How much printed linear resolution do you need to support optimal viewing from various distance?

I think I assumed a maximum resolving power of 1 minute of arc (MOA), so we want each pixel to subtend 1 MOA at the chosen viewing distance.

First, for a viewing distance of about 50cm (0.5m):
{{{
x/50cm == tan(0.000145444)

x/50cm == 0.000145444

x = 0.0072722 cm

2x = 0.0145444 cm (base of triangle, and the length subtended at 50cm

== 0.145444 mm

== 175 dpi for viewing at 50cm
}}}

For further confirmation, the page on [[http://en.wikipedia.org/wiki/Minute_of_arc#Firearms| MOA for firearms]] states 1MOA at 100m distance is 29.08mm. A simple linear scaling confirms that this is correct.


What's a simpler formula for ready reckoning?

{{{
2.54 / (A_CONST * 2 * VIEWING DISTANCE IN CM)


From left to right:
2.54 / 0.000145444 / 2 / VIEWING DISTANCE
}}}


This we arrive at:

 * 218 dpi at 40cm
 * 291 dpi at 30cm
 * 436 dpi at 20cm

The constant is: 8,731.883061522


'''8732 / VIEWING DISTANCE == necessary DPI'''

You could round this to: '''9000 / VIEWING_DISTANCE_IN_CM = necessary DPI'''


||<rowbgcolor="lightblue"> Distance || Necessary linear resolution || Nearest useful number? ||
||  10m ||   9 dpi ||  10 dpi ||
||   5m ||  18 dpi ||  20 dpi ||
||   3m ||  29 dpi ||  30 dpi ||
||   1m ||  87 dpi ||  90 dpi ||
|| 50cm || 175 dpi || 180 dpi ||
|| 40cm || 218 dpi || 225 dpi ||
|| 30cm || 291 dpi || 300 dpi ||
|| 20cm || 436 dpi || 450 dpi ||
|| 10cm || 872 dpi || 900 dpi ||


I dunno if these longer distances hold up to scrutiny - are they correct? They'd suggest that long-distance roadside billboards can be comfortably printed with lolhueg pixels that are like 4.36mm in diameter.