Atan2

In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, θ = atan2 ⁡ ( y , x ) {\displaystyle \theta =\operatorname {atan2} (y,x)} is the angle measure (in radians, with − π < θ ≤ π {\displaystyle -\pi <\theta \leq \pi } ) between the positive x {\displaystyle x} -axis and the ray from the origin to the point ( x , y ) {\displaystyle (x,\,y)} in the Cartesian plane. Equivalently, atan2 ⁡ ( y , x ) {\displaystyle \operatorname {atan2} (y,x)} is the argument (also called phase or angle) of the complex number x + i y . {\displaystyle x+iy.} (The argument of a function and the argument of a complex number, each mentioned above, should not be confused.) The atan2 {\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 ⁠ θ {\displaystyle \theta } ⁠ in converting from Cartesian coordinates ⁠ ( x , y ) {\displaystyle (x,\,y)} ⁠ to polar coordinates ⁠ ( r , θ ) {\displaystyle (r,\,\theta )} ⁠. If θ = atan2 ⁡ ( y , x ) {\displaystyle \theta =\operatorname {atan2} (y,x)} and r = x 2 + y 2 {\textstyle r={\sqrt {x^{2}+y^{2}}}} , then x = r cos ⁡ θ {\displaystyle x=r\cos \theta } and y = r sin ⁡ θ . {\displaystyle y=r\sin \theta .} If ⁠ x > 0 {\displaystyle x>0} ⁠, the desired angle measure is θ = atan2 ⁡ ( y , x ) = arctan ⁡ ( y / x ) . {\textstyle \theta =\operatorname {atan2} (y,x)=\arctan \left(y/x\right).} However, when x < 0, the angle arctan ⁡ ( y / x ) {\displaystyle \arctan(y/x)} is diametrically opposite the desired angle, and ⁠ ± π...