## 2 Theoretical Background and Adopted Polishing Cycle  
It has been shown [11; 12] that the quality and efficiency of a stone polishing process in a linear polishing machine, with one or more rotating heads moving along zigzag trajectories, as indicated in Figure 2, is controlled by a geometric condition relating the two linear velocities, \(V_{L}\) and \(V_{T}\). Condition 1 states that after a complete zigzag movement (with two linear segments), the tool center should be closer to its initial position than the tool diameter [1]: \(l<d\), with \(d=2r=\) tool diameter. Otherwise, there will be gaps of size \(l-d\) where the polishing head does not interact with the surface. In this work we suggest that a second condition should be considered, relating the linear and rotating tool speeds, thus including all the most relevant kinematic parameters involved in the process. Condition 2 states that after a complete rotation, the tool center should be closer to its initial position than the tool radius [11]: \(x<r\). Otherwise each point of the surface only interacts with a small angular fraction of the circular polishing head, and in some circumstances it may not be touched at all. Figure 1: Aspects of a linear polishing machine with rotating heads (example from PEDRINI). The combination of these two conditions determines the relations that must exist between the three velocities involved in this polishing process: the conveyor belt velocity, \(V_{L}\), the transverse head velocity, \(V_{T}\), and the rotational head velocity, \(\omega\). Defining \(b\) as the width of the conveyor belt, the time needed for the tool to move across the belt is given by \(t_{T}=b/V_{T}\) and the belt shift after a zigzag becomes \(l=2 bV_{L}/V_{T}\leq d\) (limited by condition 1). The distance \(x_{S}\) moved by the polishing head in one linear segment (half of a complete zigzag) is given by:  
\[x_{s}=\sqrt{b^{2}+\left(\frac{l}{2}\right)^{2}}=\sqrt{\left(V_{T}t_{T}\right) ^{2}+\left(V_{T}t_{T}\right)^{2}}=\frac{b}{V_{T}}\sqrt{{V_{T}}^{2}+{V_{L}}^{2} }=b\sqrt{1+\left(V_{L}/V_{T}\right)^{2}}\, \tag{1}\]  
Defining \(n_{S}=\omega t_{T}\) as the number of rotations executed during the time \(t_{T}\) by the tool head rotating at \(\omega\) rotations per unit of time, the distance \(x\) that the tool head moves in a single rotation is given by (limited by condition 2):  
\[x=\frac{x_{s}}{n_{s}}=\frac{b\ \sqrt{1+\left(V_{L}/V_{T}\right)^{2}}}{\omega\ t_ {T}}=\frac{V_{T}}{\omega}\sqrt{1+\left(V_{L}/V_{T}\right)^{2}}\leq r \tag{2}\]  
To assess the validity of condition 2 and its combination with condition 1, we acquired experimental data and compared them with equivalent simulated results. The polishing operations used in this work involved the last three stages of the polishing sequence (320, 400, and 5 Extra grit [13] sequence--see Figure 3). This sequence was selected to achieve a mirrored surface, with a significant gloss. The quantitative effectiveness of the surface polishing was then assessed experimentally measuring the gloss [14] with a TQC GL0030 glossmeter (TQC Sheen B.V., Capelle aan  
Figure 3: Sequence of the polishing cycle abrasives used in this work: (**a**) 320 grit; (**b**) 400 grit; (c) Frankfurt 5 Extra grit. Figure 2: The two polishing conditions. (**a**) \(l\leq d\); (**b**) \(x\leq r\). den Ijssel, The Netherlands)--depicted in Figure 4--at 20\({}^{\circ}\), 60\({}^{\circ}\), and 85\({}^{\circ}\) to the normal surface (the standard angles usually supported by this type of equipment). A wired grid positioned over the stone was used to define each square section where gloss was measured. The light reflected (the gloss) for a given angle can be compared with a standard scale going from 100 gloss units (GU) for a highly reflective black glass, to 0 gloss units for a perfectly matt surface [15]. Therefore, the measuring head of a glossmeter must be calibrated previously. Highly reflective materials, such as mirrors, may achieve gloss values as high as 1000, when measured at 20\({}^{\circ}\).