ATAN2

'atan2' is a two-argument function that computes the arctangent of $y/x$ given ''y'' and ''x'', but with a range of $(-pi,pi]$. It was introduced first in many computer programming languages but is now common in all fields of science and engineering too.

Contents

Definition

Variations

References

See also

External links

Other implementations/code for atan2

Definition

''atan2'' is defined using the standard arctan function, whose range is (−π/2, π/2), as follows:
:
rctan(rac y x) & qquad x > 0 \
pi + rctan(rac y x) & qquad y ge 0 , x < 0 \
-pi + rctan(rac y x) & qquad y < 0 , x < 0 \
rac{pi}{2} & qquad y > 0 , x = 0 \
-rac{pi}{2} & qquad y < 0 , x = 0 \
ext{undefined} & qquad y = 0, x = 0 \
end{cases}
(Note that the first parameter is ''y'' and ''x'' is the second.)
The so defined atan2 has domain in $Bbb\{R\}^2\; -\; \{(0,\; 0)\}$ and range in (−π, π].
If is a complex number, then
:
is its argument in the interval (-π,π].
The same result is valid if a point on a plane is represented by (x,y) in cartesian coordinates and by (r,φ) in polar coordinates.
atan2 is available in most modern programming languages. The Linux Programmer's Manual [1] says:
:"The atan2 C++ function calculates the arctangent of the two variables '''y''' and '''x'''. It is similar to calculating the arctangent of '''y'''/'''x''', except that the signs of both arguments are used to determine the quadrant of the result."
Effectively, this means that atan2(y,x) finds the counterclockwise angle in radians between the ''x''-axis and the vector '' in 2-dimensional Euclidean space, which is useful in many applications involving vectors, such as finding the direction from one point to another.
This is the preferred form ^[1] for correctly finding the sign of the result in all quadrants. See also inverse trigonometric function.
:
-operatorname{atan2}(-y, x) & qquad y < 0 \
pi - rctan(-rac y x) & qquad y ge 0 , x < 0 \
rctan(rac y x) & qquad y ge 0 , x > 0 \
rac{pi}{2} & qquad y > 0 , x = 0 \
ext{undefined} & qquad y = 0, x = 0 \
end{cases}

Variations

★ In Microsoft Excel, the atan2 function has the two arguments reversed.^[2] . OpenOffice.org Calc also reverses the arguments.

★ In the Intel Architecture assembler code, ''atan2'' is known as FPATAN (floating-point partial arctangent) instruction ^[3]. It can deal with infinities and results range from $[-pi,+pi]$ (closed interval), e.g. $operatorname\{atan2\}(infty,\; x)=+pi$. Particularly, FPATAN 'is' defined when both arguments are zero:

★ :$operatorname\{atan2\}(+0,\; +0)\; =\; +0$

★ :$operatorname\{atan2\}(+0,\; -0)\; =\; +pi$

★ :$operatorname\{atan2\}(-0,\; +0)\; =\; -0$

★ :$operatorname\{atan2\}(-0,\; -0)\; =\; -pi$

★ :This definition is related with the concept of signed zero, i.e. $pm\; 0;\; overset\{def\}\{=\}\; lim\_\{;x$
ightarrow 0^pm} x

★ On the TI-89 calculator, the equivalent function is called 'R►Pθ' and has the arguments reversed.

References

1. [2]
2. Atan2 Method
3. IA-32 Intel® Architecture Software Developer’s Manual. Volume 2A: Instruction Set Reference, A-M, 2004.

See also

★ Complex number

★ Inverse trigonometric function

External links

★ C++ Programmer's Reference

★ MATLAB Function Reference

★ Mathematica Function Reference

Other implementations/code for atan2

★ [3]

★ Bearing Between Two Points

★ Arctan and Polar Coordinates

★ What's 'Arccos'?