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Atan2
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In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real 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).
The atan2 function was first introduced in computer programming languages, but now it is also common in other fields of science and engineering.
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In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real 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).
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 and is currently found in the C programming language's math.h standard library, the Java Math library and elsewhere. Many scripting languages, such as Perl, include the C-style atan2 function.
In mathematical terms, atan2 computes the principal value of the argument function applied to the complex number x+iy. That is atan2(y, x) = Pr arg(x+iy) = Arg(x+iy). The argument can be changed by 2p corresponding to a complete turn round the origin without making any difference to the angle, the principal value is the value in the range (-p,p], that is -p < atan2(y, x) = p.
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 or a different name is used for the function. 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 associated function hypot, which is less widely defined, calculates the length of the hypotenuse of a right-angle triangle. This is easy to do using the square root function but hypot(x, y) avoids possible problems with very large or very small numbers.
Motivation
The one-argument arctangent function does not 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 p/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 p/4, even though the answer clearly should be −3p/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) = p/4 and atan2(−1, −1) = −3p/4.
Additionally, the ordinary arctangent method breaks down when required to produce an angle of ±p/2 (or ±90°). 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 p/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
For y?0:
where is the angle in [0,p/2) such that and sgn is the sign function.
And:
Notes:
- This produces results in the range (-p,p], which can be mapped to [0,2p) by adding 2p to negative values.
- Traditionally, atan2(0,0) is undefined.
- The C function, and most other computer implementations, are designed to reduce the effort 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 [-p,p] rather than raising an error or returning a NaN (Not a Number).
- Systems supporting symbolic mathematics should normally return an undefined value for atan2(0,0) or otherwise signal that an abnormal condition has arisen.
- For systems, for example IEEE floating point, implementing signed zero, infinities, or Not a Number it is usual to implement reasonable extensions which may extend the range of values produced to include -p and -0. These also may return NaN or raise an exception when given a NaN argument.
The free maths 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. This expression may be more suited for symbolic use than the definition above. It is however unsuitable for computational use; the division overflows in a neighbourhood of the negative x axis and gives an NaN or raises an error for atan2(0,0).
Variations
- In Common Lisp, where optional arguments exist, the atan function allows one to optionally supply the x coordinate as in (atan x y)
- 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, the atan2 function has the two arguments reversed. OpenOffice.org Calc also reverses the arguments.
- In the Intel Architecture assembler code, atan2 is known as FPATAN (floating-point partial arctangent) instruction. It can deal with infinities and results range from (closed interval), e.g. . Particularly, FPATAN is defined when both arguments are zero:**
- This definition is related with the concept of signed zero, i.e.
- On the TI-89 calculator, the equivalent function is called R?P? and has the arguments reversed.
The hypot function
When converting to polar coordinates the magnitude of the hypotenuse from (0, 0) to (x, y) can be calculated using:
There is a problem with very large or small values of x and y in that their squares may go out of the range of the machine representation. The hypot function when available calculates the result without causing such problems. It is not as widely defined as the atan2 function as using the square root function is easy and normally fully satisfactory. Using both together gives:
- r = hypot(x, y)
- theta = atan2(y, x)
Illustrations
The left diagram below shows values of atan2 at selected points on the unit circle. The values, in radians, are shown in blue inside the circle. The four points (1,0), (0,1), (-1,0), and (0,-1) are labeled outside the circle. Note that the order of arguments is reversed; the function atan2(y,x) computes the angle corresponding to the point (x,y).
The right diagram below show values argument of complex number with principal value from 0 to 2pi in radians measured in anticlockwise (counterclockwise) direction.
The diagram below shows values of atan2 for points of unit circle. On X axis is complex angle of point.
It starts from 0 ( point (0,1) ) and goes in anticlockwise (counterclockwise) direction through points :
- (0,1) with complex angle pi/2 in radians,
- (-1,0) with complex angle pi,
- (-1,-1) with complex angle 3pi/2,
to (1,0) with complex angle 0 = 2pi modulo 2pi
On this diagram one can see discontinuity of function atan2.
When a point z is crossing the negative real axis, for example point z goes from (0,1) through (-1,0) to (0,-1).
Its argument should go from pi/2 through pi to 3pi/2, but the output of atan2 ( principal value of argument) goes from pi/2 to pi, jumps to -pi (discontinuity), and goes to -pi/2.
The diagrams below show 3D view of respectively atan2 and over a region of the plane.
Note that for atan2, rays emanating from the origin have constant values, but for atan lines passing through the origin have constant values.
See also
External links
- (note argument order is reversed)
- at Everything2
Other implementations/code for atan2
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