## 1 Introduction
### Principle of the Stitching Linear-Scan Method  
Figure 1 shows a schematic of the setup for measurement of a small cylinder by the stitching linear-scan method [10]. The setup is composed of a linear-scan surface form stylus profilometer, a round magnet and a workpiece holder with a V-groove. A small cylinder to be measured is attached to the round magnet, on whose outer surface indexing marks with an angular interval of 45\({}^{\circ}\) are prepared over 360\({}^{\circ}\). Due to the magnetic force of the round magnet, a small cylinder can automatically be aligned to the center of the round magnet. It should be noted that there should be eccentricity of a small cylinder with respect to the round magnet. However, the influence of the eccentricity can be canceled during the following stitching operation; this is one of the advantages of employing the stitching linear-scan method [10]. The small cylinder with the round magnet is placed in the V-groove on the workpiece holder in such a way that the round magnet surface contacts the side of the work holder. Measurement of the circumferential profile of the small cylinder is carried out by repeating the 45\({}^{\circ}\) rotation of the workpiece and the linear-scan of its arc-profiles, as shown in Figure 2. For the workpiece rotation, \(n\) indexing marks (\(n\) is a small integer) prepared on the round magnet by a dividing head are utilized. By using the obtained eight arc-profiles of the workpiece, the workpiece profile over 360\({}^{\circ}\) can  
Figure 1: A schematic of the roundness measurement by the stitching linear-scan method. be obtained through the stitching process. From the obtained workpiece profile, evaluations of the roundness and diameter can be carried out. The cylinder-holding mechanism composed of the round magnet and the workpiece holder with the V-groove enables the linear-scan method to evaluate the whole length of a small cylinder, which cannot be achieved by the conventional roundness measuring instruments where a part of the cylinder in its axial direction should be held by a clamping mechanism. Furthermore, the precision positioning of a small cylinder along its axial direction can be achieved in a simple manner by utilizing the round magnet surface as the datum for the positioning. The stitching process of the obtained eight arcs contains three-step calculations: the coordinate transformation, the radial stitching, and the circumferential stitching. Figure 3 shows a schematic of the stitching procedure. First we repeat the linear-scan and rotation of a small cylinder to obtain a series of arc profiles. After that, radial stitching is carried out for the obtained eight arcs. Each arc-profile datum of a small cylinder is in the Cartesian coordinate system (\(x_{i,j}\), \(z_{i,j}\)), where \(i\) is the sampling number in each of the arcs, and \(j\) (\(j\) = 1, 2,..., 8) is the measurement order of the arc. By fitting a circle to each of the obtained arcs based on the least-squares method, the center coordinates and radius of the \(j\)th arc (\(x_{c,j}\), \(z_{c,j}\)) and \(r_{j}\), respectively, can be obtained. Then, the coordinates of the eight arcs are shifted based on the following equation to match the center coordinates of the arcs:  
\[\left( \begin{array}{c}{x^{\prime}}_{i,j}\\ {z^{\prime}}_{i,j}\end{array} \right)=\left( \begin{array}{c}x_{i,j}-x_{c,j} \\ z_{i,j}-z_{c,j}\end{array} \right) \tag{1}\]  
Furthermore, each arc profile is rotated by the following equation:  
\[\left( \begin{array}{c}{x^{\prime\prime}}_{i,j}\\ {z^{\prime\prime}}_{i,j}\end{array} \right)=\left( \begin{array}{cc}\cos j \theta&\sin j\theta\\ -\sin j\theta&\cos j\theta\end{array} \right)\left( \begin{array}{c}{x^{\prime }}_{i,j}\\ {z^{\prime}}_{i,j}\end{array} \right) \tag{2}\]  
It should be noted that each of the arcs has a different radius \(r_{j}\).