Personalize the Meaning of “Retina Display”

Apple’s Retina Display marketing broadly publicized the concept of retinal acuity, but each person’s vision differs; so, just how small do those pixels need to be for your vision?

Fortunately, inverting the well known Snellen notation (e.g. 20/20 corrected vision, 20/30 uncorrected vision, etc…) gives your personal visual acuity in minutes of arc. For example, inverting 20/20 = 1 meaning that 20/20 vision can resolve 1 arc minute sized details.  Similarly, someone with 20/60 vision has a visual acuity of 60/20 = 3.3 arc minutes; 20/15 vision can resolve 15/20 = 0.75 arc minutes.  Go ahead and calculate your own visual acuity in arc minutes.  Ready?

OK, let’s see how tiny the pixels on a screen need to be to make it a retina display for you.  To do this, we’ll calculate the smallest pixels that you can resolve at a given distance. For example, if you have 20/20, or 1 arc minute, vision and hold a smartphone 11 inches (28 cm) away, you’ll be able to resolve individual pixels if there are 313 pixels per inch (123 pixels/cm) or fewer; if it has more pixels than that per inch/cm (i.e. higher pixel density & smaller pixels), then it’s a “retina display”.

Here’s how to calculate the minimum number of pixels per distance to match your eyes (fill in your visual resolution in place of “1“:

tan(½ × 1 arc minute) × 2 × 11 inches = 0.0032 inches (or the inverse of 313 pixels per inch (ppi) or more)
tan(½ × 1 arc minute) × 2 × 28 cm       = 0.00814 cm (or 123 pixels per cm (ppcm) or more)

Spreadsheet formulas for this looks like:

<resolvable pixel> = tan(radians(0.5 * <your arc min.>/60)) * 2 * <distance>
<pixel density> = 1 / <resolvable pixel>

In more detail: to calculate the pixel size, s, opposite the viewer divide the angle, a, in half to give a right triangle with the viewing distance, d, adjacent to the angle and the length of ½ of a pixel opposite. 1 arc minute = 1/60 degree. Then with basic trigonometry:

tangent (angle) = opposite/adjacent
tangent (½ a= ½ s/d
½ s = tan( ½ a ) d
    s = tan( ½ a ) 2 d

Tangent of half of the angle time the distance equals the spacing needed

If you were looking at a television 5-½ feet away instead, then you’d only be able to resolve 52 ppi (20 ppcm):

tan(½ × 1 arc minute) × 2 ×   66 in.  = 0.0192 in. or 52 ppi
tan(½ × 1 arc minute) × 2 × 170 cm = 0.0495 cm or 20 ppcm

A 42-inch diagonal, full HD television (1920×1080) also happens to have 52 pixels per inch; therefore, when viewed from 5-½ feet or farther the pixels begin to blur together for 20/20 vision.  Homework: how close/far should you sit from your television to turn it into a “retina” display? Enjoy!

Snellen acuity Visual resolution (arc minutes) Retina display, iPhone Retina display, TV
(11 in, ppi) (28cm, ppcm) (9′, ppi) (2.75m, ppcm)
20/200 10 31 12 3 1
20/100 5 63 25 6 3
20/70 3.5 89 35 9 4
20/50 2.5 125 49 13 5
20/30 1.5 208 82 21 8
20/20 1 313 123 32 13
20/15 0.75 417 164 42 17

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