Angles Of View

If projected displays aren't hard to see, they sure are hard to read sometimes. Why this should be so isn't at all hard to understand once we see how hard it is to display lots of letters and numbers on a screen in ways in which people from all sorts of viewing angles and distances can still accurately make them out. Figuring out how to account for all of those viewing angles and distances is so complicated that the only way to simplify it is to resort to mathematics, the preeminent language of science and the tool by which so much of our comprehension of phenomena can be generalized. Since some attributes of visual displays can also be expressed in mathematical ways, this article, howsoever reluctantly, will be

Looking at Trig - Can I See You a Minute?

All of us in the Audio/Visual industry who continue to work hard and to work together to create better and better visual displays are discovering the task to contain more and more science and less and less instinct. If we all were pilot s, we'd be flying less and less by sight and more and more by instruments.

This development shouldn't surprise us all that much because most of us have seen it coming for a long, long time. Always remembering that ours is a business driven primarily by another, much larger industry ñ the computer industry ñ, we can easily see how very much more complex is the information that a computer can output to its screen today compared with what it could do only a few years ago.

Desktop publishing, web sites, presentation software, and multimedia in general have all become very, very sophisticated. "System Requirements" are routinely calling for 100MHz processors and 32 MB Ram as minimums just to get the programs to run. What is all that speed and memory (to say nothing of the huge amounts of space required from the hard disk) in aid of?

The answer, of course, is us. We're the ones who want to be able to pack more and more information onto our screens and we're the ones who want to be able to have those high density data communicated to lots of other people, in lots of other venues.

This series of articles has been stressing recently the critical significance of this data concentration in terms particularly of the visual task necessary to its assimilation. To put it simply (just one more time), it all boils down to this: We aren't looking at projection screens anymore. We're reading them. And reading something, as anyone who gives it a moment's thought can recognize, is much harder than looking at it.

When the material we are expected to read is projected and then reflected or transmitted by some screen, comprehension becomes downright demanding. Because the degree of this difficulty cannot be overestimated, efforts needed to manage it must be considerable.

Here's the problem: if you want to ensure that everybody looking at a display can discern and decipher all of the data projected upon it, you must absolutely guarantee that every single character and symbol is large enough to be reliably and accurately identified. This is not a question of font choice (although that is certainly important). Nor is it a question of the contrast or color pallette that is available (although these also can be vital). It is a question of size, pure and simple. If the characters and numbers aren't big enough to be read by the person positioned at the back of the room, she won't be able to figure them out (even if she squints). Period.

So, how do we ensure that all letters and symbols generated by all computers projected by all projectors onto all screens in all rooms before all audiences are big enough? We define the minimum size in a way that can be reliably generalized over all of those cases in all of their variations. What we actually say is:

The height of no lowercase character shall subtend less than 10 minutes of arc on the retina of any viewer.

That's the mathematically expressed rule and, once we understand it, we can use it in any and all cases always.

To see exactly why the concept of minutes of arc is so useful, consider Figure 1 (for which, once again, we are indebted to Joy Ebben, Ph.D.).



Figure 1

First let's dispense with the symbol a (the Greek letter alpha). That's the angle we're interested in and, for the moment, it is an unknown. (Mathematicians like to use Greek letters for unknowns. It makes them harder for the rest of us to understand.)

Now notice that there's a second a, the little one, pointing into the intersection of the two lines at the back of the eye. And that's the alpha we're really interested in. Fortunately, there's a provable theorem in Geometry which states that all vertical angles are equal and vertical angles are the two opposing pairs of angles which are created wherever two straight lines intersect. Thus, if we can find a way to measure or calculate the big alpha (a) we will automatically know the value of the little one way down there, inside our eyes and nearly impossible to measure.

Next we have to think about angles and how we see. If you get up out of your chair and, with your eyes open, slowly turn completely around, your eyes will have swept over everything in your horizontal field-of-view. Since that field is a complete circle we can define at as divisible into 360 equal slices, which we call degrees. But, since you don't have eyes in the back of your head, the only way you can "see" in all 360 directions is to turn completely around. Otherwise you can "see" only what's in front of you, which is only a part of 360°. How large is that part? Amazingly enough, it's a whopping 200° (for more on this, see Vol. III, No. 4 of this series).

But because 200° is not equal to the complete, 360° circle, we'll have to recognize that it's only a section of that circle and any section of a circle is called an arc.

When we look to see how much of that available 200° gets used by our eyes when read something, however, we discover that it's only the central part of our visual attention that gets used. It is encompassed by something less than 30°.

If we're reading a page of text, it ought, therefore, to be simple enough to divide each line by 30 and discover how many degrees are taken up by each character. Unfortunately, when we do that, we soon see that there are many more characters to a typical line than 30. Projected displays, for instance, often include lines that are 80 characters in length.

To deal with this sort of problem, mathematicians came up with a way to subdivide each degree into 60 smaller slices which, for numerically obvious reasons, they call minutes (and each minute may be subdivided into 60 arc seconds). This enables us to divide a 30° field-of-view not into just 30, but into eighteen hundred equal parts.

The final two things we have to think about there are easy; they're just vocabulary. The verb subtend is simply a mathematical word meaning to be opposite to and delimit. All it means here is that if we can measure the height of a symbol on a screen, we know that the angle (the big a ) "subtends" on the screen will be the same angle that is subtended on the tiny screen inside our eye which is called the retina (and there's the other piece of vocabulary).

Having now done with the math, we can return to our rule and notice that its terrifically useful virtue is that it will and does hold true for all viewing distances and for all screens. (This is so because the size of the little screen, the retina, doesn't change even as all the other variables may.)

The way that the number of requisite arc minutes was proved to be 10 had and has nothing whatsoever to do with calculation. It was established by long and arduous empirical research which entailed asking lots of audience volunteers, "Can you read this from there or do I need to make it bigger?" When people positioned on-axis to the projected characters stopped saying that the symbols were hard to make out, they were measured to be subtended by at least 10 minutes of arc.

For off-axis viewing 10 has to be increased because trying to read something from an oblique viewing angle is much harder than looking at it straight on. Try reading a book you're holding way out to your side and you'll see for yourself.

But as a benchmark for on-axis viewing, 10 minutes of arc is a solidly reliable minimum. If you take the trouble, incidentally, to calculate it out, you'll discover that 10 arc minutes translate to just over ¼ inch for every 7 feet of viewing distance. Hence a lowercase letter should be not less than .86 inches high if people seated 24 feet back are to be able to read it

Now that wasn't so hard, was it?