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Manual Reference Pages  - atan2 (3fortran)

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NAME

ATAN2(3) - [MATHEMATICS:TRIGONOMETRIC] Arctangent (inverse tangent) function

SYNOPSIS

result = atan2(y, x)

         elemental real(kind=KIND) function atan2(y, x)

          real,kind=KIND) :: atan2
          real,kind=KIND),intent(in) :: y, x

CHARACTERISTICS

o	X and Y must be reals of the same kind.
o	The return value has the same type and kind as Y and X.

DESCRIPTION

ATAN2(3) computes in radians a processor-dependent approximation of the arctangent of the complex number ( X, Y ) or equivalently the principal value of the arctangent of the value Y/X (which determines a unique angle).

If Y has the value zero, X shall not have the value zero.

The resulting phase lies in the range -PI <= ATAN2 (Y,X) <= PI and is equal to a processor-dependent approximation to a value of arctan(Y/X).

OPTIONS

o	Y : The imaginary component of the complex value (X,Y) or the Y component of the point <X,Y>.
o	X : The real component of the complex value (X,Y) or the X component of the point <X,Y>.

RESULT

The value returned is by definition the principal value of the complex number (X, Y), or in other terms, the phase of the phasor x+i*y.

The principal value is simply what we get when we adjust a radian value to lie between -PI and PI inclusive,

The classic definition of the arctangent is the angle that is formed in Cartesian coordinates of the line from the origin point <0,0> to the point <X,Y> .

Pictured as a vector it is easy to see that if X and Y are both zero the angle is indeterminate because it sits directly over the origin, so ATAN(0.0,0.0) will produce an error.

Range of returned values by quadrant:

    >                   +PI/2
    >                     |
    >                     |
    >     PI/2 < z < PI   |   0 > z < PI/2
    >                     |
    >   +-PI -------------+---------------- +-0
    >                     |
    >     PI/2 < -z < PI  |   0 < -z < PI/2
    >                     |
    >                     |
    >                   -PI/2
    >
         NOTES:

         If the processor distinguishes -0 and +0 then the sign of the
         returned value is that of Y when Y is zero, else when Y is zero
         the returned value is always positive.

EXAMPLES

Sample program:

    program demo_atan2
    real :: z
    complex :: c
     !
     ! basic usage
      ! ATAN2 (1.5574077, 1.0) has the value 1.0 (approximately).
      z=atan2(1.5574077, 1.0)
      write(*,*) ’radians=’,z,’degrees=’,r2d(z)
     !
     ! elemental arrays
      write(*,*)’elemental’,atan2( [10.0, 20.0], [30.0,40.0] )
     !
     ! elemental arrays and scalars
      write(*,*)’elemental’,atan2( [10.0, 20.0], 50.0 )
     !
     ! break complex values into real and imaginary components
     ! (note TAN2() can take a complex type value )
      c=(0.0,1.0)
      write(*,*)’complex’,c,atan2( x=c%re, y=c%im )
     !
     ! extended sample converting cartesian coordinates to polar
      COMPLEX_VALS: block
      real                :: ang, radius
      complex,allocatable :: vals(:)
      integer             :: i
     !
      vals=[ &
        ( 1.0, 0.0 ), & ! 0
        ( 1.0, 1.0 ), & ! 45
        ( 0.0, 1.0 ), & ! 90
        (-1.0, 1.0 ), & ! 135
        (-1.0, 0.0 ), & ! 180
        (-1.0,-1.0 ), & ! 225
        ( 0.0,-1.0 )]   ! 270
      do i=1,size(vals)
         call cartesian_to_polar(vals(i)%re, vals(i)%im, radius,ang)
         write(*,101)vals(i),ang,r2d(ang),radius
      enddo
      101 format(             &
      & ’X= ’,f5.2,           &
      & ’ Y= ’,f5.2,          &
      & ’ ANGLE= ’,g0,        &
      & T38,’DEGREES= ’,g0.4, &
      & T54,’DISTANCE=’,g0)
     endblock COMPLEX_VALS
    !
    contains
    !
    elemental real function r2d(radians)
    ! input radians to convert to degrees
    doubleprecision,parameter :: DEGREE=0.017453292519943d0 ! radians
    real,intent(in)           :: radians
       r2d=radians / DEGREE ! do the conversion
    end function r2d
    !
    subroutine cartesian_to_polar(x,y,radius,inclination)
    ! return angle in radians in range 0 to 2*PI
    implicit none
    real,intent(in)  :: x,y
    real,intent(out) :: radius,inclination
       radius=sqrt(x**2+y**2)
       if(radius.eq.0)then
          inclination=0.0
       else
          inclination=atan2(y,x)
          if(inclination < 0.0)inclination=inclination+2*atan2(0.0d0,-1.0d0)
       endif
    end subroutine cartesian_to_polar
    !
    end program demo_atan2

Results:

     >  radians=   1.000000     degrees=   57.29578
     >  elemental  0.3217506      0.4636476
     >  elemental  0.1973956      0.3805064
     >  complex (0.0000000E+00,1.000000)   1.570796
     > X=  1.00 Y=  0.00 ANGLE= .000000     DEGREES= .000   DISTANCE=1.000000
     > X=  1.00 Y=  1.00 ANGLE= .7853982    DEGREES= 45.00  DISTANCE=1.414214
     > X=  0.00 Y=  1.00 ANGLE= 1.570796    DEGREES= 90.00  DISTANCE=1.000000
     > X= -1.00 Y=  1.00 ANGLE= 2.356194    DEGREES= 135.0  DISTANCE=1.414214
     > X= -1.00 Y=  0.00 ANGLE= 3.141593    DEGREES= 180.0  DISTANCE=1.000000
     > X= -1.00 Y= -1.00 ANGLE= 3.926991    DEGREES= 225.0  DISTANCE=1.414214
     > X=  0.00 Y= -1.00 ANGLE= 4.712389    DEGREES= 270.0  DISTANCE=1.000000

STANDARD

FORTRAN 77

SEE ALSO

o	ATAN(3)

RESOURCES

o	arctan:wikipedia Fortran intrinsic descriptions (license: MIT) @urbanjost

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Nemo Release 3.1

atan2 (3fortran)

November 02, 2024

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