## 3 Method
### Build the Mapping Equation
Next, this paper will study the geometric relationship between the scene viewpoint plane coordinate point \((U,V)\) and the pupil center coordinate \((x_{p},y_{p},z_{p})\) in the world coordinate system:  
Suppose that the center of the sphere is converted to the origin of eyeball coordinate system, then the pupil center coordinates \((x_{p},y_{p},z_{p})\) and gaze coordinate points \((U,V)\) will be converted into \((x_{p}-x_{0},y_{p}-y_{0},z_{p}-z_{0})\) and \((U-x_{0},V-y_{0})\), and combined with the horizontal distance between the human eye and the plane of the viewpoint is L, so according to the similar conditions of the triangle, we can get:  
\[\frac{x_{p}-x_{0}}{U-x_{0}}=\frac{y_{p}-y_{0}}{V-y_{0}}=\frac{z_{p}-z_{0}}{Z} \tag{13}\]  
The above formula converts both sides of the equation to calculate the following Formula (14):  
\[ \begin{split} U&=\frac{L}{z_{p}-z_{0}}\cdot x_{p}+(1 -\frac{L}{z_{p}-z_{0}})\cdot x_{0}\\ V&=\frac{L}{z_{p}-z_{0}}\cdot y_{p}+(1-\frac{L}{z_{ p}-z_{0}})\cdot y_{0}\end{split}  \tag{14}\]  
The eyeball is a standard sphere, so it satisfies the standard equation expression of the sphere, and then combines the two conditions of Formula (11):  
\[ \begin{split}\left\{\begin{aligned} x_{p}^{2}+y_{p}^{2}+ z_{p}^{2}=R^{2}\\ x_{p}=\frac{z_{p}}{f_{x}}\cdot(x-c_{x})\\ y_{p}=\frac{z_{p}}{f_{y}}\cdot(y-c_{y})\end{aligned}\right.\end{split}  \tag{15}\]  
The final expression of \(z_{p}\) on the z-axis of the pupil center can be obtained as follows:  
\[z_{p}=\frac{R\cdot f_{x}\cdot f_{y}}{\sqrt{f_{y}^{2}\cdot(x-c_{x})^{2}+f_{x} ^{2}\cdot(y-c_{y})^{2}+f_{x}^{2}\cdot f_{y}^{2}}} \tag{16}\]  
Finally, substituting Equations (11) and (16) into Equation (14), the final expression of the mapping equation based on the homography transformation is:  
\[\begin{split} U&=\frac{R\cdot L\cdot f_{y}\cdot(x -c_{x})-x_{0}\cdot L\cdot\sqrt{f_{y}^{2}\cdot(x-c_{x})^{2}+f_{x}^{2}\cdot(y- c_{y})^{2}+f_{x}^{2}\cdot f_{y}^{2}}}{R\cdot f_{x}\cdot f_{y}-z_{0}\sqrt{f_{y}^{2} \cdot(x-c_{x})^{2}+f_{x}^{2}\cdot(y-c_{y})^{2}+f_{x}^{2}\cdot f_{y}^{2}}}+x_{ 0}\\ V&=\frac{R\cdot L\cdot f_{y}\cdot(y-c_{y})-y_{0} \cdot L\cdot\sqrt{f_{y}^{2}\cdot(x-c_{x})^{2}+f_{x}^{2}\cdot(y-c_{y})^{2}+f_{ x}^{2}\cdot f_{y}^{2}}}{R\cdot f_{x}\cdot f_{y}-z_{0}\sqrt{f  
The experimental results show that this method achieves experimental accuracy within 0.5\({}^{\circ}\), has good experimental results, and can be generalized to practical applications. In order to further prove the practicability of the mapping equation based on homography transformation, this article will continue to design comparative experiments and obtain experimental results to prove the advantages of this mapping equation. According to the current calibration method, any number of calibration points and verification points with the same number can obtain high experimental accuracy. Therefore, this section will use different experimental designs such as nine-point calibration and 16-point verification to illustrate the advantages of this method, and further illustrate the feasibility of improving the mapping equation by comparing the experimental result data with the mapping equation of classical quadratic polynomial. The experimental results obtained are shown in Figure 8.  
From the experimental results in Figure 8, it can be clearly seen that the result of advantages and disadvantages of the two method show that the accuracy of the classical quadratic polynomial mapping equation is 0.99\({}^{\circ}\), which is close to 1\({}^{\circ}\). Using the improved mapping method based on homography transformation, it can be seen that the accuracy is basically maintained at about 0.5\({}^{\circ}\). This shows that the design of different calibration points and verification points is more suitable for mapping methods based on homography transformation. Finally, according to the knowledge of the imaging principle of the projector, this article thought that this improved equation can have good experimental results in the design of small area calibration and large-scale verification.