## 3 The Proposed Method
### Brush Trajectory Model  
The shape and position of the brush's footprint on the paper will vary with the handle's coordinates. In this section, we will describe the changes in the brush's position with handle position in further detail. [PERSON] et al. [3] computationally modeled the changes in the brush spine during writing processes. Based on this idea, we will calculate the changes in the angle between the horizontal projections of the handle and tip. As the distance between the horizontal projections of the handle and tip, \(r\), will change with the height of the handle, \(z\), we have replaced \(r\) with the effective radius, \(r_{z}\), and assumed that the handle, front of the footprint, and tip of the footprint lie on the same line.  
Figure 9 shows the effects of \(L\) (the movement distance of the handle's projection) on the angle between the trajectories of the handle and brush. Let \(r_{z0}\) and \(r_{z1}\) be the effective distances between the horizontal projections of the handle and tip, and \(\phi_{0}\) and \(\phi_{1}\) the angles between the trajectories of the tip and handle. \(\overline{O_{0}P_{f}}\) and \(\overline{O_{1}P_{b}}\) are parallel to each other. Assuming \(k=p_{f}+\eta(p_{b}-p_{f}),\eta<1\) where \(\eta\) is a function of \(z\); if \(z\) is fixed, \(\phi_{1}\) will decrease as the displacement of the handle, \(L\), increases.  
\(\eta\) is obtained from measurements. As \(r_{z}\) is directly related to \(z\), and it is much easier to control \(z\) than to measure \(r\), we simply measured the correlation between \(z_{i}\) and \(\eta_{i}\). To simplify the model's parameters, \(\eta_{i}\) was only measured for \(z_{i}\) values of 1, 2, and 3 mm. Figure 10 gives the relationship between the displacement of the handle and its angle to the front of the footprint, with \(\eta\) being fixed to 0.2.  
Each Chinese character can be decomposed into one or multiple strokes, and the writing of each stroke involves the anbi, dunbi, singbi, tibi, and shoubi processes. Suppose that each stroke requires N of these actions; the pseudocode for the brush trajectory model is Algorithm 1:  
Figure 10: Relationship between handle displacement and the angle between the trajectories of the handle and front of the footprint.  
Figure 9: Changes in the angle between the handle and tip of the brush.  
```
1:The initial coordinates of the handle are \(P_{0}=(x_{0},y_{0},z_{0})\), where \(z_{0}=0\). The coordinates of the tip are tip\({}_{0}\)\(=(x_{0},y_{0})\), which are the same as those of the handle. \(\phi_{0}=0\).
2:The trajectory of the handle coordinate \(P_{i}\).
3:for\(i\gets 1\to N\)do
4:\(d_{i}=\sqrt{((x_{i}-x_{i-1})^{2}+(y_{i}-y_{i-1})^{2}}\)
5: Search for the basic footprint, \(Ink_{i}\) from the database, based on \(d_{i}\) and \(z_{i}\).
6: Calculate the corresponding \(\eta_{i}\) values based on \(z_{i}\).
7: Draw a geometric diagram based on \(z_{i-1}\), \(z_{i}\), \(d_{i}\), \(\phi_{i-1}\), and \(\eta_{i}\) to obtain \(\phi_{i}\).
8: Rotate \(Ink_{i}\) by \(\phi_{i}\) and translate it to \((x_{i}\), \(y_{i})\), and sketch the outer frame of the footprint.
9:endfor
10:return
```  
**Algorithm 1** Brush trajectory model.