# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 1 Introduction
### Mathematical Modeling Cylindricity  
The parameters and recommendations for measurement of cylindricity are described in the standard ISO 12180 [6]. The cylindrical profile can be considered as three superposed deviations: radial deviations, median line deviations, and cross-section deviations, as shown in Figure 1. Obviously, median line deviations denote out-of-straightness of the median line of a cylinder, and radial deviations and cross-section deviations denote the radius deviation and out-of-roundness of the cross-section, respectively.  
Cylindrical profile is made up of sequential cross-sectional profiles arranged in the axial direction, and each cross-sectional profile can be described as definite periodic signals with a base frequency equal to rotational frequency of the cylinder [5]. Therefore, the cylindrical profile \(r(z,\theta)\) can be described as a Fourier expansion:  
\[r(z,\theta)=r_{0}(z)+\sum_{\rho=1}^{\infty}\bigl{[}a_{p}(z)\cos(p\theta)+b_{p }(z)\sin(p\theta)\bigr{]}. \tag{1}\]  
where, \(z\) and \(\theta\) are respectively the axial and angular coordinates of the cylindrical profile, and \(r_{0}(z)\) is the radius of the cross-sectional profile. \(a_{p}(z)+jb_{p}(z)\), which denotes the \(p\)th harmonic vector of the cross-sectional profile, can be expressed with the Legendre interpolation, i.e., \(a_{p}(z)=\sum_{j=0}^{\text{M}_{a}}a_{pj}z^{j}\), and \(b_{p}(z)=\sum_{j=0}^{\text{M}_{b}}b_{pj}z^{j}\). Substituting \(a_{p}(z)\) and \(b_{p}(z)\) into Equation (1), the cylindrical profile \(r(z,\theta)\) can be rewritten as follows:  
\[\begin{split} r(z,\theta)=&\ r_{0}(z)+\Bigl{[}\Bigl{(} \sum_{j=0}^{\text{M}_{a}}a_{1j}z^{j}\Bigr{)}\cos(\theta)+(\sum_{j=0}^{\text{ M}_{b}}b_{1j}z^{j})\sin(\theta)\Bigr{]}\\ &+\sum_{\rho=2}^{\infty}\bigl{[}a_{p}(z)\cos(p\theta)+b_{p}(z) \sin(p\theta)\bigr{]}\end{split} \tag{2}\]  
Term 2 in Equation (2) belongs to the first harmonic component of the cross-sectional profile. Let \(a_{1}(z)=\sum_{j=0}^{\text{M}_{a}}a_{1j}z^{j}\) and \(b_{1}(z)=\sum_{j=0}^{\text{M}_{b}}b_{1j}z^{j}\), then, \(a_{1}(z)+jb_{1}(z)\) can be characterized as a vector from the rotating center in a cross-section of the cylinder to the least squares center (LSC) of the cross-sectional profile [5; 7]. If the linear term \((a_{10}+a_{11}z)+j(b_{10}+b_{11}z)\) about the \(Z\)-axis in \(a_{1}(z)+jb_{1}(z)\) is canceled, the cylinder as a whole will be tilted and moved, and the cylindrical profile will be not changed. Thus, the mathematical model of the cylindrical profile can be expressed by Equation (3).  
\[r(z,\theta)=r_{0}(z)+r_{1}(z,\theta)+r_{2}(z,\theta) \tag{3}\]  
where, \(r_{1}(z,\theta)=a_{1}(z)\cos(\theta)+b_{1}(z)\sin(\theta)\) is the first harmonic component of the cross-sectional profile. Obviously, a median line of the cylindrical profile, as shown Figure 1b, can be fitted through the LSC vectors \(a_{1}(z)+jb_{1}(z)\) in sequential cross-sections [5; 6; 8]. The 0 th harmonic component, \(r_{0}(z)\), represents average radius deviations of sequential cross-sectional profiles, as shown Figure 1a. The sum of the \(p\)th\((p=2\text{--}\infty)\) harmonic components, denoted by \(r_{2}(z,\theta)=\sum_{\rho=2}^{\infty}\bigl{[}a_{p}(z)\cos(p\theta)+b_{p}(z) \sin(p\theta)\bigr{]}\), represents the out-of-roundness of sequential cross-sectional profiles, as shown Figure 1c.  
Figure 1: Components of the cylindricity deviations.  
From Equation (3), a feasible method of measuring and reconstructing a cylindrical profile can be suggested. It is to accurately measure the radius deviations, roundness errors, and LSC vectors of sequential cross-sections. Moreover, the median line is fitted through the LSCs, and cylindrical profile can be reconstructed by assembling the measured radius deviation and out-of-roundness of each cross-section on the fitted median line.