# Presenting a Human Pupil Orbit Model (HPOM) for Eye-Gaze Tracking
## 2 Materials and Methods
### 3D Rotation of the Pupil Disc  
Previously, the principle that the eccentricity and rotation of the PPC changes according to the RAE and RDA using HPOM, was analysed. Based on the results of this analysis, a three-dimensional Rotation Model of a Disk (RMD) was constructed to analyse the attitude of the pupil in three-dimensional space. RMD is a model used to observe the three-dimensional attitude of the eye using images captured by a camera that observes changes due to the rotation along each axis of the disk in an environment where the xy-coordinate system of the camera and the disk are synchronized, and a constant distance is maintained along the z-axis. When the disk is perpendicular to the z-axis of the camera, the initial attitude of the RMD is set and the attitude of the disk in 3D with respect to the centre point and the shape projected  
The length of the long axis is constant at \(d_{o}\) even though \(\theta\), the rotation angle of the y-axis, varies. This is because there is no change in the distance between the camera and the z-axis in that part of the disk that was in contact with the y-axis at the point where the rotation occurred. Based on this, it was expected that the length of the long axis would always be constant even if rotation of the other axis occurred. To confirm this, the result was projected on the camera as each axis was rotated through all angles, as shown in Figure 4c. Despite the lengths of the short and long axes of the ellipse changing, as well as the angle formed with the x axis changing, the length of the long axis remains the same, i.e., \(d_{o}\).  
\[d_{r}=d_{o}\times\cos\theta \tag{3}\]  
\[\theta=\cos^{-1}\frac{d_{r}}{d_{o}} \tag{4}\]  
\[\angle\mathrm{QRP}=90^{\circ}-\mathrm{RAE} \tag{5}\]  
Figure 4: Projected result as an ellipse onto the camera after a disk has rotated each axis. (**a**) The shape of the projected disk when the rotation of the y-axis occurs. (**b**) The shape of the projected disk when the rotation of the y-axis different magnitude from (**a**) occurs. (**c**) The shape of the projected disk when the rotation of the x-, y-, and z-axis are randomly.  
Figure 5: Principle of the disk being projected onto an ellipse.  
By reflecting the RMD analysis to PPC changes according to attitude changes of the pupil disk that were analysed previously, the results can be interpreted according to our premise, as shown in Figure 6. In Figure 6, points O, P, and Q indicate the ORCP, PCP, and the position when a line is drawn perpendicular to the z-axis in the PCP, respectively. Point R is the position where the z-axis is met by a perpendicular line drawn parallel to the rotated pupil disk in PCP. Since the sum of the interior angles of a right triangle is always 180 degrees, the angle of QRP can be calculated using Equation (4). Using the same principle, it can be seen that the rotation angle \(\theta\) of the pupil disk is the same as the RAE. Therefore, if lengths of the short axis and the long axis of the PPC are known, the RAE can be calculated according to Equation (5). Thus, the three-dimensional position of the PPC can be approximated using the centre point of the PCP and the RAE.