# Validation of a Manual Methodology for Measuring Roundness and Cylindicity Tolerances
## 2 Materials and Methods  
### Theoretical and Analytical Fundamentals and Development  
#### 2.1.1 Roundness Tolerance  
ISO 1101 standard [19] defines the roundness tolerance zone of a considered plane section as the area bounded by two concentric circumferences that contain all the points of the profile, Figure 1a. As is well known, the distance between these circumferences is the roundness tolerance. The control of this tolerance is crucial for the manufacturing of rotational workpieces taking part in assemblies [20]. Several different methods have been suggested in the ISO 12181-1 standard [21] for the evaluation of roundness tolerance according to four different reference circles. These evaluation methods include: the minimum circumscribed circle (MCC), the maximum inscribed circle (MIC), the least squares circle (LSC), and the minimum zone circles (MZC). Various researchers have attempted to find analytical methods or algorithm-based ones for establishing reference features [8]. Among the cited methods, only MZC considers exactly the definition of roundness tolerance established by ANSI and ISO standards, since it involves the determination of two concentric circles by using all points of the profile, so that the radial distance between them is minimum [22], as shown in Figure 1b. However, this method requires the use of numerical procedures with calculus algorithms [8], which is not considered to be the most appropriate for educational environments. On the other hand, the authors consider that LSC is the most convenient method for understanding the concepts of tolerance, as well, it is the most used in engineering design. It allows users to apply and reinforce methodologies of statistical analysis and regression, acquired previously, focused on a real meaningful objective. Furthermore,  
Figure 1: Roundness tolerance zone definition according to (**a**) UNE-EN ISO 1101 standard [19]; (**b**) MZC method; and (**c**) LSC method. although LSC sometimes leads to higher tolerance values than MZC, which might cause one to discard of some valid parts, it is commonly used at the industrial level due to its ease and robustness [22]. In LSC, the least squares circle is fitted, and taking into consideration the measured part profile, the sum of the squares of radial distances between the circle and each point of the surface must be minimized. The solution is unique, so the center of the LSC circle can be used to define an inscribed and a circumscribed circle with respect to the profile. The radial distance between them is defined as the roundness tolerance, see Figure 1c. The analytical determination of the roundness tolerance according to LSC consists of determining the position of the LSC center since it is the reference that allows the roundness tolerance to be calculated. The surface of the evaluated profile can be discretized into a succession of points, \(P_{i}\), located from the origin of coordinates, \(O\), by a distance \(r_{i}\), forming an angle \(\theta\) with respect to the horizontal axis, Figure 2. The cartesian coordinates of each point \(P_{i}(x_{i},y_{i})\) can be obtained by Equation (1). \[\left. \begin{array}{l}x_{i}=r_{i}\cos\theta_{i}\\ y_{i}=r_{i}\sin\theta_{i}\end{array} \right\} \tag{1}\]  
The position of the LSC center, \(C(a,b)\), is defined by \(a=c\) cos\(\alpha\), \(b=c\) sin\(\alpha\), where \(c\) is the distance from the origin, \(O\), and \(\alpha\) is the angle with respect to the horizontal axis. By means of trigonometry, it is possible to obtain a relationship between the distance from each point of the profile to the origin, \(r_{i}\), the position of the center of LSC, and the radial error between the considered point and LSC, \(e_{i}\), Equation (2).