Atan2

In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, ${\displaystyle \theta =\operatorname {atan2} (y,x)}$ is the angle measure (in radians, with ${\displaystyle -\pi <\theta \leq \pi }$) between the positive ${\displaystyle x}$-axis and the ray from the origin to the point ${\displaystyle (x,\,y)}$ in the Cartesian plane. Equivalently, ${\displaystyle \operatorname {atan2} (y,x)}$ is the argument (also called phase or angle) of the complex number ${\displaystyle x+iy.}$ (The argument of a function and the argument of a complex number, each mentioned above, should not be confused.)

The ${\displaystyle \operatorname {atan2} }$ function first appeared in the programming language Fortran in 1961. It was originally intended to return a correct and unambiguous value for the angle ⁠${\displaystyle \theta }$⁠ in converting from Cartesian coordinates ⁠${\displaystyle (x,\,y)}$⁠ to polar coordinates ⁠${\displaystyle (r,\,\theta )}$⁠. If ${\displaystyle \theta =\operatorname {atan2} (y,x)}$ and ${\textstyle r={\sqrt {x^{2}+y^{2}}}}$, then ${\displaystyle x=r\cos \theta }$ and ${\displaystyle y=r\sin \theta .}$

If ⁠${\displaystyle x>0}$⁠, the desired angle measure is ${\textstyle \theta =\operatorname {atan2} (y,x)=\arctan \left(y/x\right).}$ However, when x < 0, the angle ${\displaystyle \arctan(y/x)}$ is diametrically opposite the desired angle, and ⁠${\displaystyle \pm \pi }$⁠ (a half turn) must be added to place the point in the correct quadrant.^[1] Using the ${\displaystyle \operatorname {atan2} }$ function does away with this correction, simplifying code and mathematical formulas.