## 4 Polyhedron-Based Rectilinear Projection
Thus, we define each point on the sphere as, \(pt_{i}(x_{i},y_{i},z_{i})\), where the \(i\) denotes the number of pixels in the 360-degree image. \[x =r\cos\theta\sin\phi \tag{1}\] \[y =r\sin\theta\sin\phi\] (2) \[z =r\cos\phi \tag{3}\]  
The center \(o_{n}(ac_{n},bc_{n},cc_{n})\) of each polygon is obtained by computing the mean of the vertices for each polygon as follows:  
\[o_{n}(ac_{n},bc_{n},cc_{n})=\frac{1}{t}\sum_{l=1}^{t}vt_{l}(a_{l},b_{l},c_{l}) \tag{4}\]  
The points on the sphere that are closest to the vertices of each polyhedron and the centers of polygons are computed with the Euclidian distance as Equations (5) and (6). The points that are the closest to the vertices are defined as \(V_{j}\) and the points that are the closest to the centers are defined as \(C_{n}\). \[V_{j}=\text{argmin}_{pt_{i}}\bigg{\{}\sqrt{\left(x_{i}-a_{j}\right)^{2}+\left(y _{i}-b_{j}\right)^{2}+\left(z_{i}-c_{j}\right)^{2}}\bigg{\}} \tag{5}\]  
\[C_{n}=\text{argmin}_{pt_{i}}\bigg{\{}\sqrt{\left(x_{i}-ac_{n}\right)^{2}+\left( y_{i}-bc_{n}\right)^{2}+\left(z_{i}-cc_{n}\right)^{2}}\bigg{\}} \tag{6}\]  
Figure 5: Regular and rotated dodecahedrons. (**a**) Regular dodecahedron; (**b**) Rotated dodecahedron. Figure 6 shows the points that are marked on the sphere. The black points are the closest to the vertices and the circles are the closest to the centers. The circles are used to determine the directions of the rectilinear projections. For the octahedron, instead of locating the circles, the directions are fixed as north and south latitudes at 45 degrees. With the points \(V_{j}\), we can compute the size of the viewing range. There are three types of polyhedrons. Thus, we propose three methods for computing the sizes of the viewing range for the three polyhedrons. The units for the width and height of the viewing range are represented in degrees. For the 32-hedron, we first transform the \(V_{j}\) into spherical coordinates. The vertical angle \(\phi_{V}\) and horizontal angle \(\theta_{V}\) are obtained using the inverse sine and the four-quadrant inverse tangent as follows:  
\[\phi_{V}=\text{asin}(z/r) \tag{7}\]  
\[\theta_{V}=\text{atan2}(y,x) \tag{8}\]  
We select the points that have the maximum and minimum vertical and horizontal angles. For each polygon, the difference \(dv\) between the maximum and minimum vertical angles, and the difference \(dh\) between the maximum and minimum horizontal angles are represented as in Equations (9) and (10). The height, \(hv\), and width, \(wv\), of the viewing range for each polygon is set to the maximum values in \(dv\) and \(dh\), which are greater than 0 and less than 120 degrees. Then, we obtain a wide viewing angle with low distortion. \[x =r\cos\theta\sin\phi \tag{1}\]0  
\[x =r\cos\theta\sin\phi \tag{1}\]1  
\[x =r\cos\theta\sin\phi \tag{1}\]2  
\[x =r\cos\theta\sin\phi \tag{1}\]3  
For the dodecahedron, the size of the viewing range is computed using a different method. The width \(wv\) of the viewing range is based on the distance \(V_{2}V_{5}\) between non-adjacent points, as shown in Figure 7a. The points \(V_{2}\) and \(V_{5}\) are the points \(V_{f}\) and \(V_{g}\) on the sphere in Figure 7b. Then, the \(wv\) can be computed as follows:  
\[x =r\cos\theta\sin\phi \tag{1}\]4  
The point \(mp\) is the midpoint between \(V_{3}\) and \(V_{4}\). The \(mp\) has the closest point on the sphere. That point and the \(V_{1}\) are also the points \(V_{f}\) and \(V_{g}\). Therefore, the height \(hv\) for the viewing range can be computed with Equation (13). \[x =r\cos\theta\sin\phi \tag{1}\]5 Points closest to the vertices and centers of the polygons of three polyhedrons. (**a**) Points on the 32-hedron; (**b**) Points on the dodecahedron; (**c**) Points on the octahedron. For both the 32-hedron and the dodecahedron, the \(C_{n}\) determines the direction of the rectilinear projection. The angles are obtained with Equations (7) and (8) based on the coordinates of \(C_{n}\). The width of the viewing range is the angle between two adjacent points on the Equator.