## 4 Polyhedron-Based Rectilinear Projection  
According to the properties of the two projections illustrated in the previous section, both projections incur distortions. After the rectilinear projection, the pixels near the boundary of the projected image will be stretched when the size of the viewing range is larger, specifically when the angle of the viewing range is greater than \(120\) degrees. However, the distortion of the rectilinear projected image will be reduced with appropriate decreases in the size of the viewing range. We divide the \(360\)-degree image into several areas and transform the equirectangular projected \(360\)-degree image into several rectilinear projected images. To divide the \(360\)-degree image into appropriately sized areas, we use a polyhedron-based method. Several types of polyhedrons are based on several polygonal faces, edges, and vertices. The number of projected rectilinear images is equal to the number of polygons. However, the photograph of the artwork can be taken from any angle, which indicates that the artwork in the \(360\)-degree image can be located at any position. Thus, the \(360\)-degree image must be averagely partitioned. In a previous work [75], we used \(32\)-hedron to partition the \(360\)-degree image. However, in this work, to measure the effects of the number, size, and direction of the polygons for identification, we use three types of polyhedrons for rectilinear projection: \(32\)-hedron, dodecahedron, and octahedron. Figure 4 shows three geometric models for the polyhedrons that have radii of \(1\) and the center as an origin. The \(32\)-hedron consists of \(12\) pentagons and \(20\) hexagons and is a typical model of a soccer ball. There are \(60\) vertices on the \(32\)-hedron. The dodecahedron consists of \(12\) pentagons and \(20\) vertices. The octahedron consists of eight triangles and six vertices. We define the number of polygons in each polyhedron as \(n\) and the number of vertices for each polygon as \(t\). Each vertex on each polyhedron is defined as \(vt_{j}(a_{j},b_{j},c_{j})\), where the \(j\) denotes the number of vertices in a polyhedron. The \(360\)-degree image will be divided and projected into \(n\) images. Each projected image is obtained by rectilinear projection for a portion of the \(360\)-degree image through each polygon based on its direction and size. Figure 4: Three types of polyhedrons. (**a**) \(32\)-hedron; (**b**) Dodecahedron; (**c**) Octahedron. Generally, most artwork is hung vertically on walls instead of on the ceiling or the ground. Thus, after taking 360-degree photos, the artwork is usually located above, below, or near the Equator. The regular dodecahedron is shown in Figure 5a. In this case, four polygons face toward two poles. The polygons are not efficient for rectilinear projection of the artwork. Therefore, we rotate the dodecahedron around the \(x\)-axis by 31.7 degrees (90-dihedral angle/2), so that the normal vector of a pole-faced polygon is parallel to the \(z\)-axis, as in Figure 5b. Then, only two polygons face towards the two poles. The remaining polygons can be efficiently used for rectilinear projecting of the artwork. As noted in the previous section, the further away the points are located on the plane of projection, the more distortions will occur. Accordingly, the distortions of the locations around the edges of the polyhedrons are larger than at the other locations. For the octahedron, we retain four polygons facing north and another four polygons facing south. The 360-degree image is transformed into a sphere with a radius, \(r=1\), with the center as the origin. It overlaps with the polyhedrons. Based on Figure 2b in the previous section, the rows and columns for the equirectangular projected 360-degree image can be indicated as the sphere's vertical \(\phi\) and horizontal angles \(\theta\). The spherical coordinates can be defined as \((r,\theta,\phi)\). The three dimensional Cartesian coordinates \((x,y,z)\) are represented as Equations (1)-(3).