Atan2 Explained

In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition,

\theta = \operatorname{atan 2 }(y, x) is the angle measure (in radians, with - \pi < \theta \leq \pi ) between the positive x -axis and the ray from the origin to the point (x,y) in the Cartesian plane. Equivalently, \operatorname{atan 2 }(y, x) is the argument (also called phase or angle) of the complex number x + iy.

The

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

If, the desired angle measure is $\backslash theta\; =\; \backslash operatorname(y,x)\; =\; \backslash arctan\backslash left(y\; /\; x\; \backslash right).$ However, when, the angle

\arctan(y / x) is diametrically opposite the desired angle, and ± (a half turn) must be added to place the point in the correct quadrant.^[1] Using the \operatorname{atan 2 } function does away with this correction, simplifying code and mathematical formulas.

Motivation

The ordinary single-argument arctangent function only returns angle measures in the interval $,$ and when invoking it to find the angle measure between the -axis and an arbitrary vector in the Cartesian plane, there is no simple way to indicate a direction in the left half-plane (that is, a point

(x,y) with x < 0 ). Diametrically opposite angle measures have the same tangent because y/x = ( - y) / ( - x), so the tangent y/x is not in itself sufficient to uniquely specify an angle.

To determine an angle measure using the arctangent function given a point or vector

(x, y), mathematical formulas or computer code must handle multiple cases; at least one for positive values of x and one for negative values of x, and sometimes additional cases when y is negative or one coordinate is zero. Finding angle measures and converting Cartesian to polar coordinates are common in scientific computing, and this code is redundant and error-prone.

To remedy this, computer programming languages introduced the function, at least as early as the Fortran IV language of the 1960s.^[2] The quantity is the angle measure between the -axis and a ray from the origin to a point anywhere in the Cartesian plane. The signs of and are used to determine the quadrant of the result and select the correct branch of the multivalued function .

The function is useful in many applications involving Euclidean vectors such as finding the direction from one point to another or converting a rotation matrix to Euler angles.

The function is now included in many other programming languages, and is also commonly found in mathematical formulas throughout science and engineering.

Argument order

In 1961, Fortran introduced the function with argument order

(y, x) so that the argument (phase angle) of a complex number is \operatorname{arg}z = \operatorname{atan 2 }(\operatorname{Im}z, \operatorname{Re}z). This follows the left-to-right order of a fraction written y / x, so that \operatorname{atan 2 }(y, x) = \operatorname{atan}(y / x) for positive values of x. However, this is the opposite of the conventional component order for complex numbers, z = x + iy, or as coordinates (\operatorname{Re}z, \operatorname{Im}z). See section Definition and computation.

Some other programming languages (see § Realizations of the function in common computer languages) picked the opposite order instead. For example Microsoft Excel uses

\operatorname{Atan 2 }(x,y), OpenOffice Calc uses \operatorname{arctan 2 }(x,y), and Mathematica uses \operatorname{ArcTan}[x,y], defaulting to one-argument arctangent if called with one argument.

Definition and computation

( - \pi, \pi ] , that is, .

In terms of the standard function, whose range is, it can be expressed as follows to define a surface that has no discontinuities except along the semi-infinite line x<0 y=0:

A compact expression with four overlapping half-planes is

The Iverson bracket notation allows for an even more compact expression:^[3]

Formula without apparent conditional construct:$$\backslash operatorname(y,\; x)\; =\; \backslash lim\_\backslash arctan\backslash left(\backslash frac\backslash right)\; +\; \backslash frac2\backslash sgn(y)\backslash sgn(x)\backslash left(\backslash sgn(x)-1\backslash right)$$

The following expression derived from the tangent half-angle formula can also be used to define :This expression may be more suited for symbolic use than the definition above. However it is unsuitable for general floating-point computational use, as the effect of rounding errors in $\backslash sqrt$ expand near the region (this may even lead to a division of y by zero).

A variant of the last formula that avoids these inflated rounding errors:

Notes:

• This produces results in the range .^[4]
• As mentioned above, the principal value of the argument can be related to by trigonometry. The derivation goes as follows: If, then . It follows that $$\backslash operatorname(y,\; x)\; =\; \backslash theta\; =\; 2\backslash ,\backslash theta/2\; =\; 2\backslash arctan\backslash frac.$$ Note that in the domain in question.

Derivative

As the function is a function of two variables, it has two partial derivatives. At points where these derivatives exist, is, except for a constant, equal to . Hence for or,

\begin{align} & (\partial)/(\partial x) \operatorname{atan 2 }(y, x) = (\partial)/(\arctan\left( \partial x) (y)/(x) \right) =- (y)/(x ^{2} + y ^{2}) , \\[ 5 pt] & (\partial)/(\partial y) \operatorname{atan 2 }(y, x) = (\partial)/(\arctan\left( \partial y) (y)/(x) \right) = (x)/(x ^{2} + y ^{2}) . \end{align} Thus the gradient of atan2 is given by\nabla atan2 (y,x) = \left({ - y\over x ^{2+y} ^{2},} {x\over x ^{2+y} ^{2}\right).}

Informally representing the function as the angle function (which is only defined up to a constant) yields the following formula for the total differential:

\begin{align} d \theta & = (\partial)/(\partial x) \operatorname{atan 2 }(y, x) d x + (\partial)/(\partial y) \operatorname{atan 2 }(y, x) d y \\[ 5 pt] & =- (y)/(x ^{2} + y ^{2}) d x + (x)/(x ^{2} + y ^{2}) d y. \end{align}

While the function is discontinuous along the negative -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 do not contain trigonometric functions, making it particularly useful in many applications (e.g. embedded systems) where trigonometric functions can be expensive to evaluate.

Illustrations

This figure shows values of atan2 along selected rays from the origin, labelled at the unit circle. The values, in radians, are shown inside the circle. The diagram uses the standard mathematical convention that angles increase counterclockwise from zero along the ray to the right. Note that the order of arguments is reversed; the function computes the angle corresponding to the point .

This figure shows the values of

\arctan(\tan(\theta)) along with \operatorname {atan 2 }(\sin(\theta),\cos(\theta)) for 0 \le \theta \le 2 \pi . Both functions are odd and periodic with periods \pi and 2 \pi , respectively, and thus can easily be supplemented to any region of real values of \theta . One can clearly see the branch cuts of the \operatorname {atan 2 } -function at \theta = \pi , and of the \arctan -function at \theta \in \{\tfrac{\pi}{ 2 }, \tfrac{ 3 \pi}{ 2 }\} .^[5]

The two figures below show 3D views of respectively and over a region of the plane. Note that for, rays in the X/Y-plane emanating from the origin have constant values, but for lines in the X/Y-plane passing through the origin have constant values. For, the two diagrams give identical values.

Angle sum and difference identity

The sum or difference of multiple angles to be computed by can alternately be computed by composing them as complex numbers. Given two coordinate pairs and, their angles from the positive axis will be composed (and lengths multiplied) if they are treated as complex numbers and then multiplied together, . The resulting angle can be found using a single operation, so long as the as long as the resulting angle lies in :

\operatorname{atan 2 } (y _{1,} x _{1)} \pm \operatorname{atan 2 } (y _{2,} x _{2)} = \operatorname{atan 2 } (y _{1} x _{2} \pm x _{1} y _{2,} x _{1} x _{2} \mp y _{1} y _{2),}

and likewise for more than two coordinate pairs. If the composed angle crosses the negative -axis (i.e. exceeds the range), then the crossings can be counted and the appropriate integer multiple of added to the final result to correct it.

This difference formula is frequently used in practice to compute the angle between two planar vectors, since the resulting angle is always in the range .

East-counterclockwise, north-clockwise and south-clockwise conventions, etc.

The

atan2 function was originally designed for the convention in pure mathematics that can be termed east-counterclockwise. In practical applications, however, the north-clockwise and south-clockwise conventions are often the norm. In atmospheric sciences, for instance, the wind direction can be calculated using the atan2 function with the east- and north-components of the wind vector as its arguments;^[6] the solar azimuth angle can be calculated similarly with the east- and north-components of the solar vector as its arguments. The wind direction is normally defined in the north-clockwise sense, and the solar azimuth angle uses both the north-clockwise and south-clockwise conventions widely.^[7] These different conventions can be realized by swapping the positions and changing the signs of the x- and y-arguments as follows:atan2 (y, x), (East-Counterclockwise Convention)atan2 (x, y), (North-Clockwise Convention)atan2 ( - x, - y) . (South-Clockwise Convention)

As an example, let

x _{0} = (\sqrt{ 3)/() } and y _{0} = (1)/(2) , then the east-counterclockwise format gives atan2 (y _{0} , x _{0} ) ⋅ (180)/(\pi) =30 ^{\circ} , the north-clockwise format gives atan2 (x _{0} , y _{0} ) ⋅ (180)/(\pi) =60 ^{\circ} , and the south-clockwise format gives atan2 ( - x _{0} , - y _{0} ) ⋅ (180)/(\pi) =-120 ^{\circ} .

Changing the sign of the x- and/or y-arguments and/or swapping their positions can create 8 possible variations of the

atan2 function and they, interestingly, correspond to 8 possible definitions of the angle, namely, clockwise or counterclockwise starting from each of the 4 cardinal directions, north, east, south and west.

Realizations of the function in common computer languages

The realization of the function differs from one computer language to another:

• In Microsoft Excel,^[8] OpenOffice.org Calc, LibreOffice Calc,^[9] Google Spreadsheets,^[10] and iWork Numbers,^[11] the 2-argument arctangent function has the two arguments in the standard sequence

(\operatorname{Re}, \operatorname{Im}) (reversed relative to the convention used in the discussion above).

• 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.
• On most TI graphing calculators (excluding the TI-85 and TI-86), the equivalent function is called R►Pθ and has the arguments

(\operatorname{Re}, \operatorname{Im}) .

• On TI-85 the 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: .

The

(\operatorname{Im}, \operatorname{Re}) convention is used by:

• 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 rather than raising an error or returning a NaN (Not a Number).
• In Common Lisp, where optional arguments exist, the atan function allows one to optionally supply the x coordinate: (atan ''y'' ''x'').^[12]
• In Julia, the situation is similar to Common Lisp: instead of atan2, the language has a one-parameter and a two-parameter form for atan.^[13] However, it has many more than two methods, to allow for aggressive optimisation at compile time (see the section "Why don't you compile Matlab/Python/R/… code to Julia?" ^[14]).
• 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 − and −0 when = −0. These also may return NaN or raise an exception when given a NaN argument.
• In the Intel x86 Architecture assembler code, atan2 is known as the FPATAN (floating-point partial arctangent) instruction.^[15] It can deal with infinities and results lie in the closed interval, e.g. atan2(∞, ''x'') = +/2 for finite x. Particularly, FPATAN is defined when both arguments are zero:

atan2(+0, +0) = +0;

atan2(+0, −0) = +;

atan2(−0, +0) = −0;

atan2(−0, −0) = −.

This definition is related to the concept of signed zero.

• In mathematical writings other than source code, such as in books and articles, the notations Arctan^[16] and Tan^−1[17] have been utilized; these are capitalized variants of the regular arctan and tan⁻¹. This usage is consistent with the complex argument notation, such that .
• On HP calculators, treat the coordinates as a complex number and then take the ARG. Or << C->R ARG >> 'ATAN2' STO.
• On scientific calculators the function can often be calculated as the angle given when is converted from rectangular coordinates to polar coordinates.
• Systems supporting symbolic mathematics normally return an undefined value for or otherwise signal that an abnormal condition has arisen.
• 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 can be implemented in a numerically reliable manner by the CORDIC method. Thus implementations of will probably choose to compute .

See also

• hypot

External links

• ATAN2 Online calculator
• Java 1.6 SE JavaDoc
• atan2 at Everything2
• PicBasic Pro solution atan2 for a PIC18F

Other implementations/code for atan2

• Web site: Bearing Between Two Points . 21 February 2022 . https://web.archive.org/web/20201118024355if_/http://mathforum.org/library/drmath/view/55417.html . 18 November 2020 . dead.
• Web site: Arctan and Polar Coordinates . 21 February 2022 . https://web.archive.org/web/20181018102242if_/http://mathforum.org/library/drmath/view/54114.html . 18 October 2018 . dead.
• Web site: What's 'Arccos'? . 21 February 2022 . https://web.archive.org/web/20170906082259if_/http://mathforum.org/library/drmath/view/54172.html . 6 September 2017 . dead.

Notes

1. Web site: The argument of a complex number. Winter 2011. Santa Cruz Institute for Particle Physics.
2. Book: Organick , Elliott I. . 1966 . A FORTRAN IV Primer . Addison-Wesley . Some processors also offer the library function called ATAN2, a function of two arguments (opposite and adjacent). . 42.
3. Assuming the definitions undefined ⋅ 0 = 0 , undefined ⋅ 1 = undefined , andz + undefined = undefined for any z.
4. One can apply the periodicity of the result to map to another desired range, e.g. mapping to by adding to the negative results.
5. Web site: Wolf Jung: Mandel, software for complex dynamics. www.mndynamics.com. 20 April 2018.
6. Web site: Wind Direction Quick Reference. NCAR UCAR Earth Observing Laboratory.
7. 10.1016/j.renene.2021.03.047. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function. 2021. Zhang. Taiping. Stackhouse. Paul W.. MacPherson. Bradley. Mikovitz. J. Colleen. Renewable Energy. 172. 1333–1340. 233631040. free. 2021REne..172.1333Z .
8. Web site: Microsoft Excel Atan2 Method. 14 June 2014 . Microsoft.
9. Web site: LibreOffice Calc ATAN2. Libreoffice.org.
10. Web site: Functions and formulas – Docs Editors Help. support.google.com.
11. Web site: Numbers' Trigonometric Function List. Apple .
12. Web site: CLHS: Function ASIN, ACOS, ATAN. LispWorks.
13. Web site: Mathematics · The Julia Language. docs.julialang.org.
14. Web site: Frequently Asked Questions · The Julia Language. docs.julialang.org.
15. IA-32 Intel Architecture Software Developer’s Manual. Volume 2A: Instruction Set Reference, A-M, 2004.
16. Book: Principles of Digital Image Processing: Fundamental Techniques. Wilhelm. Burger. Mark J.. Burge. 7 July 2010. Springer Science & Business Media. 978-1-84800-191-6. 20 April 2018. Google Books.
17. Book: Glisson, Tildon H.. Introduction to Circuit Analysis and Design. 18 February 2011. Springer Science & Business Media. 9789048194438. 20 April 2018. Google Books.