Over on Register Hardware, they’ve just published a piece I wrote about how far away from an HD TV you should be sitting. The answer is fairly simple, and not really much of a surprise, but we decided to start with the figures from some BBC research and work it out for ourselves, from first principles.
There are a few figures in the feature, and I thought that (since I was taught at O level maths) it would be a good idea to show how we got there.
First up, I said the height of a screen, H, is given using this formula:
How did I get there? Pythagoras. If the screen is a standard 16:9 ratio, and we call D the diagonal, then the screen is 9x high and 16x wide. Thanks to the Greeks we know that
Which we can rewrite as:
And we already know that the height, H, is 9x, which gives us the original formula.
The next bit required some trigonometry, and my memory was jogged by someone on the office who did maths a lot more recently than me, with the phrase ‘SOHCAHTOA.’
We know (from the BBC research, and other sources) that the eye can see detail that covers 1 minute of arc, or 1/60th of a degree. Imagine a triangle touching the screen, one pixel high, with the eye a distance d away, along the middle axis of the triangle.
I’d draw a diagram, but I’m rubbish at those. We can make this into two right angle triangles, dividing along that axis, so now the height is half a pixel, the side at right angles to the screen is d, and the angle at the other end is 1/120th of a degree.
SOHCAHTOA tells us that the tangent of the angle is the ratio of the opposite side (half a pixel height) to the adjacent side (the distance from screen to eye), so we can say
We’ve already worked out what H is equal to, further up the page, and tan(1/120) is a constant too. So, working all that out, we can substitute the values to come up with
And to allow for people starting with a screen size in inches, and wanting a distance in metres, we use the factor 0.0254, which results in this formula for the viewing distance (d, in metres) from a particular screen size (D, in inches):
You can, as one of the readers has pointed out, simplify this a bit further if you want, but then it wouldn’t look so scientific, would it?