ATAN2(3) FreeBSD Library Functions Manual ATAN2(3)


atan2, atan2f, atan2l, carg, cargf, carglarc tangent and complex phase angle functions


Math Library (libm, -lm)


#include < math.h>

atan2( double y, double x);

atan2f( float y, float x);

long double
atan2l( long double y, long double x);

#include < complex.h>

carg( double complex z);

cargf( float complex z);

long double
cargl( long double complex z);


The atan2(), atan2f(), and atan2l() functions compute the principal value of the arc tangent of y/ x, using the signs of both arguments to determine the quadrant of the return value.

The carg(), cargf(), and cargl() functions compute the complex argument (or phase angle) of z. The complex argument is the number theta such that z = r * e^(I * theta), where r = cabs(z). The call carg(z) is equivalent to atan2(cimag(z), creal(z)), and similarly for cargf() and cargl().


The atan2(), atan2f(), and atan2l() functions, if successful, return the arc tangent of y/ x in the range [-pi, +pi] radians. Here are some of the special cases:
atan2( y, x) := atan( y/x) if x > 0,
sign( y)*(pi - atan( |y/x|)) if x < 0,
0 if x = y = 0, or
sign( y)*pi/2 if x = 0 ≠ y.


The function atan2() defines "if x > 0," atan2( 0, 0) = 0 despite that previously atan2( 0, 0) may have generated an error message. The reasons for assigning a value to atan2( 0, 0) are these:
  1. Programs that test arguments to avoid computing atan2( 0, 0) must be indifferent to its value. Programs that require it to be invalid are vulnerable to diverse reactions to that invalidity on diverse computer systems.
  2. The atan2() function is used mostly to convert from rectangular (x,y) to polar (r,theta) coordinates that must satisfy x = r∗cos theta and y = r∗sin theta. These equations are satisfied when (x=0,y=0) is mapped to (r=0,theta=0). In general, conversions to polar coordinates should be computed thus:

    r := hypot(x,y);  ... := sqrt(x∗x+y∗y) 
    theta := atan2(y,x).
  3. The foregoing formulas need not be altered to cope in a reasonable way with signed zeros and infinities on a machine that conforms to IEEE 754; the versions of hypot(3) and atan2() provided for such a machine are designed to handle all cases. That is why atan2( ±0, -0) = ±pi for instance. In general the formulas above are equivalent to these:

    r := sqrt(x∗x+y∗y); if r = 0 then x := copysign(1,x);


The atan2(), atan2f(), atan2l(), carg(), cargf(), and cargl() functions conform to ISO/IEC 9899:1999 (“ISO C99”).
July 31, 2008 FreeBSD