## 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
\[\begin{split}\theta_{Kr}&=\arctan\biggl{(}\frac{1 }{2f\theta}-\tan\theta_{Km}\biggr{)},\ \ \ \theta_{Km}\leq\arctan\biggl{(}\frac{1}{2f\theta}\biggr{)}\\ \theta_{Kr}&=\text{Nonavailable},\ \Equation (4) shows that the reflection angle, \(\theta_{Kr}\) depends on the incidence angle, \(\theta_{Km}\). In particular, in case the incidence angle of the ray becomes less than or equal to the value of \(\arctan(1/(2\#))\), the focal point exists within the mirror pitch of \(P\). Otherwise, the focal point may be placed beyond the mirror pitch of \(P\), which means that dual-viewing zones cannot be properly generated in this case. Thus, the maximum value of one-half of the diverging beam angle of the projector must be less than or equal to \(\arctan(1/(2\#))\). Here, it is noted that the viewing zone is defined as the movable range in the viewing space where the viewer can see a full-resolution image [31]. Figure 8 shows an optical configuration for analyzing the dual-viewing zones of the proposed system generated from the DV-EIA and CMA under the far-field condition. As mentioned above, the DV-EIA is generated by a combined use of the L-EIA and R-EIA with the parameter of \(\alpha\). As seen in Figure 8, the red and blue areas represent the ray zones reflected from the L-EI and R-EI, respectively. In addition, black rays represent the boundary rays of the left and right viewing zones. When the total number of convex mirrors is set to be \(2K+1\), each of the left and right viewing angles can be decided by the \(K\)th and \(-K\)th \(CM\), respectively. The left viewing zone can be generated by overlapping of all L-EIs of the DV-EIA and its boundary can be decided by the virtual propagation of the _Ray_-2 on the \(K\)th convex mirror and boundary ray on the \(0\)th convex mirror, whereas the right viewing zone is generated by overlapping of all R-EIs of the DV-EIA and its boundary is decided by the virtual propagation of the _Ray_-2 on the \(-K\)th convex mirror and boundary ray on the \(0\)th convex mirror. Thus, the left and right viewing angles of \(\theta_{left}\) and \(\theta_{right}\) can be given by Equation (5). \[ \begin{split}\theta_{left}&=\theta_{Kr}+\arctan \left(\left(\alpha-\frac{1}{2}\right)^{P}_{f}\right)\\ \theta_{right}&=\theta_{Kr}+\arctan\left(\left( \frac{1}{2}-\alpha\right)^{P}_{f}\right)\end{split}  \tag{5}\]  
As seen in Equation (5), viewing angles depend on the value of \(\alpha\). That is, as the value of \(\alpha\) increases, the relative portion of the L-EI in the DV-EI increases, which results in an increase of the corresponding left viewing angle. Whereas when the value of \(\alpha\) decreases, the right viewing angle gets extended. For the case of \(\alpha=0.5\), the left and right viewing angles become exactly the same, which corresponds to the conventional display system. In Figure 8, \(D\) represents the distance from the CMA for the left and right viewing zones to be separated along the \(z\)-direction in the real space, which can be derived by Equation (6). \[D=\frac{(K-0.5)P}{\tan(\theta_{Kr})} \tag{6}\]  
Figure 8: Optical configuration for analyzing the dual-viewing zones of the proposed system. As seen in Equation (6), the separation distance of \(D\) mostly depends on the pitch of the convex mirror (\(K\)) and the reflection angle of the _Ray_-2 on the \(K\)th convex mirror (\(\theta_{K\prime}\)). Thus, viewers standing beyond the separation distance in the real space can see each of those reconstructed 3-D objects on the two different directions.