atan2

Graph of the tangent function from −

π

to +

π

with the corresponding signs of x and y for atan2(y, x). The green arrows show two results of the atan2 function dependent on the sign of the arguments.

In a variety of computer languages, the function atan2 is the arctangent function with two arguments. The purpose of using two arguments instead of one is to gather information on the signs of the inputs in order to return the appropriate quadrant of the computed angle, which is not possible for the single-argument arctangent function. It also avoids the problems of division by zero.

For any real number (e.g., floating point) arguments $x$ and $y$ not both equal to zero, $atan2(y, x)$ is the angle in radians between the positive $x$ -axis of a plane and the point given by the coordinates $(x, y)$ on it. The angle is positive for counter-clockwise angles (upper half-plane, $y > 0$ ), and negative for clockwise angles (lower half-plane, $y < 0$ ).

History and motivation

The atan2 function was first introduced in computer programming languages, but now it is also common in other fields of science and engineering. It dates back at least as far as the FORTRAN programming language^[1] and is currently found in C's math.h standard library, the Java Math library, .NET's System.Math (usable from C#, VB.NET, etc.), the Python math module, the Ruby Math module, and elsewhere. Many scripting languages, such as Perl, include the C-style atan2(y,x) function.^[2]

In mathematical terms, $atan2$ computes the principal value of the argument function applied to the complex number $x +i y$ . That is, $atan2(y, x) = Pr arg(x +i y) = Arg(x +i y)$ . The argument can be changed by $2π$ (corresponding to a complete turn around the origin) without making any difference to the angle, but to define $atan2$ uniquely one uses the principal value in the range $(- π, π]$ . That is, $- π < atan2(y, x) \leq π$ .

The atan2 function is useful in many applications involving vectors in Euclidean space, such as finding the direction from one point to another. A principal use is in computer graphics rotations, for converting rotation matrix representations into Euler angles.

In some computer programming languages, the order of the parameters is reversed (for example, in some spreadsheets) or a different name is used for the function (for example, Mathematica uses ArcTan[x,y]). On scientific calculators the function can often be calculated as the angle given when $(x, y)$ is converted from rectangular coordinates to polar coordinates. The one-argument arctangent function cannot distinguish between diametrically opposite directions. For example, the anticlockwise angle from the $x$ -axis to the vector $(1, 1)$ , calculated in the usual way as $arctan(1 / 1)$ , is $π / 4$ (radians), or $45°$ . However, the angle between the $x$ -axis and the vector $(-1, -1)$ appears, by the same method, to be $arctan(-1 / -1)$ , again $π / 4$ , even though the answer clearly should be $- 3π / 4$ , or $-135°$ .

The $atan2$ function takes into account the signs of both vector components, and places the angle in the correct quadrant. Thus, $atan2(1, 1) = π / 4$ and $atan2(-1, -1) = - 3π / 4$ .

Additionally, the ordinary arctangent method breaks down when required to produce an angle of $\pm π / 2$ (or $\pm90°$ ). For example, an attempt to find the angle between the $x$ -axis and the vector $(0, 1)$ requires evaluation of $arctan(1 / 0)$ , which fails on division by zero. In contrast, $atan2(1, 0)$ gives the correct answer of $π / 2$ .

When calculations are performed manually, the necessary quadrant corrections and exception handling can be done by inspection, but in computer programs it is extremely useful to have a single function that always gives an unambiguous correct result.

Definition and computation

In terms of the standard $arctan$ function, whose range is $(- π / 2, π / 2)$ , it can be expressed as follows:

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

A more compact expression with four overlapping half-planes is

\operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y>0,\\-{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y<0,\\\arctan \left({\frac {y}{x}}\right)\pm \pi &{\text{if }}x<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

Notes:

This produces results in the range $(-π, π]$ , which can be mapped to $[0, 2π)$ by adding $2π$ to negative results.
Traditionally, atan2(0, 0) is undefined.
- The C function atan2, and most other computer implementations, are designed to reduce the effort of transforming cartesian to polar coordinates and so always define atan2(0, 0). On implementations without signed zero, or when given positive zero arguments, it is normally defined as 0. It will always return a value in the range $[-π, π]$ rather than raising an error or returning a NaN (Not a Number).
- Systems supporting symbolic mathematics normally return an undefined value for $atan2(0, 0)$ or otherwise signal that an abnormal condition has arisen.
For systems implementing signed zero, infinities, or Not a Number (for example, IEEE floating point), it is common to implement reasonable extensions which may extend the range of values produced to include − $π$ and −0. These also may return NaN or raise an exception when given a NaN argument.
For systems implementing signed zero (for example, IEEE floating point), $atan2(-0, x)$ , $x$ < 0 returns the value − $π$ . $atan2(+0, x)$ , $x$ < 0 still returns + $π$ .

The free math library FDLIBM (Freely Distributable LIBM) available from netlib has source code showing how it implements atan2 including handling the various IEEE exceptional values.

For systems without a hardware multiplier the function $atan2$ can be implemented in a numerically reliable manner by the CORDIC method. Thus implementations of $atan(y)$ will probably choose to compute $atan2(y, 1)$ .

The following expression derived from the tangent half-angle formula can also be used to define $atan2$ :

\operatorname {atan2} (y,\,x)={\begin{cases}2\arctan {\biggl (}{\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}{\biggr )}&{\text{if }}x>0{\text{ or }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

This expression may be more suited for symbolic use than the definition above. However it is unsuitable for floating point computational use, as the effect of rounding errors in $\scriptstyle {\sqrt {x^{2}+y^{2}}}$ is blown up near the region $x < 0, y = 0$ (this may even lead to a division of y by zero). The formula gives a NaN or raises an error for $atan2(0, 0)$ , but this is not an issue since $atan2(0, 0)$ is not defined.

A variant of the last formula which avoids rounding errors being blown up is sometimes used in high precision computation:

\operatorname {atan2} (y,\,x)={\begin{cases}2\arctan {\biggl (}{\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}{\biggr )}&{\text{if }}x>0,\\2\arctan {\biggl (}{\frac {{\sqrt {x^{2}+y^{2}}}-x}{y}}{\biggr )}&{\text{if }}x\leq 0{\text{ and }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

Variations and notation

In Common Lisp, where optional arguments exist, the atan function allows one to optionally supply the x coordinate: (atan y x).^[3]
In Mathematica, the form ArcTan[x, y] is used where the one parameter form supplies the normal arctangent. Mathematica classifies ArcTan[0, 0] as an indeterminate expression.
In Microsoft Excel,^[4] OpenOffice.org Calc, LibreOffice Calc,^[5] Google Spreadsheets,^[6] iWork Numbers,^[7] and ANSI SQL:2008 standard,^[8] the atan2 function has the two arguments reversed.
In the Intel Architecture assembler code, atan2 is known as the FPATAN (floating-point partial arctangent) instruction.^[9] It can deal with infinities and results lie in the closed interval $[-π, π]$ , e.g. atan2(∞, x) = + $π$ /2 for finite x. Particularly, FPATAN is defined when both arguments are zero:
atan2(+0, +0) = +0;

atan2(+0, −0) = + $π$ ;

atan2(−0, +0) = −0;

atan2(−0, −0) = − $π$ .

This definition is related to the concept of signed zero.

On most TI graphing calculators (excluding the TI-85 and TI-86), the equivalent function is called R►Pθ and has the arguments reversed.
On TI-85 the $arg$ function is called angle(x,y) and although it appears to take two arguments, it really only has one complex argument which is denoted by a pair of numbers: $x + y i = (x, y)$ .
In mathematical writings other than source code, such as in books and articles, the notations Arctan^[10] and Tan⁻¹^[11] have been utilized; these are uppercase variants of the regular arctan and tan⁻¹. This usage is consistent with the complex argument notation, such that $Atan(y, x) = Arg(x + y i)$ .
On HP calculators, treat the coordinates as a complex number and then take the ARG. Or << C->R ARG >> 'ATAN2' STO.

Illustrations

atan2 at selected points.

The figure alongside shows values of atan2 at selected points on the unit circle. The values, in radians, are shown inside the circle. The diagram uses the standard mathematical convention that angles increase counterclockwise, and zero is to the right. Note that the order of arguments is reversed; the function $atan2(y, x)$ computes the angle corresponding to the point $(x, y)$ .

The figure below shows values of $atan2$ for points on the unit circle. On the $x$ -axis is the angle of the points, starting from $0$ (point $(1, 0)$ ) and going counterclockwise, through points:

$(0, 1)$ with angle $π / 2$ (in radians);
$(-1, 0)$ with angle $π$ ;
$(0, -1)$ with angle $3π / 2$ ;

to $(1, 0)$ with angle $0 = (2π n mod 2π)$ . One can clearly see the branch cut of the $atan2$ function.^[12]

atan2 on the unit circle.

The two figures below show 3D view of respectively $atan2(y, x)$ and $arctan(y / x)$ over a region of the plane. Note that for $atan2(y, x)$ , rays emanating from the origin have constant values, but for $arctan(y / x)$ lines passing through the origin have constant values. For $x > 0$ , the two diagrams give identical values.

Identities

Addition sum and difference identity

Main article: List of trigonometric identities § Angle sum and difference identities

Sums of $\operatorname {atan2}$ may be collapsed into a single operation according to the following identity

\operatorname {atan2} (y_{1},x_{1})\pm \operatorname {atan2} (y_{2},x_{2})=\operatorname {atan2} (y_{1}x_{2}\pm y_{2}x_{1},x_{1}x_{2}\mp y_{1}y_{2})

...provided that $\operatorname {atan2} (y_{1},x_{1})\pm \operatorname {atan2} (y_{2},x_{2})\in (-\pi ,\pi ]$ .

The proof involves considering two cases, one where $y_{2}\neq 0$ or $x_{2}>0$ and one where $y_{2}=0$ and $x_2 < 0$ .

We only consider the case where $y_{2}\neq 0$ or $x_{2}>0$ . To start, we make the following observations:

$-\operatorname {atan2} (y,x)=\operatorname {atan2} (-y,x)$ provided that $y\neq 0$ or $x>0$ .
$\operatorname {Arg} (x+iy)=\operatorname {atan2} (y,x)$ , where $\operatorname {Arg}$ is the complex argument function.
$\theta =\operatorname {Arg} e^{i\theta }$ whenever $\theta \in (-\pi ,\pi ]$ , a consequence of Euler's formula.
$\operatorname {Arg} (e^{i\operatorname {Arg} \zeta _{1}}e^{i\operatorname {Arg} \zeta _{2}})=\operatorname {Arg} (\zeta _{1}\zeta _{2})$ .

To see (4), we have the identity $e^{i\operatorname {Arg} \zeta }={\bar {\zeta }}$ where ${\bar {\zeta }}=\zeta /\left|\zeta \right|$ , hence $\operatorname {Arg} (e^{i\operatorname {Arg} \zeta _{1}}e^{i\operatorname {Arg} \zeta _{2}})=\operatorname {Arg} ({\bar {\zeta _{1}}}{\bar {\zeta _{2}}})$ . Furthermore, since $\operatorname {Arg} \zeta =\operatorname {Arg} a\zeta$ for any positive real value $a$ , then if we let $\zeta =\zeta _{1}\zeta _{2}$ and $a=\left|\zeta _{1}\right|\left|\zeta _{2}\right|$ then we have $\operatorname {Arg} ({\bar {\zeta _{1}}}{\bar {\zeta _{2}}})=\operatorname {Arg} (\zeta _{1}\zeta _{2})$ .

From these observations have following equivalences:

{\begin{aligned}\operatorname {atan2} (y_{1},x_{1})\pm \operatorname {atan2} (y_{2},x_{2})&{}=\operatorname {atan2} (y_{1},x_{1})+\operatorname {atan2} (\pm y_{2},x_{2})&{\text{by (1)}}\\&{}=\operatorname {Arg} (x_{1}+iy_{1})+\operatorname {Arg} (x_{2}\pm iy_{2})&{\text{by (2)}}\\&{}=\operatorname {Arg} e^{i(\operatorname {Arg} (x_{1}+iy_{1})+\operatorname {Arg} (x_{2}\pm iy_{2}))}&{\text{by (3)}}\\&{}=\operatorname {Arg} (e^{i\operatorname {Arg} (x_{1}+iy_{1})}e^{i\operatorname {Arg} (x_{2}\pm iy_{2}))})\\&{}=\operatorname {Arg} ((x_{1}+iy_{1})(x_{2}\pm iy_{2}))&{\text{by (4)}}\\&{}=\operatorname {Arg} (x_{1}x_{2}\mp y_{1}y_{2}+i(y_{1}x_{2}\pm y_{2}x_{1}))\\&{}=\operatorname {atan2} (y_{1}x_{2}\pm y_{2}x_{1},x_{1}x_{2}\mp y_{1}y_{2})&{\text{by (2)}}\end{aligned}}

Derivative

For more details on this topic, see Winding number.

As the function $atan2$ is a function of two variables, it has two partial derivatives. At points where these derivatives exist, $atan2$ is, except for a constant, equal to $arctan(y / x)$ . Hence for $x > 0$ or $y \neq 0$ ,

{\begin{aligned}&{\frac {\partial }{\partial x}}\operatorname {atan2} (y,\,x)={\frac {\partial }{\partial x}}\arctan \left({\frac {y}{x}}\right)=-{\frac {y}{x^{2}+y^{2}}},\\[5pt]&{\frac {\partial }{\partial y}}\operatorname {atan2} (y,\,x)={\frac {\partial }{\partial y}}\arctan \left({\frac {y}{x}}\right)={\frac {x}{x^{2}+y^{2}}}.\end{aligned}}

Informally representing the function $atan2$ as the angle function $θ (x, y) = atan2(y, x)$ (which is only defined up to a constant) yields the following formula for the total derivative:

{\begin{aligned}\mathrm {d} \theta &={\frac {\partial }{\partial x}}\operatorname {atan2} (y,\,x)\,\mathrm {d} x+{\frac {\partial }{\partial y}}\operatorname {atan2} (y,\,x)\,\mathrm {d} y\\[5pt]&=-{\frac {y}{x^{2}+y^{2}}}\,\mathrm {d} x+{\frac {x}{x^{2}+y^{2}}}\,\mathrm {d} y.\end{aligned}}

While the function $atan2$ is discontinuous along the negative $y$ -axis, reflecting the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number.

In the language of differential geometry, this derivative is a one-form, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, i.e., a function), and in fact it generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

The partial derivatives of $atan2$ do not contain trigonometric functions, making it particularly useful in many applications (e.g. embedded systems) where trigonometric functions can be expensive to evaluate.

References

↑ Organick, Elliott I. (1966). A FORTRAN IV Primer. Addison-Wesley. p. 42. Some processors also offer the library function called ATAN2, a function of two arguments (opposite and adjacent).
↑ The Linux Programmer's Manual says:
"The atan2() function calculates the arc tangent of the two variables y and x. It is similar to calculating the arc tangent of y / x, except that the signs of both arguments are used to determine the quadrant of the result."
↑ "CLHS: Function ASIN, ACOS, ATAN". LispWorks.
↑ "Microsoft Excel Atan2 Method". Microsoft.
↑ "LibreOffice Calc ATAN2". Libreoffice.org.
↑ "Google Spreadsheets' Function List". Google.
↑ "Numbers' Trigonometric Function List". Apple.
↑ "ANSI SQL:2008 standard". Teradata. Archived from the original on 2015-08-20.
↑ IA-32 Intel Architecture Software Developer’s Manual. Volume 2A: Instruction Set Reference, A-M, 2004.
↑ https://books.google.com/books?id=2LIMMD9FVXkC&pg=PA234&dq=four+quadrant+inverse+tangent+mathematical+notation&hl=en&sa=X&ei=Q2Y4UaGTAcmzyAHsooCoBw&ved=0CDgQ6AEwAg#v=onepage&q=four%20quadrant%20inverse%20tangent%20mathematical%20notation&f=false
↑ https://books.google.com/books?id=7nNjaH9B0_0C&pg=PA345&dq=four+quadrant+inverse+tangent+mathematical+notation&hl=en&sa=X&ei=Q2Y4UaGTAcmzyAHsooCoBw&ved=0CDIQ6AEwAQ#v=onepage&q=four%20quadrant%20inverse%20tangent%20mathematical%20notation&f=false
↑ Computation of the external argument by Wolf Jung

External links

Java 1.6 SE JavaDoc
atan2 at Everything2
PicBasic Pro solution atan2 for a PIC18F

Other implementations/code for atan2

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