Angles Of View
As we have observed previously, there are many different perspectives from which one may contemplate and analyze display systems. Some of these concentrate on projector considerations, lumen output or requisite throw distance, for example. Others focus on screen considerations, aspect ratio or gain, for instance. Still others pay attention to audience parameters, widths of the front and back rows, and so forth. One concept connected to all of these is the idea of half-angle. This article seeks to present some ways of thinking about and
Figuring Half-Angles - An Idea Worth IllustratingJust as all manufacturers of projection devices want to show us their lumen outputs, so, too, are we screen manufacturers similarly eager to expose our various surfaces' gains. Although both specifications intend to reveal something useful about their devices' "brightness", they each in fact are greatly improved by elaboration.
In the case of projection output lumens, the emendation has been the conversion to the concept of ANSI Lumens and its requisite averaging of nine multiple readings taken throughout a projector's full field. (As an aside, it is perhaps worth noting that "ANSI Lumens" is rather a misnomer. None of the ANSI IT7 standards refers to "ANSI Lumens". To be technically correct, the unit should actually be called "ANSI flux." But don't hold your breath.)
For its part, the concept of screen gain has undergone no such evolution and remains the consequence of only a single pair of on-axis measurements. While that may have been of some practical use back in the dark, dim days of early video projection, today it is by itself inadequate as a basis for skilled screen surface selection. What is missing (at least overtly) from the specification of a screen's gain, you see, is any description of its dispersion. This is how come all responsible screen manufacturers publish, in addition to gain specs, gain curves.
Figure 1 is a representative example of such a curve and describes a screen whose gain happens to be 1.5. By looking at the graph, we can discover all of this screen's additional dispersion properties and clearly see that its 1.5 gain is measurably reduced for all viewers positioned anywhere off its central projection axis. Even more usefully, for any given viewing position we can gauge what exactly that new, lower gain will be.
In this particular case, we can determine that viewers looking at the center of the screen from an angle of about 25º will see that center as being just half as bright as will other viewers seated exactly perpendicular to it. 25º, then, is the established half-angle for this 1.5 gain, neutral gray, rear projection screen.
Why we care about determining a screen's half-angle rather than, say, its quarter-angle or its two-thirds angle, results from the physiology of our own visual systems. It turns out that as long as a diminution in perceived brightness across a display screen is fairly smooth and not discontinuous, our eye:brain interfaces will not really take notice. Past the threshold where the perceived brightness is reduced by more than 50% , however, we can (and will) see those portions of the image as being dimmer. The calculation of each screen surface's half-angle, then, is what underlies the issuance of manufacturers' recommended viewing angle or viewing cone specifications.
Although all of this may seem straightforward enough, there are a few consequences of these facts which merit further scrutiny. In all of the diagrams which follow, the screen surfaces are arbitrarily assumed to have half-angles of 30º. The choice to make them rear projection displays is equally arbitrary and has been made only so that projector and viewer can graphically be kept spatially distinct. The principles here illustrated would be equally valid for any front screen with gain.
Figure 2 shows a plane view of the half-angle idea and, if we think about it, a reasonable design conclusion is that we ought to try and have all of our audience positioned inside the 60º cone formed by the two 30º half-angles. As with the measurement of gain itself, however, we need to note that all we have measured so far is the centermost light ray from the projector and perhaps there are others also worthy of our attention. So let's have a look at Figure 3.
Here we trace another light ray, the one passing through an outside edge of the screen, and we note that if we are to position a viewer within its 30º half-angle, we have to seat him inside a much smaller and asymmetrically off-axis cone.
Or do we?
To find out for sure, lets look at another couple of sketches to see what light they can shed upon the question. Figure 4 illustrates that the brightest light ray which our off-axis viewer can (will) see is not the centermost light ray but is in fact whichever is the ray aimed by the projector directly into his eyes. The amplitude of that ray defines the brightest portion of the image as seen by this off-axis viewer and, thus, the half angle for that viewer will be ± 30º on either side of that ray.
A common real life scenario which should make this point particularly clear is the case of an operator seated in a command-and-control facility which often include a matrix of screens more than one row high. Figure 5 shows in elevation the inclination of the centermost light ray in the vertical axis and its 30º half-angles. Note that their bisection is not perpendicular to the screen.
Figure 6 adds an upper screen which is geometrically identical to the lower but, because of its greater elevation is seen much differently by the operator. Notice particularly that the path of the centermost light ray emanating from the upper projector travels well above the operator's head .
At first glance this significant brightness differential between the two displays might appear visually insuperable. But in fact it is not. What saves the day is that the operator can't regard both screens at once. When he directs his attention to the bottom screen his viewing geometry is described by Figure 5. But when he looks up at the top screen, what he sees is instead Figure 7.
Now, in Figure 7, what are the critical areas of the screen for the designer to try to keep within 50% of each other's brightness? Obviously the answer is at the top and at the bottom. The center and whatever may be its specified gain become, for this analysis at least, irrelevant. It is the bottommost light ray which is directed straight at this viewer's eyes and it is the topmost ray (traveling parallel) to the ceiling which is directed farthest away.
Half-angles, then, can be looked at from many vantage points and usefully measured in many ways. While a gain curve (Figure 1) can tell us how to make Figure 2, note that its sketch results exclusively from the positioning of the projector relative to the screen. Figures 3 through 7, on the other hand, extend to include and account for viewers and, once again, their various and varying angles of view.