}}{v}=\frac{z-\overline{z}}{1} \tag{9}\]  
The straight line described by Equation (9) can be expressed as the intersection of two planes, Equation (10). \[r:\ \left\{ \begin{array}{l}x=\overline{x}+u(z-\overline{z})\\ y=\overline{y}+v(z-\overline{z})\end{array} \right.