Angles Of View

You’re designing a display system. You’ve understood the client’s needs and you’ve figured out which projector to recommend. You’ve picked out the right screen and you’ve made sure its surface won’t be too small. Now all you’ve got t o calculate is how big the projected characters and symbols need to be. Since many members of the audience will not be seated normal to screen center, you know you’ll have to consider some additional design factors if all of your client’s texts are going to be reliably legible. In an effort to help with this process, here’s:

How to Cosine a Document

We have said in earlier articles that ensuring a minimum character size is essential to the creation of readable displays. We have also worried that too many people don’t worry about what happens when they fill a projection screen with data from a computer screen.

When you sit in front of your computer, the distance separating your eyes from its monitor is typically between 20 and 24 inches. When you measure the diagonal of your display, you’ll find that it is at least 50% of your viewing distance. When you notice that you’re positioned effectively dead on axis to your screen, you’ll understand why you not only like what you’re looking at, but why you also have no trouble reading it.

It’s not clear that the manufacturers of your computer and its software had your visual comfort exclusively in mind when they combined their products in just this convenient way. But it is clear that, to date at least, the resulting geometry is fortuitous and forgiving. All of us in the large screen display business have been for years wishing we could match it.

Imagine the luxury: no seat in the house is further back than two screen diagonals and no viewing angle is greater than 20º. Within those parameters, can we make a customer’s large screen display look as good as her monitor? You bet we can.

The problem comes when she wants us to do as well when there are people seated three, four, and five diagonals away from the screen and at viewing angles which are 30º, 40º, and 50º off-axis. Can we still make it look like her monitor? For she herself maybe, seated at the head of the table or in the center of the second row. But for most of her people the display isn’t going to look at all like their monitors do even though they’re the people who’ve got to assimilate the presentation. And it’s our job to help them.

If the material being presented were old style stuff, title slides and simple graphs (but not graphics), worrying about reading it would be no big deal. But think about what you can display on your computer screen today versus what was available only a few operating systems ago. Multitasking and multimedia are the watchwords of the day and who among us can resist them?

We have established previously that if an audience member is expected to read the information displayed on a screen, the height of all lowercase characters must subtend at least 10 minutes of arc on that viewer’s retina. A less rigorous way of saying that is to state that there must be ¼ inch of lowercase character height for every seven feet on-axis viewing distance.

If that’s correct for on-axis reading, what about off-axis angles of view? How do you know what it takes to read from those? The answer is found in the following equations:



Figure 1

The first formula solves for minutes of arc for on-axis reading in still another way — it divides the product of 3438 and the character height (in inches) by the viewing distance (in inches). Hence, if the height of a symbol is, say, .65 inches and the viewing distance is 25 feet, then .65 x 3438 = 2234.7 / (25 x 12 ) = it subtends only 7.45 minutes of arc and is, therefore, not big enough. Working the numbers backward, we see that 300 x 10 = 3000 / 3438 = .88 inches of symbol height, which is big enough.

Figuring out how big to make symbols for off-axis viewing is a little more complicated. Notice that the viewing distance in both equations is along the line-of-sight. So it gets measured the same way but will always be longer for a view er positioned at the edge of a row of seats than it will be for a viewer at the center. Unlike w₁, however, w₂ depends for its size on two additional variables, a and the superscript K. As indicated, a equals the off-axis angle from which the character will be perceived. (You’ll have to wait to hear about K but, for the moment, we’ll neutralize it by assuming it to equal 1.)



Figure 2

Recalling that all of these computations are based on pretty straightforward triangles like the ones illustrated, once again, in Figure 2, we can understand that the angle a is also an internal angle of a triangle and thus will have a cosine.

If we want to, we can recall that a cosine of an acute angle in any right triangle is the ratio of the length of the adjacent leg to the hypotenuse. But we really don’t have to know that to solve the equation. Instead we can simply activate the "scientific" calculator that lurks in our computer’s Accessories directory and ask it to tell us what the cosines are of whatever angles we wish.

Thus, when we iterate through the off-axis viewing equation for the example we used above (a row of seats 25 feet back from a screen), we discover that the requisite height of a character increases as is plotted in Figure 3.



Figure 3

Slight increases are necessary when the angle is small, but by about 30º, the .88 inch symbol height has surpassed 1 inch and by 45º, the growth accelerates even faster, exceeding 2 inches by 60º. Even though this is fairly intuitive, it is extremely helpful to see exactly how the relationships between the variables may be controlled.

That leaves us to discuss that mysterious exponent,K, in the second equation. Interestingly enough, it turns out that the human eye doesn’t respond to the demands of off-axis viewing exactly according to the cosine law. Instead , it does just a little better than that. Thus K, which we’ll call the human compensation factor, is actually equal not to 1, but to about .9.

On the other hand, Dr. Joy Ebben, who was one of the first to work all of this out (and to whom we are indebted for Figures 1 and 2), is cautious about letting the K factor influence her design calculations. It’s true, she says, that the eye performs a little better than you think it will at large angles; but relying on that fact removes even a modest fudge factor from both the computations and their outcome. Thus, she counsels, let K = 1. (It certainly makes the calculation easier.)

Large viewing distances and large viewing angles combine to make small characters unreadable. That much we’ve always known. But, if we are to keep our display systems from failing their purchasers, we now have to know more. Arduous though it may be, the results of a careful and mathematical analysis are now always worthwhile. Indeed, without them, display systems can fail and there could be no end in sight.