<blockquote>
<p>What is your actual question? Are you asking others to confirm your process/formulas?</p>
<p>Correct, specifically whether the diagonal fov is always the correct method?</p>
</blockquote>
<p>No, an incorrect approximation is never &quot;always the correct method&quot;.</p>
<p>Even with the basic <a href="https://en.wikipedia.org/wiki/Pinhole_camera_model" rel="nofollow noreferrer">pinhole camera model</a> (the basis for almost all of the better approximations) position on the focal plane is never linear with incident angle. Instead, you draw similar triangles.</p>
<p>Here is a simple 2D analogy, usually people use 3D, <a href="https://en.wikipedia.org/wiki/Projective_geometry" rel="nofollow noreferrer">Projective Geometry</a> and <a href="https://en.wikipedia.org/wiki/Homography_(computer_vision)" rel="nofollow noreferrer">Homography</a> and the matrix math it requires.</p>
<p><span class="math-container">$$\frac{w}{d}=\frac{x}{f}$$</span></p>
<p><span class="math-container">$$x=f\frac{w}{d}$$</span></p>
<p>where <span class="math-container">$d$</span> and <span class="math-container">$f$</span> are the distances along the optical axis from the source and image plane to the lens pupil, and <span class="math-container">$w$</span> and <span class="math-container">$x$</span> are distances from the axis to the points on those planes which are perpendicular to the axis.</p>
<p><a href="https://i.sstatic.net/o8ueC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/o8ueC.png" alt="pinhole camera approximation" /></a></p>
<p>Most higher order methods that incorporate lens distortions star with the pinhole camera approximation, and quantify the imaging system's deviations from it.</p>
<p><em>no arctangents were harmed in the making of the <a href="https://en.wikipedia.org/wiki/Projective_plane" rel="nofollow noreferrer">projective plane</a>.</em></p>

CC BY-SA 4.0
2018-11-06T04:02:41.097Z
2018-11-06T10:10:33.803Z
2020-06-17T09:47:40.653Z
uhoh
<p>I was debating with someone online (I know, great way to get nowhere fast) regarding the size of Earth as it appears in the famous 'Earthrise' photos from Apollo 8.
Below is an accurate (almost pixel-perfect) calculation of the apparent size Earth should appear. In this instance it checks out, however if there is a better approach in calculating this and similar images I would be interested to know : </p>

<hr>

<p>The photograph was shot on 70mm film using a 250mm lens. We can calculate the what the Field of View should be for the image, using the following formula :</p>

<pre><code>         FOV (rectilinear) = 2 * arctan (frame size/(focal length * 2))
</code></pre>

<p>i.e.
             FOV (rectilinear) = 2 * arctan (70/(250 * 2))
             = a FOV of 15.93 degrees.</p>

<p>From our point of view on Earth, the average angular diameter of the moon is 0.5 degrees. From the Moon, the Earth has an angular diameter of about 1.9 .</p>

<p>From this it can be calculated how many pixels in size the Earth should appear in the photograph :</p>

<p>Diameter of Earth in Pixels = Photo's Diagonal Resolution * (Earth's Angular Diameter / FOV)</p>

<p>i.e.</p>

<p>Diameter of Earth in Pixels = 3841.87 * (1.9 / 15.93)</p>

<p>= 458.23 pixels</p>

<hr>

<p>For reference, The full, un-cropped version of the 'Earthrise' photo that the above calculation is basing the image resolution upon can be found at this location :</p>

<p><a href="https://www.nasa.gov/images/content/297755main_GPN-2001-000009_full.jpg" rel="nofollow noreferrer">https://www.nasa.gov/images/content/297755main_GPN-2001-000009_full.jpg</a></p>

<p>I imagine that this could be further refined by deriving the correct angular diameter of Earth from the exact distance of the moon at the time the photo was taken, as well as the altitude of the Apollo module above the moon.</p>

CC BY-SA 4.0
2018-11-05T18:58:55.363Z
What is your actual question? Are you asking others to confirm your process/formulas?
zephyr
CC BY-SA 4.0
2018-11-06T00:51:08.760Z
Correct, specifically whether the diagonal fov is always the correct method?
razeezar
CC BY-SA 4.0
2018-11-06T03:17:06.267Z
For photogrametry questions; how to understand and calculate the correspondence between objects and images in hand-held cameras, I would recommend that you ask in [Photography SE](https://photo.stackexchange.com/) where they will tell you about camera lens distortions. For questions specifically about photographs taken by astronauts during space missions, it is possible that [Space Exploration SE](https://space.stackexchange.com/) would also be a good site. I don't think the question is as good of a fit here as it would be in one of those two sites, though I don't think it is off-topic here.
uhoh
CC BY-SA 4.0
2018-11-06T03:17:44.910Z
However, you should take what you wrote in your comment and edit the original question to clarify your question, so that it is clear and self-contained.
uhoh
CC BY-SA 4.0
2018-11-06T03:21:27.597Z
Certainly assuming the focal length *is exactly 250 mm* is not completely justified, that's only a nominal focal length. And assuming that position on the film is exactly linear with angle is not necessarily justified either. The simplest pinhole-camera projection method would have you draw similar triangles and use something along the lines of $w/d=\Delta x/f$. Again, that's better discussed in Photography SE than here, or even Physics or Math SE, depending on the flavor of the answer you'd like.
uhoh
CC BY-SA 4.0
2018-11-06T06:20:49.503Z
Razeezar, please be aware that decisions on whether to keep your question open or to close it as off-topic are based on the *content of the question*, not on what you might have added as a comment. I strongly recommend that you [edit] your question to add the information in your comment. :-)
Chappo Hasn't Forgotten
<the-moon><photography><nasa>
Calculating the Size of Earth in the Apollo 8 Earthrise Photo
razeezar
# Calculating the Size of Earth in the Apollo 8 Earthrise Photo

(Asked by: username_0 on 2018-11-05)

I was debating with someone online (I know, great way to get nowhere fast) regarding the size of Earth as it appears in the famous 'Earthrise' photos from Apollo 8.
Below is an accurate (almost pixel-perfect) calculation of the apparent size Earth should appear. In this instance it checks out, however if there is a better approach in calculating this and similar images I would be interested to know :

---

The photograph was shot on 70mm film using a 250mm lens. We can calculate the what the Field of View should be for the image, using the following formula :

```
         FOV (rectilinear) = 2 * arctan (frame size/(focal length * 2))

```

i.e.
FOV (rectilinear) = 2 \* arctan (70/(250 \* 2))
= a FOV of 15.93 degrees.

From our point of view on Earth, the average angular diameter of the moon is 0.5 degrees. From the Moon, the Earth has an angular diameter of about 1.9 .

From this it can be calculated how many pixels in size the Earth should appear in the photograph :

Diameter of Earth in Pixels = Photo's Diagonal Resolution \* (Earth's Angular Diameter / FOV)

i.e.

Diameter of Earth in Pixels = 3841.87 \* (1.9 / 15.93)

= 458.23 pixels

---

For reference, The full, un-cropped version of the 'Earthrise' photo that the above calculation is basing the image resolution upon can be found at this location :

<https://www.nasa.gov/images/content/297755main_GPN-2001-000009_full.jpg>

I imagine that this could be further refined by deriving the correct angular diameter of Earth from the exact distance of the moon at the time the photo was taken, as well as the altitude of the Apollo module above the moon.

**Tags:** the-moon, photography, nasa

> **Comments:**
> **username_1**: What is your actual question? Are you asking others to confirm your process/formulas?  
> **username_0**: Correct, specifically whether the diagonal fov is always the correct method?  
> **username_2**: For photogrametry questions; how to understand and calculate the correspondence between objects and images in hand-held cameras, I would recommend that you ask in [Photography SE](https://photo.stackexchange.com/) where they will tell you about camera lens distortions. For questions specifically about photographs taken by astronauts during space missions, it is possible that [Space Exploration SE](https://space.stackexchange.com/) would also be a good site. I don't think the question is as good of a fit here as it would be in one of those two sites, though I don't think it is off-topic here.  
> **username_2**: However, you should take what you wrote in your comment and edit the original question to clarify your question, so that it is clear and self-contained.  
> **username_2**: Certainly assuming the focal length *is exactly 250 mm* is not completely justified, that's only a nominal focal length. And assuming that position on the film is exactly linear with angle is not necessarily justified either. The simplest pinhole-camera projection method would have you draw similar triangles and use something along the lines of $w/d=\Delta x/f$. Again, that's better discussed in Photography SE than here, or even Physics or Math SE, depending on the flavor of the answer you'd like.  
> **username_3**: username_0, please be aware that decisions on whether to keep your question open or to close it as off-topic are based on the *content of the question*, not on what you might have added as a comment. I strongly recommend that you [edit] your question to add the information in your comment. :-)  

---

## 1 answers

### Answer by username_2 on 2018-11-06. Score: 1

> What is your actual question? Are you asking others to confirm your process/formulas?
>
> Correct, specifically whether the diagonal fov is always the correct method?

No, an incorrect approximation is never "always the correct method".

Even with the basic [pinhole camera model](https://en.wikipedia.org/wiki/Pinhole_camera_model) (the basis for almost all of the better approximations) position on the focal plane is never linear with incident angle. Instead, you draw similar triangles.

Here is a simple 2D analogy, usually people use 3D, [Projective Geometry](https://en.wikipedia.org/wiki/Projective_geometry) and [Homography](https://en.wikipedia.org/wiki/Homography_(computer_vision)) and the matrix math it requires.

$$\frac{w}{d}=\frac{x}{f}$$

$$x=f\frac{w}{d}$$

where $d$ and $f$ are the distances along the optical axis from the source and image plane to the lens pupil, and $w$ and $x$ are distances from the axis to the points on those planes which are perpendicular to the axis.

[![pinhole camera approximation](https://i.sstatic.net/o8ueC.png)](https://i.sstatic.net/o8ueC.png)

Most higher order methods that incorporate lens distortions star with the pinhole camera approximation, and quantify the imaging system's deviations from it.

*no arctangents were harmed in the making of the [projective plane](https://en.wikipedia.org/wiki/Projective_plane).*


---

