## 4 Experimental Results
### Experimental Procedures
#### 4.2.1 Building the Brush Trajectory Model  
The image measurement process can be divided into two parts: (1) Measuring the basic footprints that correspond to changes in the outer frame of the brush footprints and (2) measuring the footprint trajectories that correspond to different angles between the trajectories of the brush and handle. These measurements have been described below. (I) Basic footprints:  
The marks made by a brush will change with the motions and z-axis height of the handle. Figure 14 is a DLT-corrected image of brush footprints, and there are seven basic footprints on the left side of this figure. In these footprints, the handle was pressed downwards from \(z=0\) mm to  
Figure 12: Refillable brush pens. Figure 13: Process flow of the experiment. \(z=1\) mm and horizontally displaced by 1 to 7 mm. The left end of each footprint is the position of the brush tip at \(z=0\), and the + symbol represents the \((x,y)\) coordinates of the handle when \(z=1\) mm. The seven little dots at the top left of this figure are reference points for determining the coordinates of the handle, and the smooth curve around the boundaries of each footprint were formed by drawing four Bezier curves through the footprint's feature points. The seven basic footprints on the right side of Figure 14 are the same as those on the left, except that the handle was pressed from \(z=0\) to \(z=2\) mm instead of \(z=1\) mm. Figure 15a represents the \(P_{t}\), \(P_{max}\), \(P_{f}\), and \(P_{min}\) coordinates of 18 footprints that were produced by displacing the handle by 1 mm downwards (to z = 1 mm) and 3 mm horizontally. Likewise, Figure 15b also shows the \(P_{t}\), \(P_{max}\), \(P_{f}\) and \(P_{min}\) coordinates of 18 footprints, but with the handle being horizontally displaced by 4 mm instead of 3 mm. Two phenomena may be observed in Figure 14. First, given a constant \(z\)-value, the width of the footprint will narrow as the handle's horizontal displacement increases, until a certain distance is reached. Secondly, an increase in handle displacement also increases the distance between \(P_{f}\) and the handle until a certain distance is reached. During the writing of a stroke, each new brush tip position will be covered by the front of the previous footprint. To measure the outer frame of the brush, one may employ a platform like that in [10]. In this work, we used diagrams like Figure 14 to chart out the  
Figure 14: Measurement of basic footprints (Brush Pen #1). Figure 15: Parameters of the basic footprints. (**a**) Horizontal handle displacement of 3 mm. (**b**) Horizontal handle displacement of 4 mm. variations in the basic footprint, and the data provided by these diagrams were organized to illustrate how the parameters of the brush footprints vary with the horizontal and vertical displacements of the handle, as shown in Figure 16. Here, it is shown that the outer frame of the footprint stabilizes after horizontal displacements of 3, 5, and 6 mm when z = 1, 2, and 3 mm. Therefore, the footprint database contains \(3+5+6=14\) basic footprints. Footprint trajectories:  
A bent line was drawn by the system (Figure 17) so that information about the relationship between the horizontal projection of the handle and the brush could be obtained. The sequence of coordinates for the handle is as follows:  
\[(m_{0x},m_{0y},z_{0})\rightarrow(m_{1x},m_{3y},z_{1})\rightarrow(m_{1x},m_{5y},z_{1})\rightarrow\]  
\[(m_{9x},m_{7y},z_{2})\rightarrow(\frac{m_{1x}+m_{9x}}{2},\frac{m_{5y}+m_{7y}}{ 2},z_{0})\]  
In this sequence, \(z_{0}=0\) was configured so that the tip of the brush just touches the paper, and \(z_{0}\) was used as the reference point for height. The projection of the handle's trajectory on the horizontal plane is indicated by the blue arrows in Figure 17a. Figure 17b is the DLT-corrected version of this image. Some blank space was reserved outside of the 2000 \(\times\) 2000 square to allow the parameters to be recorded on the sheet.