What is the difference between atan and atan2 in C++?

Question

From school mathematics we know that the tangent has the definition

tan(α) = sin(α) / cos(α)

and we differentiate between four quadrants based on the angle that we supply to the functions. The sign of the sin, cos and tan have the following relationship (where we neglect the exact multiples of π/2):

  Quadrant    Angle              sin   cos   tan
-------------------------------------------------
  I           0    < α < π/2      +     +     +
  II          π/2  < α < π        +     -     -
  III         π    < α < 3π/2     -     -     +
  IV          3π/2 < α < 2π       -     +     -

Given that the value of tan(α) is positive, we cannot distinguish, whether the angle was from the first or third quadrant and if it is negative, it could come from the second or fourth quadrant. So by convention, atan() returns an angle from the first or fourth quadrant (i.e. -π/2 <= atan() <= π/2), regardless of the original input to the tangent.

In order to get back the full information, we must not use the result of the division sin(α) / cos(α) but we have to look at the values of the sine and cosine separately. And this is what atan2() does. It takes both, the sin(α) and cos(α) and resolves all four quadrants by adding π to the result of atan() whenever the cosine is negative.

Remark: The atan2(y, x) function actually takes a y and a x argument, which is the projection of a vector with length v and angle α on the y- and x-axis, i.e.

y = v * sin(α)
x = v * cos(α)

which gives the relation

y/x = tan(α)

Conclusion: atan(y/x) is held back some information and can only assume that the input came from quadrants I or IV. In contrast, atan2(y,x) gets all the data and thus can resolve the correct angle.

std::atan2 allows calculating the arctangent of all four quadrants. std::atan only allows calculating from quadrants 1 and 4.

The actual values are in radians but to interpret them in degrees it will be:

• atan = gives angle value between -90 and 90
• atan2 = gives angle value between -180 and 180

For my work which involves computation of various angles such as heading and bearing in navigation, atan2 in most cases does the job.

Another thing to mention is that atan2 is more stable when computing tangents using an expression like atan(y / x) and x is 0 or close to 0.

atan(x) Returns the principal value of the arc tangent of x, expressed in radians.

atan2(y,x) Returns the principal value of the arc tangent of y/x, expressed in radians.

Notice that because of the sign ambiguity, a function cannot determine with certainty in which quadrant the angle falls only by its tangent value (atan alone). You can use atan2 if you need to determine the quadrant.

I guess the main question tries to figure out: "when should I use one or the other", or "which should I use", or "Am I using the right one"?

I guess the important point is atan only was intended to feed positive values in a right-upwards direction curve like for time-distance vectors. Zero is always at the bottom left, and thigs can only go up and right, just slower or faster. atan doesn't return negative numbers, so you can't trace things in the 4 directions on a screen just by adding/subtracting its result.

atan2 is intended for the origin to be in the middle, and things can go backwards or down. That's what you'd use in a screen representation, because it DOES matter what direction you want the curve to go. So atan2 can give you negative numbers, because its zero is in the center, and its result is something you can use to trace things in 4 directions.