How much of a cylinder can you see from a single point of view?

I'm creating wraps for trash cans and recycling bins to label them appropriately. They'll wrap 360 degrees around the bins. I want the text on the bins to be readable from a single point-of-view (so someone doesn't need to walk around the bin to read the whole thing). Assuming the bins are cylinders, is there a way to represent what percentage of the cylinder would be viewable from a given point of view?

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It depends how close to the bin you are. I think a more significant question is how distorted the text will be, which probably comes down to a matter of taste. – Ben Millwood Feb 5 '13 at 15:05
Assuming top and bottom don't matter, if a cylinder is bisected by a plane, an ellipse is obtained. What fraction of an ellipse is visible from a single point-of-view in that plane? – alancalvitti Feb 5 '13 at 15:05

When you stand $N$ radiuses away from the surface of the cylinder, you can see this fraction of its surface: $$\frac{1}{\pi}\cos^{-1}\left(\frac{1}{N+1}\right)$$ -- or $\arccos(1/(N+a))/180^\circ$ if your arccosine function produces answers in degrees rather than radians.

Practically: When you stand outside the cylinder, you can see a fraction of its circumference that depends on how far from it you are. The fraction is always less than 180°, but approaches it closely as your viewpoint moves farther away from the cylinder.

On the other hand, you need to get your head fairly close to the trash can (closer than one radius to the surface) in order to see less than a third of the circumference.

So if your goal is just that an observer at a reasonable distance can see at least one copy of every letter of the text you write, writing it three times will be sufficient. But then you risk that the viewer will have to piece the text together from to halves as in:

    |                                                                |
|TED THREE TIMES AROUND THE CAN.        THIS TEXT IS REPEATED THR|
|                                                                |


If you want at least one full copy of the entire notice to be visible from anywhere, four or five repetitions would be advisable.

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A very practical answer. – Ross Millikan Feb 5 '13 at 15:24
Henning, this is an awesome answer. Thanks so much for your practical advice. – Brett Cassette Feb 5 '13 at 15:49
+1 for the great answer but I believe the correct plural form of "radius" is "radii", not "radiuses". – wltrup Aug 4 '15 at 20:55