Maths/formula behind atan2 function

Hi all,

I understand issues around this function have been discussed in great detail and I'm very thankful for the many helpful contributions on this topic (particularly from Roger Stafford) to date... they have helped me immensely with my work. But I was hoping to ask just one more question about it (sorry!).

I understand that creating the vector cross product and scalar dot product allows me to pass the sin and cosine to atan2 function to calculate the angle between 2 3D vectors (important for accuracy and also practical if vectors aren't normalised - as I understand it anyway). What I am unsure of is the maths or formula behind the atan2 function. I would like to know what it is and was wondering if anyone could help?

My guesses are that it might be something like:

atan2 = 2arctan(y/x)

where y = dot product and x = norm of cross product if x > 0.

Or

atan2 = 2arctan(y/(sqrt(x.^2+y.^2)+x)

where y = dot product and x = norm of cross product if x > 0.

Hopefully someone can tell me if I'm wrong/right and point me in the right direction.

Thanks in advance!

[Stephen23]

Wikipedia has a very good page on atan2:

https://en.wikipedia.org/wiki/Atan2

Answer 1

[John D'Errico]

atan2(x,y) is effectively atan(y/x), with the exception that it caters to all 4 quadrants. So, while we have

y/x == (-y)/(-x)

it is not true that

atan2(y,x) == atan2(-y,-x)

So you see these are the same:

atan(1/3)
ans =
      0.32175

atan2(1,3)
ans =
      0.32175

Yet ...

atan2(-1,-3)
ans =
      -2.8198

The difference is important when you need a 4 quadrant result. This is a major reason why atan2 exists. As well, atan2 is a bit more robust to problems. So, while atan would have survived this one:

atan2(1,0)
ans =
       1.5708

atan(1/0)
ans =
       1.5708

0/0 would cause a problem.

atan(0/0)
ans =
   NaN

atan2(0,0)
ans =
     0

So atan2 really is just atan, but with care taken to handle all 4 quadrants properly. The range of atan should be [-pi/2,pi/2]. But the range of atan2 is (-pi,pi]. (I might not have gotten the open and and closed intervals right there, but you should get the idea.)

The complete code (for scalar x and y) might look something like this:

if x == 0 && y == 0
  result = 0;
else
  offset = 0;
  if x < 0
    offset = pi;
  end
  result = atan(y/x) + offset;
end

[DC]

Thanks John. Informative, concise and clear! And exactly what I needed.