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Abstract
A method to construct a virtual annular projector array that acts as numerous light sources to produce 360°-viewable 3D images on a round table is proposed. The conventional method requires multiple projectors and a conical screen for its 3D imaging principle but is limited physically by the projector arrangement. The proposed approach significantly increases the number of projectors virtually by inserting cylindrical mirrors into the optical paths used in the conventional method. This paper describes the multiplication principle and a prototype 3D display produces 3D images that are approximately 10 times denser than those produced by the conventional method.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Glasses-free tabletop 3D displays have been proposed in several different forms [1–8]. Unlike volume-swept displays, which reproduce 3D images as if they were located inside a showcase [9–11], tabletop 3D displays make virtual 3D objects appear as though they are on an empty flat tabletop surface. These displays generally have an annular viewing area located above the table and present an image perspective on the tabletop surface for each viewing direction.
Display panels and projectors are the major candidate technologies for tabletop 3D display implementation. The display panel-based approaches have generally used an optical element array composed of lenslets [2] or diffraction gratings [3] to control the directions of travel of the light rays. Projector-based approaches have been proposed that used: (1) a few high-refresh-rate projectors combined with a rotating optical component [4–7], and (2) multiple projectors combined with a static optical component [1,8].
An advantage of approach (1) is that it requires a small number of components for implementation. However, it is generally difficult to display video-rate 3D animation with natural color depth using high-speed devices, and the mechanical parts are both bulky and noisy. In contrast, approach (2) offers the advantageous ability to display full-color interactive animations without the need for such devices.
Our previously developed tabletop 3D display [1] represents one instance of approach (2). This display produced 360°-viewable 3D images on a round table using a conical screen and 288 projectors that were placed beneath the table. This principle allows viewers located at any viewpoint to observe the light projected onto the screen by several projectors positioned in front of the viewers, and the light produced by each projector is observed as a slit-like image on the screen. The observed image on the retina is then formed as an integration of a plurality of these slit-like images in a row. Therefore, the number of slit-like images that appears in the observed image affects the final 3D image quality, and this number varies with the number of the projectors present in front of the viewers.
In this paper, we propose a method to construct a virtual annular projector array that increases the array density significantly by inserting cylindrical mirrors into the optical paths between the projectors and the screen. This paper first describes the principle by which the proposed method can increase both the number and the angular density of the circularly arranged projectors virtually. Then, a practical implementation of the scheme is demonstrated and several experimental results are presented. Finally, we confirm that the number of slit-like images in the observed 3D image increases in accordance with the virtual multiplication of the light sources.
2. Slit-like images observed on the tabletop 3D display
2.1 Conventional 3D imaging principle
Figure 1 illustrates the 3D imaging principle of the 360°-viewable tabletop 3D display described in [1] (hereafter referred to as the conventional method).
Fig. 1. 3D image reproduction via the conventional method (Ref. [1], Fig. 1). The right side depicts only three representative projectors as examples, but approximately 20 projectors were present in this area in the practical implementation in [1].
The conventional method uses a concave conical rear-projection screen with multiple projectors arranged around it in a circle. The screen acts as an anisotropic diffuser in that it diffuses the incident light widely in the vertical direction only, while diffusing the light only slightly in the horizontal (circumferential) direction. This means that when an image is projected onto the outer surface of the conical screen and observed through this screen, only a vertical slit-like portion of the image (i.e., the slit-like image) can be seen by the viewer. Because multiple projectors are located around the circumference, an eye situated at any viewpoint observes several sequential slit-like images that are derived from each light source. For example, an eye located at ${E_a}$ can only see light rays along the directions of the red solid lines coming from each projector. Simultaneously, a separate eye at ${E_b}$ can see the different light ray groups along the green dashed line directions from each projector.
2.2 Problem: bokeh and crosstalk caused by insufficient light source density
To reproduce any 3D point light source at ${P_a}$, all light rays that pass through ${P_a}$ and exit in any direction should be presented ideally. This principle can be used to form the 3D surfaces of the virtual objects by preparing a sufficient number of projectors and conveying the appropriate light properties in each slit-like image. The conventional tabletop 3D display contained 288 projectors that were arranged at equal intervals of 1.25°. However, the observed image of a 10-cm-wide 3D scene was composed of only 21 slit-like images in the configuration used in [1].
To analyze the relationship between the projector array configuration and the number of slit-like images observed in the 3D images, we also developed a simulator for this 3D imaging principle [12]. Figures 2(a)–(c) show the simulation results for the observations of [1].
Fig. 2. Simulated observation results for slit-like images when using the conventional method. The values shown are the angular intervals between the circularly arranged projectors in the simulator described in [12]. The ordinary observation provided by the conventional method [1] looks like (b) because of blurring of the image in (a) by diffusers. Actual observation results showing crosstalk can also be found in Fig. 2 in [12].
If the horizontal diffusion power of the screen and the angular interval between the projectors were mismatched, black grating lines would appear among the slit-like images, as shown in Fig. 2(a), because of the lack of light sources between them. To fill these gaps, the conventional method used an approach in which diffusers were applied to broaden each of the slit-like images (Fig. 2(b)). However, this approach is essentially inadequate because the diffusers blur the observed images and produce considerable bokeh and crosstalk (Fig. 2(c)).
The ideal approach to address this problem is to increase the number of projectors used and arrange these projectors densely around the same circle. Figures 2(d)–(f) show the simulated observation images obtained when the arrangement interval is narrower than that in Fig. 2(a). These results suggest that preparation of denser arrays of light sources per unit angle increases the number of slit-like images that appear in the observed image and thus produces shaper and clearer 3D images.
However, the physical size of each projector sometimes limits the ability to arrange the projectors more densely. Use of a longer projection distance may ease this limitation, but more space will be required for the array and a round table that is excessively large is inconvenient for practical use. Additionally, increasing the number of projectors simply raises the display fabrication cost. For example, the 3D images shown in Figs. 2(a) and (b) can be produced using only 288 projectors, but the images in Figs. 2(d)–(f) require 480, 720 and 1,440 projectors, respectively. Therefore, this approach does not provide a practical solution.
2.3 Solution: increase the number of slit-like images rather than use diffusion
This paper proposes another novel approach as a solution to this problem; it involves construction of a dense virtual projector array by introduction of cylindrical mirrors.
A similar idea that used cylindrical mirrors for a tabletop 3D display was also proposed in [13], but the aim in that case was to generate 360°-viewable 3D images using the same number of slit-like images as the conventional method while also using a minimum number of projectors. Although 360°-viewable 3D images were produced using only 24 projectors in that case, additional diffusers were also required for the observation stage. Therefore, the observed images were still blurred and the crosstalk problem remained, as shown in Fig. 2(b).
In contrast, this paper aims to produce sufficiently dense slit-like images that do not involve crosstalk, as shown in Fig. 2(f). We also investigate the principle by which the proposed method increases the number of light sources virtually and demonstrate the effectiveness of this method.
3. Multiplying the number of light sources virtually
3.1 Projectors that contribute to the formation of 3D images
We outline the principle by which virtual projector arrays with more than the prepared number of projectors are organized by insertion of cylindrical mirrors. See also the Appendix for further details.
Figure 3(a) illustrates the relationship between the reproduced 3D images and the range of the projectors that contribute to formation of the 3D images. In the conventional method, when an arbitrary viewpoint ${P_e}$ on the annular viewing area and the screen shape were determined, the spherical displayable volume V, which guarantees that all 3D images contained inside that volume can be observed from any direction, can then be obtained [14].
Fig. 3. Configurations and optical paths for reproduction of 3D images in the conventional and proposed methods. Note that these are schematic representations of the horizontal plane view and the vertical direction is not considered here for simplicity. The light rays appear to be reflected only at the inner surfaces of the cylindrical mirrors because the projectors are positioned below the mirrors and cast the rays onto the inner surfaces of the mirrors in the practical system (see also Fig. 4).
Any surface on the 3D shapes inside V can be assumed to be a cloud composed of an infinite number of point light sources reflecting the ambient light rays. Then, a proportion of these light rays penetrates V and reaches the eye location at ${P_e}$. By computing all traveling paths of these rays and conveying the exact light properties appropriately, it is ultimately possible to represent the identical state of this light field as if the 3D objects are present, and a stereoscopic effect is obtained because of existence of binocular disparity.
In Fig. 3(a), multiple projectors that act as sources to represent these rays are arranged in a circle. Here, by considering the ray ${L_0}$, which is tangential to V, we can see that only the projectors arranged within the range of the red arrow ${A_0}$ can contribute to the formation of the observed image at ${P_e}$. For example, it can be considered that 21 of the 288 projectors existed in the range of ${A_0}$ in the prototype in [1].
3.2 Function of cylindrical mirrors
In the proposed method, cylindrical mirrors that share a central axis with each of the above circles are inserted into the optical paths used in the conventional method.
Figure 3(b) shows an example in which Mirror 1 is inserted. In this new configuration, a group of rays passing through V still contributes to the formation of the observed image at ${P_e}$, but they depart from different locations within the projector array. These locations are then obtained by back tracing of the rays. For example, ${L_0}$ intersects with Mirror 1 at ${M_1}$ and the corresponding reflected ray ${L_1}$ reaches the point ${P_1}$ on the projector array. This means that when light ray ${L_1}$ conveys the same information as ${L_0}$, the virtual light rays of ${L_0}$ that are emitted from ${P_0}$ can be generated by projection from ${P_1}$. It thus becomes obvious here that a group of rays that passes through V can be produced using the projectors arranged within the range of the green arrow ${A_1}$.
Figure 3(c) shows another example in which Mirror 2 is added and then two reflections occur. By considering the light ray ${L_2}$, which is the reflected ray of ${L_1}$ at ${M_2}$ on Mirror 2, and the intersection point ${P_2}$ between ${L_2}$ and the projector array, we find that the blue arrow ${A_2}$ represents the range that contributes to the formation of V in the two-reflection case.
3.3 Multiplication ratio of the virtual projector array
The center angles ${\theta _0},\,{\theta _1}$ and ${\theta _2}$ are proportional to arcs ${A_0},\,{A_1}$ and ${A_2}$, respectively, in Fig. 3. These angles are expressed as Eqs. (1)–(3) using $\theta $, which is half of the viewing angle of V, and the angles ${\alpha _1}$ and ${\alpha _2}$, which are formed with respect to the normals of the reflected and incident light rays, respectively. Note that ${\phi _0},{\phi _1}$, and ${\phi _2}$ are equal, as described in the Appendix, so these angles are expressed using the common symbol $\phi $.
These expressions show that the range of the projectors is expanded by an amount related to double the reflection angles. It is obvious that more projectors will exist within the expanded range if the array spacing remains the same, and the number of light sources that will contribute to the formation of V will then increase.
The multiplication ratios of the virtual light sources are expressed as ${\theta _1}/{\theta _0}$ and ${\theta _2}/{\theta _0}$. These factors can be manipulated by varying the radii of the cylindrical mirrors, where smaller radii yield larger values of both ${\alpha _1}$ and ${\alpha _2}$.
4. Implementation
4.1 Configuration of the prototype tabletop 3D display
Our prototype demonstrates that the number of slit-like images that appears in the observation image is actually increased by adding only the cylindrical mirrors to the configuration used for the conventional method. The conical screen (10 cm radius, 12 cm depth) and the array of 288 projectors (liquid crystal on silicon (LCoS), 960×540 pixels, 7.4 mm wide) were the same as those used in [1].
The substrate for the cylindrical mirrors was an off-the-shelf glass pipe with a thickness of 5 mm. We selected two different pipe sizes with better conditions (lower distortion and fewer scratches). To create a mirror, aluminum was deposited on the inner surface of the pipe, and a white color reflectance rate of more than 85% was achieved.
Figure 4 shows the arrangement of these components. The overhead camera is used for the calibration process, which is described later in the paper.
4.2 Appropriate size for cylindrical mirrors
To use as many of the prepared projectors as possible, it is necessary to widen arcs ${A_1}$ and ${A_2}$. In other words, the radii of the cylindrical mirrors should be minimized to enlarge both ${\alpha _1}$ and ${\alpha _2}$. However, the mirror that casts the reflected light rays directly onto the screen must at least be larger than the screen diameter. Additionally, the mirror that receives the light rays projected from the projectors directly must be large enough to reflect all the light rays that reach the screen.
Under these restrictions, the single mirror configuration was not a suitable solution because ${A_1}$ can hardly exceed 180°. However, use of two or more mirrors relaxes these constraints. It is then easy for ${A_2}$ to be more than 180° and close to 360°.
Using these considerations as a basis, our prototype was implemented using the two-reflection condition and the radii of Mirrors 1 and 2 were selected to be ${r_1} = 12$ cm and ${r_2} = 7.5$ cm, respectively, because of the commercial availability of the glass pipes required.
4.3 Measurement of traveling light rays and light-field rendering
As the basis for the light-field reproduction, each pixel that is projected from a projector corresponds to a specific light ray traveling in the 3D space. However, some positioning and orientation errors must occur when the components are being arranged and it is impractical to attempt to eliminate these installation errors manually. Additionally, a brief software calibration such as a linear interpolation based on a few sampling points is unlikely to work well because the rays are reflected from the curved surface several times before arriving at the screen. Therefore, we measured all light ray vectors in the prototype in the same manner as the measurements in [13].
For the measurements, a flat rear-projection screen is first placed on the tabletop surface, rather than the conical screen. When a projector casts a pixel, a bright spot appears on the flat screen after two reflections and this is captured by the overhead camera. The 3D coordinates of the spot can then be computed as ${X_1} = ({{x_1},{y_{1,}}0} )$ based on the camera parameters and the level of the known plane. Another coordinate point ${X_2} = ({{x_2},{y_{2,}}20} )$ is also obtained by changing the level of the flat screen to $z = 20$ mm. Finally, each 3D ray vector corresponding to a specific pixel is computed using ${X_2} - {X_1}$ by iterating this measurement procedure for all the pixels.
Thereafter, the projection images for the 288 projectors can be calculated in the same manner used in the conventional rendering method [14].
5. Experiments
5.1 Results for measured light ray vectors
First, the measured light ray vectors are visualized as shown in Fig. 5. We computed each projection image by drawing a grid with 2 cm intervals on a horizontal plane using the vectors. Figure 5(a) shows a projection result for the images from 288 projectors and Fig. 5(b) shows an example of the source image.
Fig. 5. Measurement results for optical paths. (a) Grid pattern drawn using the measured light ray vectors on the horizontal plane. (b) Source image from a projector used to draw the grid.
In this prototyping approach, only the pixels located inside the heart-shaped area shown in Fig. 5(b) can be projected onto the conical screen. The screen area was approximately 400×200 pixels and the other pixels must be black to prevent production of stray light rays. Although many curved lines are shown in Fig. 5(b), the result shown in Fig. 5(a) is composed solely of straight lines from the provided grid.
These figures also show that multiple reflections from the curved mirrors allow separate pixels from the source image to pass through the same point on the horizontal plane. For example, magenta pixels appear in the upper left and lower center areas in Fig. 5(b), but these pixels comprise the same right-side grid that is shown in Fig. 5(a). Note that the rays that pass through at the same point go outward in different directions.
These results allowed us to confirm that the light ray vectors in the present state of the hardware were obtained correctly and could thus be used appropriately for rendering applications.
5.2 Observation results for slit-like images and confirmation of parallax
In the next step, the conical screen was settled at the center of the table in place of the flat screen and the source images to illuminate the surface area of a 10-cm-diameter 3D sphere were projected. To allow the principle of the proposed method to be understood easily, each of the projection data were painted in different colors according to the relevant projector’s location. For example, all pixels that penetrated the sphere from the projector located at 12 o’clock were all depicted in red and the data from the projectors located at 4 and 8 o’clock were depicted in green and blue, respectively. A gradated color scheme was assigned in between the projectors.
Figure 6(a) shows the experience of an observation of the prototype tabletop 3D display from the 6 o’clock direction. Under this operating principle, the viewer sees the inner surface of the conical screen from an oblique angle above. The dotted circle indicates the displayable volume V.
Fig. 6. Results of projection of colored data on the conical screen. (a) Overview of the screen without projection. (b) View of the image produced at 6 o’clock. (c) Corresponding view at 9 o’clock.
Figure 6(b) shows the observation results obtained from the projected color data that were photographed at the same viewpoint used in Fig. 6(a). We can see colorful lines inside the same dotted circle and the pattern is very dense because it is composed of uncountable slit-like images without black gratings. The appearance of color gradation implies that multiple projectors contributed to the formation of this image. As noted above, the projector at 12 o’clock projected the red data and the red portion then appeared as a narrow vertical line at the center of the photograph. The other colors were the results of projections from different projectors. Because the blue and green portions were also observed in Fig. 6(b), it was also confirmed that almost all the prepared 288 projectors were used to form this viewpoint image. As discussed in Section 6 below, the lines contained inside the circle would consist of more than 200 slit-like images in practice. In contrast, the conventional method could only generate 21 slit-like images along with black gratings inside the same area, as shown in Fig. 2(a).
Figure 6(c) was photographed from the 9 o’clock viewpoint and it confirmed that the gradation had shifted to the left. Note that the gradation pattern does not represent a texture on the virtual 3D sphere. This shift to the left means that the projector at 12 o’clock that projects the red sphere was in charge of the left area at the viewpoint shown in Fig. 6(c), although it was at the center in the viewpoint shown in Fig. 6(b). The vertical length of the red slit-like image had also changed, thus indicating that the different pixels of the projection source images were observed from different viewpoints.
Based on these results, it is expected that 3D images can be reproduced by computing the light field and then providing the appropriate light properties to each pixel.
5.3 Reproduced 360°-viewable 3D images
Finally, several results for the reproduction of 3D images are presented.
The 3D scenes represented in the images contained some virtual objects (a color checkered cube or three objects) on a blue specular disk. In this prototype system, more than 200 slit-like images can be seen continuously in the horizontal direction and because each slit contained approximately 200 pixels in the vertical direction, the resolution of the observed 3D images was considered to be close to approximately 200×200 pixels. For reference, the rendered images of the scenes using 200×200 pixels are shown in Fig. 7.
Figures 8 and 9 show the observation results of the reproduced images photographed from the 8, 6 and 4 o’clock directions around the tabletop 3D display. The observation results for the full 360° around the table are also presented in Visualization 1.
In these photographs, 3D images of dense slit-like images with reduced crosstalk were observed without the use of additional diffusers or increases in the actual number of projectors used, as simulated in Fig. 2. These reductions in crosstalk were also confirmed by comparison with the 3D image obtained via the conventional method and presented in Fig. 2 in [12]. In addition, motion parallax could be confirmed in the circumferential direction of the table because the relationship between the three objects changed appropriately in each directional observation.
During the shooting of these images, three projectors were unlit, but these projectors provide a good example to explain the principle of the operation. For example, one of the projectors was located at approximately 12 o’clock and a thin black line can be seen at the center of the middle-column photographs; this is the actual width of one slit-like image in this prototype setup. In the right-column photographs, the black line moved to the right because the area in charge of the projector was changed by the movement of the viewpoint. These black lines can also be seen in Visualization 1 and may help in understanding the situation in this case.
6. Discussion
The results above verified that the proposed method can increase the number of slit-like images substantially during observation, despite the fact that it used the same projector array that was used in the conventional method. The multiplication ratio can be modified easily via the hardware settings and calculated using the relationship presented in the Appendix.
The original projector array in [1] arranged 288 projectors at 1.25° intervals around a circle with ${r_p} = 34$ cm. When a V with ${r_v} = 5$ cm is seen at an observation distance of ${r_e} = 65$ cm, the following values apply: $\theta = {4.4^ \circ },\,\phi = {8.4^ \circ }$, and $2{\theta _0} = 25.7^\circ $. Therefore, the number of projectors arranged in ${A_0}$ was approximately 21 (25.7°/1.25°).
In the experiment in this work, the proposed method involved introduction of mirrors with ${r_1} = 12$ cm and ${r_2} = 7.5$ cm to the projector array. In this case, the relationship in the Appendix indicates that ${\alpha _1} = 23.9^\circ $ and ${\alpha _2} = 38.2^\circ $, so $2{\theta _1} = 121^\circ $ and $2{\theta _2} = 274^\circ $. It can then be estimated from these parameters, that the slit-like images seen in Figs. 6, 8 and 9 are composed of 219 (274°/1.25°) lines.
By adding these mirrors, the virtual projector array densities are multiplied by ${A_1}/{A_0} = 4.7$ and ${A_2}/{A_0} = 10.7$ times when compared with the original. In other words, 1,354 or 3,082 projectors would need to be arranged in a straightforward manner to achieve the same conditions when using the conventional method.
When using these 3D imaging principles, the horizontal resolution of the observed image is dependent on the number of slit-like images used. The number of horizontal pixels (directional light rays) cast by each projector was the same (400 pixels) in both the conventional method and the proposed method because identical projector arrays were used in the two cases, and all rays were designed to pass through the V of a sphere with a diameter of approximately 10 cm. In the conventional method, the number of pixels that appeared in one slit-like image was approximately 19 (400 pixels/21) pixels, but it should be noted that this image also included considerable bokeh and crosstalk caused by the diffusers. In the proposed method, the pixel number was 1.8 (400 pixels/219) pixels with reduced bokeh and crosstalk. To obtain a sharp observation image equivalent to 400 pixels in the horizontal direction, it is necessary to generate the same number (i.e., 400) of slit-like images. To achieve this, the conventional method would require 5,603 (360°/(25.7°/400)) projectors. However, the proposed method can generate the required image easily by simply modifying the sizes of the cylindrical mirrors and the number of reflections according to the equations derived in this work.
A primary advantage of the proposed method is that the density of the slit-like images can be increased simply and virtually without changing the basic hardware specifications. The same rendering approach is also available, so the prototype can achieve these higher-density results even when the same number of rendering computers is used.
Another advantage of the proposed method is that the observation style remains the same as before because the configurations above the table remain unchanged, while the viewer simply feels that the number of projectors has increased. Although there is a limitation where more vertical space is required underneath the table to accommodate the height of the mirrors, this space was 55 cm in the present experimental conditions. This is sufficiently lower than the ordinary table height of 70 cm and implementation in the manner of a round table is still easy.
This method does not require the use of additional diffusers and many slit-like images can be seen directly, but the drawback that individual differences and artifacts were conspicuous is also clarified. For example, an LCoS projector generally shows considerable color deviation; the factory-guaranteed color temperature of our projectors was approximately 6,500 ± 1,000 K. Additionally, residual errors in the measurements of the light ray vectors generally cause misalignments between the slit-like images. The conventional method used diffusers to fill in the black portions observed in the final 3D images, as shown in Figs. 2(a)–(c). This procedure acts as a Gaussian filter, so these noises were absorbed and were not obvious in the prior system. These issues can be resolved by performing color calibration and accurate system measurements.
7. Conclusion
This paper has proposed a new method to provide a sufficiently dense light source array to enhance the image quality of a 360°-viewable tabletop 3D display consisting of a circular projector array and a conical screen. The proposed method can construct a virtual projector array by introducing cylindrical mirrors into the system to multiply the number of projectors to exceed the original number of prepared units. As a result, the number of slit-like images that appeared in the observations increased remarkably. The method was demonstrated using a prototype system and 3D images with reduced crosstalk could be obtained without use of additional diffusers.
This paper also clarifies the principle by which a virtual projector array is organized such that the multiplication ratio can be modified arbitrarily based on the number of cylindrical mirrors and their radii. The proposed implementation could build a virtual projector array that is composed of approximately 10 times as many units when compared with the setup for the conventional method and 3D images composed of more than 200 slit-like images were reproduced successfully.
Although the prototype demonstrated the functionality of the proposed method well, a few problems remained. As one of the image quality criteria, the crosstalk issue will be resolved using this approach, but further improvements in the image quality such as enhanced resolution and sharpness could not be compared with the corresponding properties of the conventional method at this stage as a result of limitations caused by the hardware available to perform the experiment. For example, one slit-like image obtained from the method of [1] could contain approximately 400 pixels vertically, although the image was blurred. However, the prototype could project a limited number of approximately 200 pixels onto a screen of the same height because the projection distance was extended. This limitation can be resolved simply by fabricating a custom projection lens system, but this was too costly for the proof of principle experiment. This system will be optimized as part of the future development of the method and more detailed studies to address the required image quality improvements will then be conducted.
Appendix
In the conventional method, the relationship between the spherical displayable volume of the 3D image ($V$), the circle of the projector array and a viewpoint ${P_e}$ located on the annular viewing area is illustrated as shown in Fig. 3(a); the radii of the corresponding circles in this case are ${r_v},\,{r_p}$ and ${r_e}$, respectively, and the common center is O. The light ray vector ${L_0}$ is tangential to V and crosses the projector array at ${P_0}$. ${P_c}$ is the projector located straight in front of the viewpoint, and ${P_e},\,O$, and ${P_c}$ are all located on the same line.
In this configuration, when the angles $\angle O{P_e}{P_0},\,\angle {P_0}O{P_c}$, and $\angle {P_e}{P_0}O$ are labeled as $\theta $, ${\theta _0}$ and ${\phi _0}$, respectively, the angular range of the projector array $\angle {P_0}O{P_c} = {\theta _0}$ is expressed using Eq. (4) from the exterior angle theorem for the triangle $\triangle O{P_e}{P_0}$.
The proposed method involves the insertion of cylindrical mirrors into this configuration. Figure 3(b) illustrates the result of the addition of Mirror 1 of radius ${r_1}$ to the system in Fig. 3(a). When ${\alpha _1}$ represents the equal angles of incidence and reflection of ${L_0}$ with respect to the normal at ${M_1},{P_1}$, is the point reached by the reflected ray ${L_1},P_c^{\prime}$, is the projector position opposite to that of ${P_c}$ and ${\phi _1}$ is the angle of $\angle {M_1}{P_1}O$, where the angular range $\angle {P_1}OP_c^{\prime} = {\theta _1}$ is expressed easily using Eq. (5) from the summation of the two exterior angles $\triangle O{M_1}{P_e}$ and $\triangle O{M_1}{P_1}$ by considering an extension line from the normal at ${M_1}$.
The results of the two reflections produced by adding Mirror 2 at the radius ${r_2}$ to the setup in Fig. 3(b) is derived in the same manner. In Fig. 3(c), ${M_2}$ is the reflection point of ${L_1}$ on the second mirror and the reflected ray ${L_2}$ reaches ${P_2}$ on the projector array. ${\alpha _2}$ is the equal angles of incidence and reflection, and ${\phi _2}$ is $\angle {M_2}{P_2}O$. Then, $\angle {P_2}O{P_c} = {\theta _2}$ is expressed using Eq. (6) based on the summation of the three exterior angles $\triangle O{M_1}{P_e},\triangle O{M_1}{M_2}$ and $\triangle O{M_2}{P_2}$.
Here, by considering three triangles that have each vertex angle of ${\phi _0},\,{\phi _1}$ and ${\phi _2}$ and bases that are perpendicular from O to ${L_0},\,{L_1}$ and ${L_2}$, we realize that these are congruent right-angled triangles with the same side lengths; i.e., ${\phi _0} = {\phi _1} = {\phi _2}$. Using the common label $\phi $ for these angles, Eqs. (1) to (3) are then obtained.
Finally, the angles $\theta ,\,\phi ,\,{\alpha _1}$ and ${\alpha _2}$ are obtained as follows: $\theta = {r_v}/{r_e},\,\phi = {r_v}/{r_p},{\alpha _1} = {r_v}/{r_1}$ and ${\alpha _2} = {r_v}/{r_2}$.
Disclosures
The author declares no conflicts of interest.
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