No, there are many problems using an angular size

Really? I am yet to find one which isn't caused by people making mistakes.

The only thing that comes close is the limit of resolution where very small objects cannot be resolved and either appear larger than they are (if they are bright enough) or they aren't noticed at all (if they aren't bright enough).

If the distance is twice as far away, the object looks half the size.

Only as an approximation for distant objects.

If the object is close enough then doubling the distance will not cut the size in half.

The simple way to show this is to first consider an object 1024 m away which takes up 1 degree. (that is roughly 18 m tall).

Now we repeatedly half the distance and double the angular size and consider what happens. (the faster way is to note it is based upon the assumption that the size times the distance is constant, and thus we can just flip it and get it to be 1024 degrees, but lets do it the long way for completeness and to avoid any arguments)

512 m gives 2 degrees.

256 m gives 4 degrees.

128 m gives 8 degrees.

64 m gives 16 degrees.

32 m gives 32 degrees.

16 m gives 64 degrees.

8 m gives 128 degrees.

4 m gives 256 degrees, more than covering an entire side of your vision. This is only possible if you are inside it.

2 m gives 512 degrees. This is more than physically possible. Even if you were inside an object with no opening, it would only be 360 degrees.

1 m gives 1024 degrees. This is just insanity.

So clearly objects cannot simply half the angle angle when the distance is doubled.

If you think I am being unfair with my initial conditions then just take any object, and do the same. A tall building viewed from a distance is a good start.

The origin for this is the small x approximation for the tangent function.

When x is small, tan(x)~=x.

In reality, tan(a)=h/d.

The small x approximation means that a~=h/d, but only when a is small, which requires h/d to be small.

We can even show the problems with this.

Lets take a 16 m tall object.

At 1 m it is 86.4 deg

2 m it is 82.9 deg. Notice this is nothing like halving. In fact, it is more like subtracting the first angle from 90, then doubling it, then subtracting that from 90 to get the second.

4 m it is 76.0

8 m it is 63.4

16 m it is 45.0

32 m it is 26.6

64 m it is 14.0

128 m it is 7.1, now this is more like halving

256 m it is 3.6

512 m it is 1.8

1024 m it is 0.9.

Notice how this only predicts the angle is halved at large distances.

At small distances it is much larger than half the size at twice the distance.

So no, the reality of distant objects not having their size reduced to half when the distance is doubled is entirely in line with modern understanding of how light and seeing works.

**As shown in the picture, however, the angle of light entering the eye is not reduced by half.**

**But the tangent of the angle is only reduced by half.**

The inverse tangent (which is what I am assuming you are using) is that angle.

If they don't match, you have done something wrong.