The 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 in converting from Cartesian coordinates (x, y) to
polar coordinates(r, θ). If and , then and
If x > 0, the desired angle measure is However, when x < 0, the angle is
diametrically opposite the desired angle, and ±π (a half
turn) must be added to place the point in the correct
quadrant.[1] Using the 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 x-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 with ).
Diametrically opposite angle measures have the same tangent because so the tangent is not in itself sufficient to uniquely specify an angle.
To determine an angle measure using the arctangent function given a point or vector mathematical formulas or computer code must handle multiple cases; at least one for positive values of and one for negative values of and sometimes additional cases when 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 atan2 function, at least as early as the
Fortran IV language of the 1960s.[2] The quantity atan2(y,x) is the angle measure between the x-axis and a ray from the origin to a point (x, y) anywhere in the Cartesian plane. The
signs of x and y are used to determine the
quadrant of the result and select the correct branch of the
multivalued functionArctan(y/x).
The atan2 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 atan2 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 atan2 function with argument order so that the
argument (phase angle) of a complex number is This follows the left-to-right order of a fraction written so that for positive values of However, this is the opposite of the conventional component order for complex numbers, or as coordinates See section
Definition and computation.
The function atan2 computes the
principal value of the
argument function applied to the
complex numberx + iy. That is, atan2(y, x) = Pr arg(x + iy) = Arg(x + iy). The argument could be changed by an arbitrary multiple of 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, that is, −π < atan2(y, x) ≤ π.
In terms of the standard arctan function, whose range is (−π/2, π/2), 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
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 expand near the region x < 0, y = 0 (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 (−π, π].[note 2]
As mentioned above, the principal value of the argument atan2(y, x) can be related to arctan(y/x) by trigonometry. The derivation goes as follows: If (x, y) = (r cos θ, r sin θ), then tan(θ/2) = y / (r + x). It follows that
Note that √x2 + y2 + x ≠ 0 in the domain in question.
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 ≠ 0,
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 differential:
While the function atan2 is discontinuous along the negative x-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.
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 atan2(y, x) computes the angle corresponding to the point (x, y).
This figure shows the values of along with for . Both functions are odd and periodic with periods and , respectively, and thus can easily be supplemented to any region of real values of . One can clearly see the
branch cuts of the -function at , and of the -function at .[3]
The two figures below show 3D views of respectively atan2(y, x) and arctan(y/x) over a region of the plane. Note that for atan2(y, x), rays in the X/Y-plane emanating from the origin have constant values, but for arctan(y/x)lines in the X/Y-plane passing through the origin have constant values. For x > 0, the two diagrams give identical values.
To see (4), we have the
identity where , hence . Furthermore, since for any positive real value , then if we let and then we have .
From these observations have following equivalences:
Corollary: if and are 2-dimensional vectors, the difference formula is frequently used in practice to compute the angle between those vectors with the help of , since the resulting computation behaves benign in the range and can thus be used without range checks in many practical situations.
East-counterclockwise, north-clockwise and south-clockwise conventions, etc.
The 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 function with the east- and north-components of the wind vector as its arguments;[4] 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.[5] These different conventions can be realized by swapping the positions and changing the signs of the x- and y-arguments as follows:
(East-Counterclockwise Convention)
(North-Clockwise Convention)
. (South-Clockwise Convention)
As an example, let and , then the east-counterclockwise format gives , the north-clockwise format gives , and the south-clockwise format gives .
Changing the sign of the x- and/or y-arguments and/or swapping their positions can create 8 possible variations of the 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
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 .
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 + iy = (x, y).
The 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 [−π, π] 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 yx).[11]
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.[12] 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?" [13]).
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 when y = −0. These also may return NaN or raise an exception when given a NaN argument.
In the
Intel x86 Architectureassembler code, atan2 is known as the FPATAN (floating-point partial arctangent) instruction.[14] 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.
In mathematical writings other than source code, such as in books and articles, the notations Arctan[15] and Tan−1[16] 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 Atan(y, x) = Arg(x + iy).
On
HP calculators, treat the coordinates as a complex number and then take the ARG. Or << C->R ARG >> 'ATAN2' STO.
Systems supporting symbolic mathematics normally return an undefined value for atan2(0, 0) 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 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).
^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).