2-Argument Arctangent¶

Definition¶

The 2-argument arctangent, denoted as $\operatorname{atan2}$ , is a function defined by

\operatorname{atan2}(y,x)= \begin{cases} \displaystyle \,\arctan(y/x),& x>0,\\[2pt] \displaystyle \,\arctan(y/x)+\pi,& x<0 \text{ and }y\geq 0,\\[2pt] \displaystyle \,\arctan(y/x)-\pi,& x<0 \text{ and }y< 0,\\[2pt] \displaystyle \,\pi/2,& x=0 \text{ and }y> 0,\\[2pt] \displaystyle \,-\pi/2,& x=0 \text{ and }y< 0,\\[2pt] \displaystyle \,\text{undefined},&x=0 \text{ and }y= 0. \end{cases}

Interpretation¶

The function $\operatorname{atan2}$ answers the question: in which direction does the point $(x,y)$ lie, seen from the origin? It returns the angle in $(-\pi,\pi]$ measured from the positive $x$ -axis, which is the principal argument of the complex number $x+\mathrm{i}y$ $i y$ .

Coordinate axes with a point (x, y) in the second quadrant and its opposite point (-x, -y), which has the same ratio y/x; the angle atan2(y, x) from the positive x-axis tells them apart.

The blue point $(x,y)$ lies in the second quadrant, with $x<0$ and $y>0$ $0$ . The grey point $(-x,-y)$ has the same ratio $y/x$ , so the inverse tangent gives both points the same angle in $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ , namely the direction of the grey point. For the blue point the definition adds $\pi$ , half a turn, which gives the correct direction.

Drag the point $(x,y)$ into the left half-plane: $\arctan(y/x)$ only sees the slope, while $\operatorname{atan2}(y,x)$ adds or subtracts $\pi$ to give the true direction.

Each case of the definition repairs one defect of $\arctan(y/x)$ . For $x>0$ the point lies in the right half-plane, where the range $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ of $\arctan$ is already correct. For $x<0$ the angle is shifted by $\pi$ , upwards when $y\ge0$ and downwards when $y<0$ $0$ , so that the result stays in $(-\pi,\pi]$ ; points on the negative $x$ -axis get $\pi$ , not $-\pi$ . For $x=0$ the ratio $y/x$ is not defined, and the angle $\pm\tfrac{\pi}{2}$ is read off directly. At the origin there is no direction at all, so no value is assigned.

In practice, $\operatorname{atan2}$ is how the polar form of $x+\mathrm{i}y$ is found: $r=\sqrt{x^2+y^2}$ and $\varphi=\operatorname{atan2}(y,x)$ $atan2 (y, x)$ . Python provides it as math.atan2(y, x). Note the order of the arguments: $y$ comes first, as in the ratio $y/x$ .

Key Results¶

The range of $\operatorname{atan2}(y, x)$ is $(-\pi, \pi]$ , covering all four quadrants.
$\operatorname{atan2}(y, x)$ is continuous at every point $(x, y) \neq (0, 0)$ that does not lie on the negative real axis. Crossing the negative real axis ( $x <$ $x<0$ $0$ , $y=0$ $0$ ), it jumps by $2\pi$ : just below the axis its values are close to $-\pi$ , on and just above the axis they are close to $\pi$ .

Examples¶

$\operatorname{atan2}(1, -1) = 3\pi/4$ (second quadrant).
$\operatorname{atan2}(-1, -1) = -3\pi/4$ (third quadrant, within range $(-\pi, \pi]$ ).

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: atan2, two-argument arctangent, arctangent with two arguments, arctan with two arguments, arctangent with 2 arguments, arctan with 2 arguments, arctan2 Symbols: phi, pi, square root

Last updated: 22 October 2026 19:04