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## Sunday, February 18th, 2018

### From How Far to Watch TV?

Categories: [ Science ]

The distance l from the TV depends on the desired horizontal viewing angle a, the screen's diagonal d and the number N of pixels on a row. Additionally, we will assume the screen's aspect ratio r to be 16:9 and the human eye's smallest angle that can be seen e to be 31.5 arcseconds.

Let R be the ratio between the diagonal and the width of the screen:

R = √(1 + 1/r2)

We can then write a relationship between a, N and e:

tan(a / 2) = NR tan(e / 2)    (1)

From (1) we can deduce that for any given e, there is a maximum horizontal viewing angle amax above which pixels can theoretically be distinguished.

For N = 1920 (FullHD), amax = 19.1°. With a 4K screen, amax = 37.2°.

We can also write a relationship between horizontal viewing angle, screen diagonal and distance:

d / l = 2 R tan(a / 2)    (2)

The ideal value or a is a matter of debate, but THX defines a horizontal viewing angle of at least 36° (the screen viewed from the rear seat of a THX theatre), while SMPTE suggests 30°. A value of 20° is also mentioned.

With a 4K screen, amax = 37.2° and (2), we draw that the ideal distance is 1.30 times the screen's diagonal. For example:

• 132 cm for a 40" screen
• 165 cm for a 50" screen
• 197 cm for a 60" screen

With a = 30°, the ideal screen distance is 1.63 times the screen's diagonal. For example:

• 132 cm for a 32" screen
• 166 cm for a 40" screen
• 207 cm for a 50" screen
• 248 cm for a 60" screen

With a FullHD screen and a compromise angle a = 20°, the ideal distance is 2.47 times the screen diagonal. For example:

• 201 cm for a 32" screen
• 251 cm for a 40" screen
• 314 cm for a 50" screen

EDIT: The value of e is valid for a high contrast between two pixels. Most images do not have such a high contrast, and therefore a value of e = 1 arcminute is a reasonnable assumption in practice.

From this follows that for N = 1920 (FullHD), amax = 35.5° (1.36 times the screen diagonal). With a 4K screen, amax = 65.3° (0.68 times the screen diagonal). This also gives a reasonnable value for standard definition PAL TV with N = 1024, amax = 19.4° (2.55 times the screen diagonal).

That would allow for larger horizonat viewing angles, such as 45° (1.05 times the screen diagonal) or 60° (0.75 times the screen diagonal) when viewing a 4K screen. At such short distances one must however take into account the possible lack of comfort due to the physical closeness of smaller screens.

[ Posted on February 18th, 2018 at 18:41 | no comment | ]