# Presenting a Human Pupil Orbit Model (HPOM) for Eye-Gaze Tracking
## 2 Materials and Methods
### Presumptive Background of HPOM
#### 2.6.1 ORCP Inference  
When the HPOM is a sphere, the three-dimensional position of the PCP according to how gaze direction changes is reflected on the surface of the sphere orbit centred on the ORCP. Figure 7a shows positions of the ORCP and PCP in 3D space resulting from randomly acquiring n PCPs (\(P_{n}\)) as the gaze direction changes. Since the distance between the ORCP and PCP is always constant, the position where the ORCP can exist can be expressed as shown in Figure 7b. This is done with respect to each point P and is expressed by Equation (6) with a sphere equation. Spheres centred at each point, P, intersect each other. All spheres intersect at the ORCP. Figure 6: Relationship between the RAE of HPOM and the rotation angle of the pupil. The intersection of all spheres centred on each point P in 3D space can be obtained by considering the intersecting surfaces of the spheres. However, since we can't directly acquire the z-axis position of the PCP, it is difficult to estimate the ORCP despite calculating the equation of the three-dimensional plane where the spheres meet. In order to overcome this, the variables related to the z-axis position in (6) are summarized as shown in Equation (8) using Equation (7). This shows the positional relationship between the ORCP and PCP on the xy-plane in the spherical coordinate system. The transformation process from Equation (6) to Equation (8) is described as the process of calculating the xy section of the sphere centred for each point P at the z-axis position of the ORCP, the result of which are shown in Figure 7c. By Equation (8), the position where ORCP can exist around each point P is limited to a circle on the xy-plane. This can be expressed as the equation of a circle (Equation (9)). Circles centred at each point P have radii proportional to RAE. All circles intersect at ORCP and have a maximum of two intersections with each of the other circles. A circle with a different RDA always intersects with each other only at the ORCP regardless of the size of their RAE. For this reason, at least three circles centred on the point P are required to estimate the ORCP. The ORCP where all circles intersect can be obtained by considering a straight line passing through that intersection point where each circle intersects the others. Figure 7d shows a straight line (Lnm) passing through the intersection of circles centered on points Pn and Pm. All straight lines intersect at one point and all circles intersect at the same point. Based on this, in order to calculate a straight line passing through the intersection point of the circles, Equation (9) of the circle centred on each point P is combined and arranged as shown in Equation (10). By using Equation (10), the radius of the unknown HPOM can be eliminated. However, the calculation result of Equation (10) is arranged in the form of a straight-line equation to obtain Equation (11) when the difference of the constant K in Equation (10), i.e., the coefficient of the quadratic term, is 0. On the other hand, in the form  
Figure 7: (**a**) The spherical orbital formed by PCP. (**b**) Candidate positions of the centre point of a spherical orbit located at a certain distance from each PCP. (**c**) Candidate positions obtained from the z-axis position of ORCP. (**d**) Lines passing through intersections of circles. of the new circle equation shown in Equation (12), when K is not 0, this resolves. Since at least one of the intersection points of the two circles is always an ORCP, the equation of the newly calculated circle also forms an intersection point in the ORCP. Therefore, equations of all the circles, newly obtained through the simultaneous process, are divided and normalized by K in Equation (11), which gives the coefficient of each quadratic term. When equations of the normalized circle are simultaneously processed, the quadratic term is eliminated, and all can be arranged in the equation of a straight line (Equation (11)).