Direct Geodetic Problem

The “direct” geodetic problem is: given the latitude and longitude of one point and the azimuth and distance to a second point, determine the latitude and longitude of that second point.  The solution can be obtained using the algorithm by Polish American geodesist Thaddeus Vincenty [1]. A modern Fortran implementation is given below:

subroutine direct(a,f,glat1,glon1,glat2,glon2,faz,baz,s)

use, intrinsic :: iso_fortran_env, wp => real64

implicit none

real(wp),intent(in)  :: a        !semimajor axis of ellipsoid [m]
real(wp),intent(in)  :: f        !flattening of ellipsoid [-]
real(wp),intent(in)  :: glat1    !latitude of 1 [rad]
real(wp),intent(in)  :: glon1    !longitude of 1 [rad]
real(wp),intent(in)  :: faz      !forward azimuth 1->2 [rad]
real(wp),intent(in)  :: s        !distance from 1->2 [m]
real(wp),intent(out) :: glat2    !latitude of 2 [rad]
real(wp),intent(out) :: glon2    !longitude of 2 [rad]
real(wp),intent(out) :: baz      !back azimuth 2->1 [rad]

real(wp) :: r,tu,sf,cf,cu,su,sa,csa,c2a,x,c,d,y,sy,cy,cz,e

real(wp),parameter :: pi  = acos(-1.0_wp)
real(wp),parameter :: eps = 0.5e-13_wp

r = 1.0_wp-f
tu = r*sin(glat1)/cos(glat1)
sf = sin(faz)
cf = cos(faz)
baz = 0.0_wp
if (cf/=0.0_wp) baz = atan2(tu,cf)*2.0_wp
cu = 1.0_wp/sqrt(tu*tu+1.0_wp)
su = tu*cu
sa = cu*sf
c2a = -sa*sa+1.0_wp
x = sqrt((1.0_wp/r/r-1.0_wp)*c2a+1.0_wp)+1.0_wp
x = (x-2.0_wp)/x
c = 1.0_wp-x
c = (x*x/4.0_wp+1.0_wp)/c
d = (0.375_wp*x*x-1.0_wp)*x
tu = s/r/a/c
y = tu
do
  sy = sin(y)
  cy = cos(y)
  cz = cos(baz+y)
  e = cz*cz*2.0_wp-1.0_wp
  c = y
  x = e*cy
  y = e+e-1.0_wp
  y = (((sy*sy*4.0_wp-3.0_wp)*y*cz*d/6.0_wp+x)*d/4.0_wp-cz)*sy*d+tu
  if (abs(y-c)<=eps) exit
end do
baz = cu*cy*cf-su*sy
c = r*sqrt(sa*sa+baz*baz)
d = su*cy+cu*sy*cf
glat2 = atan2(d,c)
c = cu*cy-su*sy*cf
x = atan2(sy*sf,c)
c = ((-3.0_wp*c2a+4.0_wp)*f+4.0_wp)*c2a*f/16.0_wp
d = ((e*cy*c+cz)*sy*c+y)*sa
glon2 = glon1+x-(1.0_wp-c)*d*f
baz = atan2(sa,baz)+pi

end subroutine direct

References

  1. T. Vincenty, “Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations”, Survey Review XXII. 176, April 1975. [sourcecode from the U.S. National Geodetic Survey].

 

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