Does the short focal length of the eyes enlarge the moon?

The real cause of the moon illusion?

As shown in the figure, the blue line is the lens, w is the height of the object, x is the height of the image, v is the image distance, u is the object distance and f is the focal length. The red line is the light path.

The relationship between u, v, f is

1 / u + 1 / v = 1 / f

and so

f = uv / (v + u) (1)

The observer's eyes are constant, so v is fixed. The distance between the observer and the object is constant, so u is also fixed. As soon as v and u are fixed, it can be known from equation (1) that f is also fixed. If v is fixed and u decreases, f also decreases.

Knowledge from similar triangles:

x / w = (v-f) / f = v / f-1

and so

x = w (v / f-1) (2)

According to formula (2), x increases when v and w are fixed, when f decreases.

If the observer observes the moon on the horizon due to the influence of mountains and trees, f is smaller than when observing the moon at the zenith. According to formula (2) we can know that x increases as f decreases. The viewer will feel that the moon is larger and closer on the horizon than the moon at the zenith.

I think that's the reason for the moon illusion.

Simple calculation

If you look at nearby trees with your eyes:

u = 200 m (assuming 200 m from the tree)

v = 0.024 m (eyeball diameter, assumed image length)

w = 10 m (provided the tree is 10 m high)

f = uv / (v + u)

= 0.0239971 m

x = w (v / f-1)

= 0.0012 m = 1.2 mm (height of the tree picture)

When you look at the zenith moon (without the influence of trees on the ground) with your eyes:

u = 380000000 m (distance from the observer to the moon)

v = 0.024 m

w = 3476000 meters (diameter of the moon)

f = uv / (v + u) = A (we set this focal length to A)

x = w (v / f-1)

= 0.000219537 m = 0.219537 mm

If you observe the moon towards the horizon with the focal length of the observation tree:

f = 0.0239971 m

x = w (v / f-1)

= 420.067 m

The observation of the moon image on the zenith is 0.219537 mm, and the observation of the moon image on the horizon is 420.067 m, which shows a big difference between the two. So if you use a focal length less than A, the moon will be "enlarged".

Of course, the eyes do not generally observe the moon with a focal length of 0.0239971 m. Because the picture may not be clear. If the image of the moon is not clear at this focal length, the eyes adjust the focal length. Set a focal length that is clear for imaging. This focal length is less than A, but it is the focal length for clear imaging. Since the moon is far away, the depth of field of the moon imaging is very large. Therefore, there is a focal length that is smaller than A and can reproduce clearly. The moon illusion is therefore caused by a relatively short focal length. I think that's why the moon illusion.

**Does the short focal length of the eyes enlarge the moon?**

reference

https://en.wikipedia.org/wiki/Moon_illusion