May someone explain math.atan2 and it's capabilities? [closed]

The mathematics behind arc tangent

I will assume working knowledge of...

• tangent (hereby referred to as tan),
• cosine (hereby referred to as cos),
• sine (hereby referred to as sin),
• radians,
• and the four-quadrant 2D Cartesian coordinate system.

You can read about basic trigonometric functions, radians, and the four-quadrant 2D Cartesian coordinate system with the links provided in this sentence.

In trigonometry, we can tell the sign of trigonometryFunction(x) on some number x simply by looking at which quadrant x is in.

• sin is positive for quadrants I and II, and negative for quadrants III and IV. In other words, the quadrants above the x-axis are where sin is positive.
• cos is positive for quadrants I and IV, and negative for quadrants II and III. In other words, the quadrants to the right of the y-axis are where cos is positive.

From trigonometry, we define tan to be the following function: tan(x) = sin(x) / cos(x).

This brings about a rather interesting relationship, because this means that the sign of tan is dependent on the sign of sin and cos, because tan is *defined in terms of * sin and cos. We can derive the sign of tan(x) by first deriving the sign of sin(x) and cos(x), and then dividing the sign of sin(x) by the sign of cos(x).

With this relationship in mind, we get the following:

• tan is positive for quadrants I and III, and negative for quadrants II and IV. In other words, the quadrants where the line y = -x resides within are where tan is positive.

These sign quadrants are not particularly important for many things, but they are incredibly important when calculating the values of the inverse trigonometric functions.

As the name suggests, the basic inverse trigonometric functions are used to find the angle fed into a trigonometric function that produces a number bounded between -1 and 1. They are typically called arc sine (hereby referred to as asin), arc cosine (hereby referred to as acos), and arc tangent (hereby referred to as atan).

This brings a common misunderstanding, because then people start thinking that the following relationships are true...

• asin(sin(x)) = x, given an angle x bounded from 0 to 2pi.
• acos(cos(x)) = x, given an angle x bounded from 0 to 2pi.
• atan(tan(x)) = x, given an angle x bounded from 0 to 2pi.

Unfortunately, these relationships do not hold for all x. The reasoning lies behind the words inverse trigonometric functions and the quadrant signs that we derived above.

First, notice that the signs for the quadrants must be either negative or positive for all of the basic trigonometric functions.

Second, notice that there are exactly two positive quadrants and two negative quadrants for all of the basic trigonometric functions.

Third (and this one takes a bit of thinking), notice that the range of the absolute value of the trigonometric functions repeat multiple times within the domain 0 to 2pi (exactly twice for this domain, to verify plot the graphs of tan(x), sin(x), and cos(x)). This is best illustrated with multiples of k such that pi/4 + k*pi/2 that lie in the domain (so pi/4, 3pi/4, 5pi/4, 7pi/4).

Notice that if you take those four numbers and use the basic trigonometric functions on them, you get the following result...

• sin produces 1/sqrt(2), 1/sqrt(2), -1/sqrt(2), and -1/sqrt(2), respectively.
• cos produces 1/sqrt(2), -1/sqrt(2), -1/sqrt(2), and 1/sqrt(2), respectively.
• tan produces 1, -1, 1, and -1, respectively.

Notice that the results of each trigonometric function produces only two different numbers as outputs, even though we gave four different numbers as inputs.

If you know anything about functions in mathematics, you know that each input into a function must produce only a single output. The general idea is that the number of unique inputs must be greater than or equal to the number of unique outputs.

So, the results above are okay for trigonometric functions because we gave it four numbers, and got two numbers. Four is less than two. Good.

However, if we switch the results around like so to get the possible results of the basic inverse trigonometric functions...

• asin given 1/sqrt(2), -1/sqrt(2) produces (pi/4, 3pi/4) and (5pi/4, 7pi/4) respectively.
• acos given 1/sqrt(2), -1/sqrt(2) produces (pi/4, 7pi/4) and (3pi/4, 5pi/4) respectively.
• atan given 1, -1 produces (pi/4, 5pi/4) and (3pi/4, 7pi/4) respectively.

Notice how for each input, we get two outputs. This violates the definition of a function.

However, we still want to be able to get the equivalent original angle, and inverse trigonometric functions are our best bet. How can we turn our inverse trigonometric not-really-functions into functions?

The answer lies in restricting the range. Notice that our range was originally bound from 0 to 2pi. What if we chose our range carefully such that we throw away half of our outputs and end up with only one output per input?

• For asin, we will pick the range -pi/2 to pi/2.
• For acos, we will pick the range 0 to pi.
• For atan, we will pick the range -pi/2 to pi/2.

Notice that if you overlay each inverse trigonometric function's range with the domain of its corresponding trigonometric function, the range only covers a single positive quadrant and a single negative quadrant. This is how we effectively throw out half of our outputs; it turns out that the reason we were getting two outputs per input was because we were including the answers from two quadrants at a time!

Why we need another arc tangent function

When we throw out half of our outputs from before, we lose half of the information. This means that we aren't quite sure what quadrant the angle was in; atan could have given us the right angle or it could have given us an angle pi radians off. Ouch.

Let me give you an example that I think is the best example of a perfect opportunity to use atan2 over atan.

Let <x, y> be a two-dimensional vector on a Cartesian plane where the vector's value along the domain is x and the vector's value along the range is y.

Can you find the angle <-1, sqrt(3)> makes with the x-axis? This relies on the geometric definition of tangent...

• the tangent of an acute angle in a right triangle is equivalent to the length of the opposite leg divided by the length of the adjacent leg.

We compute atan(-sqrt(3)/1) and we get -pi/3... which is the wrong answer. Why? -pi/3 lies in the fourth quadrant, while <-1, sqrt(3)> is in the second quadrant.

Huh. That's strange. How about, <1, -sqrt(3)>? We computer atan(sqrt(3)/-1) and we get -pi/3 as well... and that's the right answer.

Notice that atan(sqrt(3)/-1) is equivalent to atan(-sqrt(3)/1) because sqrt(3)/-1 is equal to -sqrt(3)/1, which is also equal to -sqrt(3).

This brings us to the conclusion that if the input of atan involves negative numbers in its numerator or denominator, then atan may give us a result off by pi.

This is because the range of atan only covers the first and fourth quadrants. If our input is in the second or third quadrant, our atan will be off.

In case you didn't notice, this is bad. Really bad (see why in the section below).

We need something more exact; something that doesn't require us to do more calculations to verify our output is right or not.

Why the issue of throwing out half of the range does not affect sine or cosine

In sin or cos, this isn't as much of a problem because they only take into account either the x or the y axis according to the geometric definition of sin and cos:

• the sine of an acute angle in a right triangle is equivalent to the length of the opposite leg divided by the length of the hypotenuse.
• the cosine of an acute angle in a right triangle is equivalent to the length of the adjacent leg divided by the length of the hypotenuse.

Notice that both involve dividing by the length of the hypotenuse. This will always be positive, because of the Pythagorean theorem, where a^2 + b^2 = c^2.

Since the hypotenuse is always positive, then a negative input will indicate that...

• in the case of sin, the x-axis is negative
• in the case of cos, the y-axis is negative

Therefore, we can always tell if the input is off by pi for sin and cos.

In the case of tangent, we cannot get that same detailed information because both the numerator and the denominator of the input may be negative. Therefore, we are at a loss and atan just cannot give us the exact angle the input represents.

Introducing atan2

atan2 is a function that takes two inputs, so it looks like atan(y, x).

What makes atan2 incredibly easy to remember is that it behaves exactly like the problem above; given a vector <x, y>, get the angle it makes with the x-axis.

This means that if we do atan2(sqrt(3), -1), we get 2pi/3. This is perfect! 2pi/3 is in the second quadrant, and so is <-1, sqrt(3)>.

Quirks of atan2

Because of its definition, atan2 has a few quirks. I've listed all that comes to mind below.

• In the case that x is zero, atan2 still works and will correctly return the angle the vector makes with the positive x-axis (as opposed to atan, which will throw an error because you attempted to divide by 0).
• In the case that both x and y are zero, atan will return 0 (non-intuitive quirk of atan2).

See my math.atan2 here.