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.Dd July 31, 2008
.Dt ATAN2 3
.Os
.Sh NAME
.Nm atan2 ,
.Nm atan2f ,
.Nm atan2l ,
.Nm carg ,
.Nm cargf ,
.Nm cargl
.Nd arc tangent and complex phase angle functions
.Sh LIBRARY
.Lb libm
.Sh SYNOPSIS
.In math.h
.Ft double
.Fn atan2 "double y" "double x"
.Ft float
.Fn atan2f "float y" "float x"
.Ft long double
.Fn atan2l "long double y" "long double x"
.In complex.h
.Ft double
.Fn carg "double complex z"
.Ft float
.Fn cargf "float complex z"
.Ft long double
.Fn cargl "long double complex z"
.Sh DESCRIPTION
The
.Fn atan2 ,
.Fn atan2f ,
and
.Fn atan2l
functions compute the principal value of the arc tangent of
.Fa y/ Ns Fa x ,
using the signs of both arguments to determine the quadrant of
the return value.
.Pp
The
.Fn carg ,
.Fn cargf ,
and
.Fn cargl
functions compute the complex argument (or phase angle) of
.Fa z .
The complex argument is the number theta such that
.Li z = r * e^(I * theta) ,
where
.Li r = cabs(z) .
The call
.Li carg(z)
is equivalent to
.Li atan2(cimag(z), creal(z)) ,
and similarly for
.Fn cargf
and
.Fn cargl .
.Sh RETURN VALUES
The
.Fn atan2 ,
.Fn atan2f ,
and
.Fn atan2l
functions, if successful,
return the arc tangent of
.Fa y/ Ns Fa x
in the range
.Bk -words
.Bq \&- Ns \*(Pi , \&+ Ns \*(Pi
.Ek
radians.
Here are some of the special cases:
.Bl -column atan_(y,x)_:=____  sign(y)_(Pi_atan2(Xy_xX))___
.It Fn atan2 y x No := Ta
.Fn atan y/x Ta
if
.Fa x
> 0,
.It Ta sign( Ns Fa y Ns )*(\*(Pi -
.Fn atan "\*(Bay/x\*(Ba" ) Ta
if
.Fa x
< 0,
.It Ta
.No 0 Ta
if x = y = 0, or
.It Ta
.Pf sign( Fa y Ns )*\*(Pi/2 Ta
if
.Fa x
= 0 \(!=
.Fa y .
.El
.Sh NOTES
The function
.Fn atan2
defines "if x > 0,"
.Fn atan2 0 0
= 0 despite that previously
.Fn atan2 0 0
may have generated an error message.
The reasons for assigning a value to
.Fn atan2 0 0
are these:
.Bl -enum -offset indent
.It
Programs that test arguments to avoid computing
.Fn 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.
.It
The
.Fn atan2
function is used mostly to convert from rectangular (x,y)
to polar
.if n\
(r,theta)
.if t\
(r,\(*h)
coordinates that must satisfy x =
.if n\
r\(**cos theta
.if t\
r\(**cos\(*h
and y =
.if n\
r\(**sin theta.
.if t\
r\(**sin\(*h.
These equations are satisfied when (x=0,y=0)
is mapped to
.if n \
(r=0,theta=0).
.if t \
(r=0,\(*h=0).
In general, conversions to polar coordinates
should be computed thus:
.Bd -unfilled -offset indent
.if n \{\
r	:= hypot(x,y);  ... := sqrt(x\(**x+y\(**y)
theta	:= atan2(y,x).
.\}
.if t \{\
r	:= hypot(x,y);  ... := \(sr(x\u\s82\s10\d+y\u\s82\s10\d)
\(*h	:= atan2(y,x).
.\}
.Ed
.It
The foregoing formulas need not be altered to cope in a
reasonable way with signed zeros and infinities
on a machine that conforms to
.Tn IEEE 754 ;
the versions of
.Xr hypot 3
and
.Fn atan2
provided for
such a machine are designed to handle all cases.
That is why
.Fn atan2 \(+-0 \-0
= \(+-\*(Pi
for instance.
In general the formulas above are equivalent to these:
.Bd -unfilled -offset indent
.if n \
r := sqrt(x\(**x+y\(**y); if r = 0 then x := copysign(1,x);
.if t \
r := \(sr(x\(**x+y\(**y);\0\0if r = 0 then x := copysign(1,x);
.Ed
.El
.Sh SEE ALSO
.Xr acos 3 ,
.Xr asin 3 ,
.Xr atan 3 ,
.Xr cabs 3 ,
.Xr cos 3 ,
.Xr cosh 3 ,
.Xr math 3 ,
.Xr sin 3 ,
.Xr sinh 3 ,
.Xr tan 3 ,
.Xr tanh 3
.Sh STANDARDS
The
.Fn atan2 ,
.Fn atan2f ,
.Fn atan2l ,
.Fn carg ,
.Fn cargf ,
and
.Fn cargl
functions conform to
.St -isoC-99 .