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# Smartphones

## Smartphones

Typical distance between Smartphone and eyes is approximatly l=40cm.

Model x y x:y d[“] l[cm] Res(l=40cm)
Samsung Galaxy A40 1080 2340 6:13 5.9 19.98 540×1170
Nokia 1 480 854 9:16 4.5 40.11 450×800

Calculate l in bash using gnuplot:

echo 'x=1080; y=2340; d=5.9; print "l = ", (2*d*2.54) / (sqrt(x**2+y**2)*tan( (2/60.0) * pi/180.0) )," cm" ;' | gnuplot 2>&1
echo 'x=480; y=854; d=4.5; print "l = ", (2*d*2.54) / (sqrt(x**2+y**2)*tan( (2/60.0) * pi/180.0) )," cm" ;' | gnuplot 2>&1

Calculate x:y in bash using gnuplot:

echo 'x=1080.0;y=2340.0; do for [c=1:30] { print c,":",c*y/x ;}' | gnuplot 2>&1
echo 'x=480.0;y=854.0; do for [c=1:30] { print c,":",c*y/x ;}' | gnuplot 2>&1


## Calculation via human eyes resolution

What is the minimal distance l on wich the human eye can separeate each pixel?

The resolution of human eyes is alpha=2”. The diagonal of the display is d.

  Display:
_____
|\    |
| \   |
|  \D | y
|   \ |
|____\|
x

It applies for the pixel diagonal D: D²=x²+y² ⇒ D=sqrt(x²+y²) The metric size of a pixel p is : p=d/D ⇒ p=d/sqrt(x²+y²)

human resolution:

| pixel 2 p
__alpha___________________| pixel 1 p
l

We know: tan(alpha)= 2p / l. So it applies for l: l= 2p / tan(alpha)

It follows: l = 2d/( sqrt(x²+y²) * tan(alpha) )

Please note that 2“ is a small angle, so it can be linearized well by small angle approximation. 