This is the transcript and images from my video Deriving the Angular Size Formula.

There is a lot of talk about perspective. The term is often used as a to conveniently explain random observations. However, perspective has a clear meaning and shouldn’t be used inappropriately to explain whatever you cannot otherwise explain. It’s actually quite simple: as an object gets farther away it appears smaller.

When things are farther away you can see more things in the same field of view, so more stuff has to get crammed into that same field of view.

It’s very important to understand that perspective causes an object to appear to shrink in size as it gets farther away. This always happens uniformly. The vertical size shrinks at the same rate as the horizontal size. There are other phenomenon that cause other visual effects, but perspective always acts in a completely predictable and uniform manner.

This causes things below true level to rise up to the line of true level as they recede. Also, things above true level will lower down to the line of true level as they recede.

It’s critical to remember that things below true level will never appear above true level and things above true level will never appear below true level.

The good news is this effect is completely consistent and predictable. If we know an object’s elevation and distance we can use basic triangulation to accurately predict the angular elevation above or below true level where an object will appear. We can extend this triangulation to derive a formula to accurately predict an object’s angular size if we know it’s actual size and distance. I will be deriving this formula in this video.

We will start with Batman as he surveys Gotham city form the top of a tall building. Several blocks away is the Bat signal on top of the police headquarters. We know the diameter of the bat signal and how far away it is from Batman, now we want to calculate the expected angular size.

We start by defining a right triangle with one point at batman’s eyes, another point in the center of the bat signal and the third point on the edge of the bat signal.

We will be considering the angle near batman and call it Theta. The sides of a right triangle have specific names. The longest side is called the “hypotenuse”, the side near theta is called “adjacent” and the side opposite theta is conveniently called “opposite”.

The ratios between the lengths of the sides are trigonometric functions. These are actually just the ratios of the different sides of a right triangle. Sine is the length of the opposite divided by the hypotenuse, cosine is adjacent divided by hypotenuse and tangent is opposite divided by adjacent.

We know the diameter of the bat signal, let’s call it “g”. If we divide it in half this is the length of the “opposite” side of the triangle. This is just “g over 2”.

We also know the distance from Batman to the bat signal which is the length of the “adjacent” side of the triangle, let’s call it “r”.

When we divide the length of the opposite by the length of the adjacent side we get one of these special ratios called “tangent”.

Since we are interested in the value of theta we need to do some basic algebra to get theta all alone on one side of the equation. To get this we do the opposite of tangent with is called “Arctangent”. This gives us “theta equals arctangent of g over 2 all over r”.

Since theta only represents half of the angular size of the bat signal we need to double the result to get the full angular size of the bat signal.

Now we have the full formula to calculate the perspective angular size of an object: a = 2 arc tangent of g over 2 all over r, where g is the actual size of the object, r is the distance to the object and a is the angular size of the object.

So, when someone attributes an observation to “perspective” you can ask how they used the formula of angular perspective to confirm the observation and predict similar observations. If they cannot use this formula or it doesn’t apply to their claims they are probably just trying to use the word “perspective” to explain away an inconvenient observation.

I plan a followup to this video where I will use the formula to confirm and predict real world observations. Be sure to subscribe and click the bell notification so you know when future videos come out.