<p>According to a review by Legge &amp; Bigelow the arc or degrees of visual angle ($\alpha$) is,</p>

<p>$$
\alpha = 57.3 \times S/D,
$$</p>

<p>where S is height of object and D is distance to object. [1] $S/D$ is the small angles approximation of $2 \times arctan(S/2D)$ which follows form geometry.</p>

<p><a href="https://i.sstatic.net/168C3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/168C3.png" alt="enter image description here"></a></p>

<p><strong>Image 1</strong>: the equation comes straight from trigonometric definitions.</p>

<p>Since humans continuously scan the page with the fovea, small angles assumption is a good approximation for entire pages. The factor $57.3$ is simply $180/\pi$ or conversion form radians to angles.</p>

<p>So given that we do not know the nature of $tg$, then its hard to say. Looks a bit fishy, but if tg is some sort of inversion of S then its plausible.</p>

<p>[1] G. Legge &amp; C Bigelow. 2011. Jornal of vision. <a href="http://jov.arvojournals.org/article.aspx?articleid=2191906" rel="nofollow noreferrer">Does print size matter for reading? A review of findings from vision science and typography</a>. </p>

CC BY-SA 3.0
2017-08-04T15:43:36.317Z
2017-08-04T15:48:47.340Z
2017-08-04T15:48:47.340Z
joojaa
<p>I was told that human's visual sharpness is 60 pixels per degree. As such, pixel per inch is approximately given by:</p>

<pre><code>PPI = 1/ (Distance * tg((1/60*pi/180) )
</code></pre>

<p>Is this formula fully correct? (I saw similar, but slightly different versions). How is this formula derived? Maybe some step by step description?</p>

CC BY-SA 3.0
2017-08-05T21:05:01.283Z
Could you either include a link to the source of this formula, or define what the terms in it refer to?
trichoplax is on Codidact now
<computational-geometry><2d><pixel-graphics><pixels><human-vision>
human visual: relation of Distance and DPI
angel_30
# human visual: relation of Distance and DPI

(Asked by: username_0 on 2017-08-03)

I was told that human's visual sharpness is 60 pixels per degree. As such, pixel per inch is approximately given by:

```
PPI = 1/ (Distance * tg((1/60*pi/180) )

```

Is this formula fully correct? (I saw similar, but slightly different versions). How is this formula derived? Maybe some step by step description?

**Tags:** computational-geometry, 2d, pixel-graphics, pixels, human-vision

> **Comments:**
> **username_1**: Could you either include a link to the source of this formula, or define what the terms in it refer to?  

---

## 1 answers

### Answer by username_2 on 2017-08-04. Score: 3 | Accepted answer

According to a review by Legge & Bigelow the arc or degrees of visual angle ($\alpha$) is,

$$
\alpha = 57.3 \times S/D,
$$

where S is height of object and D is distance to object. [1] $S/D$ is the small angles approximation of $2 \times arctan(S/2D)$ which follows form geometry.

[![enter image description here](https://i.sstatic.net/168C3.png)](https://i.sstatic.net/168C3.png)

**Image 1**: the equation comes straight from trigonometric definitions.

Since humans continuously scan the page with the fovea, small angles assumption is a good approximation for entire pages. The factor $57.3$ is simply $180/\pi$ or conversion form radians to angles.

So given that we do not know the nature of $tg$, then its hard to say. Looks a bit fishy, but if tg is some sort of inversion of S then its plausible.

[1] G. Legge & C Bigelow. 2011. Jornal of vision. [Does print size matter for reading? A review of findings from vision science and typography](http://jov.arvojournals.org/article.aspx?articleid=2191906).


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