# Considerations of Curvature for a Near-Eye Holo-Video Display
## 3 Circular Screen
### Theory  
We now outline the principles and equations defining our system. The curved surface is placed in front of the eye, and the radius of curvature (ROC _r_) is somewhere behind the eye (anywhere between infinity and directly at the center of the eye). The flat case is the same as the curved case with the ROC set at infinity. The angles and distances for the setup are defined in Figure 7.  
With a known radius of curvature and arclength, the closest drawable distance can be found. A similar method can be used to find the other two parameters. The equations, listed below, fall out directly from the Pythagorean Theorem, the Law of Cosines, and the Law of Sines. We use these equations to create Figures 6, 8 and 9 by defining one parameter (ROC \(r\) or the closest drawable point, for example) and solving for the rest.  
\[\theta_{3}=\frac{a}{r} \tag{1}\]  
\[l=a\cdot\frac{\sin\theta_{3}}{\theta_{3}\cdot\sin\theta_{2}} \tag{2}\]  
\[\theta_{2}=\frac{\pi-\theta_{3}}{2} \tag{3}\]  
\[\theta_{1}=\pi-\theta_{2}-\theta_{MAX}\cdot\frac{\pi}{180} \tag{4}\]  
\[\theta_{4}=\pi-\theta_{2} \tag{5}\]  
\[\theta_{5}=\pi-\theta_{4}-\theta_{1} \tag{6}\]  
\[d=l\cdot\frac{\sin\theta_{1}}{\sin\theta_{5}} \tag{7}\]  
Figure 8: (**a**) In this scenario, we set maximum deflection angle at \(\theta_{MAX}=2^{\circ}\), and we place a curved  
display (arbitrarily) 5 cm in front of the eye and hold that fixed position. As the radius of curvature of the display increases past the eye, the FOV rapidly decreases until it converges to the 4\({}^{\circ}\) flat-screen case. **(b)** Here, we hold the FOV = 180\({}^{\circ}\) and the distance of the eye from the device to 1.5 cm. If we allow for the possibility of an increased deflection angle in the future, we see that, even at high deflection angles, the possibility of moving the eye from the ROC center and maintaining our high FOV is minimal.  
The variables in these equations are as follows: \(d\), as previously decided, is the closest drawable point, assuming we are using the maximum deflection allowed by the system. \(l\) is the chord underneath the length \(a\), which runs along the region subtended by \(\theta_{3}\) and represents half the allowable aperture of our system (given this closest drawable point and predetermined deflection angle). \(2\theta_{3}\) is equivalent to the FOV for the system, and \(r\) is the radius of curvature. All other angles have no practical real-world meaning and serve only to find the final variables we seek.  
Figure 10: A display with a radius of curvature (ROC) placed behind the eye will have some angular view that the eye can perceive. The point (on either side of the eye) at which this no longer is true is where the furthest ray towards the eye no longer falls on the eye.  
Figure 9: (**a**) The largest aperture allowed for the device is displayed here with respect to the radius of curvature (when \(\theta_{MAX}=2^{\circ}\)). The closest point the display can focus to is held at 25 cm (from the eye). (**b**) The closest point \(d\) the display can focus to is displayed here with respect to the radius of curvature (when \(\theta_{MAX}=2^{\circ}\)). The aperture size (created by the active region of the device) is held at 2.2 mm. (**c**) Figures 6a and 10a are 2D cross-sections of this plot; they are shown as dashed lines. The intersection point is chosen for the preferred instantiation in Section 6.