## 2 Methods
### Direct Projection of the DV-EIA onto the CMA for the DV 3-D Display
#### 2.3.2 Analysis of the Dual-Viewing Zones and Angles  
Thus, under this circumstance, dual-viewing zones and angles of the proposed system can be analyzed. If these parallel rays containing the information data of the DV-EIA, are projected into the surfaces of each convex mirror, left and right-view components of the DV-EIs are separated and reflected back into their viewing directions with their viewing angles. Then, two different 3-D object images are integrated from all those surfaces of the CMA and displayed in their viewing zones. Figure 7 shows an optical configuration of the bundle of parallel rays coming from the projector and incident to the \(K\)th convex mirror based on the ray-tracing model. As seen in Figure 7, a bundle of red-color parallel rays with information of one component of the DV-EIA, is assumed to be incident  
Figure 6: Optical geometry for the ray-optical analysis of the proposed system based on the parallel-ray-approximation (PRA) method. onto the \(K\)th convex mirror. Green-color rays represent the reflected beams on the surface of the \(K\)th convex-mirror for their corresponding incident rays, forming rays through the backward extension of the reflected ones. Here,\(f_{m}\) denotes a focal length of the convex mirror, and \(\theta_{p}\) represents the half beam angle of the projector being located far away from the CMA plane. The _Ray_-1, representing the center ray of the bundle of rays incident to the \(K\)th convex mirror with the incidence angle of \(\theta_{Km}\), is reflected back from the surface of the \(K\)th convex mirror to make the reflected ray with the reflection angle of \(\theta_{Km}\). This reflection of the incident ray is equivalent to its virtual propagation into the convex mirror as seen in Figure 7. The _Ray_-2 and _Ray_-3 also represent the parallel rays incident to both edges of the \(K\)th convex mirror. Just like the _Ray_-1 case, _Ray_-2 and _Ray_-3 are also reflected back from the convex mirror to make their reflected rays, which are equivalent to their virtually propagated rays into the convex lens. All those virtually propagated rays into the convex lens come across on the same points called a focal point of the convex lens as seen In Figure 7.  
Based on the ray-tracing model, all virtually propagated rays for each convex mirror are focused on its focal point at the focal plane of the CMA as seen in Figure 7. Since the angle of a bundle of parallel rays incident to the \(K\)th convex mirror is related to the relative position of the convex mirror (\(KP\)) and the distance from the projector to the CMA (\(L\)) under the far-field condition, the incidence angle of the center ray to the \(K\)th convex mirror, \(\theta_{Km}\) can be calculated based on ray-optics, and given by Equation (3). \[\theta_{Km}=\arctan\biggl{(}\frac{KP}{L}\biggr{)} \tag{3}\]  
Even though the incidence angle to the corresponding convex mirror can be increased as the number of \(K\) increases, its maximum angle value may be practically bounded by the diverging angle of the projector, \(2\theta_{p}\). The whole bunch of rays between the _Ray_-2 and _Ray_-3 incident to the \(K\)th convex mirror corresponds to one DV-EI of the DV-EIA. This DV-EI is composed of a pair of L-EI and R-EL. Each of the left and right viewing zones can be generated by the reflected propagations of the L-EI at the \(K\)th convex mirror and R-EI at the \(K\)th convex mirror, respectively. Thus, the virtual propagation angle of the _Ray_-2 needs to be calculated for calculating the viewing zones of the proposed system because the reflection angle of the _Ray_-2 affects the viewing angle for each viewing zone. Here, the reflection angle \(\theta_{Kr}\) of the _Ray_-2 on the \(K\)th convex mirror can be derived as Equation (4).