If we assume that the Earth is a sphere of radius \(r\), the shortest distance between two sites (\([\phi_1,\lambda_1]\) and \([\phi_2,\lambda_2]\)) on the surface is the great-circle distance, which can be computed in several ways:
Law of Cosines equation:
$$
d = r \arccos\bigl(\sin\phi_1\cdot\sin\phi_2+\cos\phi_1\cdot\cos\phi_2\cdot\cos(\Delta\lambda)\bigr)
$$
This equation, while mathematically correct, is not usually recommended for use, since it is ill-conditioned for sites that are very close to each other.
Using the haversine equation:
$$
d = 2 r \arcsin \sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right)+\cos{\phi_1}\cdot\cos{\phi_2}\cdot\sin^2\left(\frac{\Delta\lambda}{2}\right)}
$$
The haversine one is better, but can be ill-conditioned for near antipodal points.
Using the Vincenty equation:
$$
d = r \arctan \frac{\sqrt{\left(\cos\phi_2\cdot\sin(\Delta\lambda)\right)^2+\left(\cos\phi_1\cdot\sin\phi_2-\sin\phi_1\cdot\cos\phi_2\cdot\cos(\Delta\lambda)\right)^2}}{\sin\phi_1\cdot\sin\phi_2+\cos\phi_1\cdot\cos\phi_2\cdot\cos(\Delta\lambda)}
$$
This one is accurate for any two points on a sphere.
Code
Fortran implementations of these algorithms are given here:
Law of Cosines:
pure function gcdist(r,lat1,long1,lat2,long2) &
result(d)
implicit none
real(wp) :: d !! great circle distance from 1 to 2 [km]
real(wp),intent(in) :: r !! radius of the body [km]
real(wp),intent(in) :: long1 !! longitude of first site [rad]
real(wp),intent(in) :: lat1 !! latitude of the first site [rad]
real(wp),intent(in) :: long2 !! longitude of the second site [rad]
real(wp),intent(in) :: lat2 !! latitude of the second site [rad]
real(wp) :: clat1,clat2,slat1,slat2,dlon,cdlon
clat1 = cos(lat1)
clat2 = cos(lat2)
slat1 = sin(lat1)
slat2 = sin(lat2)
dlon = long1-long2
cdlon = cos(dlon)
d = r * acos(slat1*slat2+clat1*clat2*cdlon)
end function gcdist
Haversine:
pure function gcdist_haversine(r,lat1,long1,lat2,long2) &
result(d)
implicit none
real(wp) :: d !! great circle distance from 1 to 2 [km]
real(wp),intent(in) :: r !! radius of the body [km]
real(wp),intent(in) :: long1 !! longitude of first site [rad]
real(wp),intent(in) :: lat1 !! latitude of the first site [rad]
real(wp),intent(in) :: long2 !! longitude of the second site [rad]
real(wp),intent(in) :: lat2 !! latitude of the second site [rad]
real(wp) :: dlat,clat1,clat2,dlon,a
dlat = lat1-lat2
clat1 = cos(lat1)
clat2 = cos(lat2)
dlon = long1-long2
a = sin(dlat/2.0_wp)**2 + clat1*clat2*sin(dlon/2.0_wp)**2
d = r * 2.0_wp * asin(min(1.0_wp,sqrt(a)))
end function gcdist_haversine
Vincenty:
pure function gcdist_vincenty(r,lat1,long1,lat2,long2) &
result(d)
implicit none
real(wp) :: d !! great circle distance from 1 to 2 [km]
real(wp),intent(in) :: r !! radius of the body [km]
real(wp),intent(in) :: long1 !! longitude of first site [rad]
real(wp),intent(in) :: lat1 !! latitude of the first site [rad]
real(wp),intent(in) :: long2 !! longitude of the second site [rad]
real(wp),intent(in) :: lat2 !! latitude of the second site [rad]
real(wp) :: c1,s1,c2,s2,dlon,clon,slon
c1 = cos(lat1)
s1 = sin(lat1)
c2 = cos(lat2)
s2 = sin(lat2)
dlon = long1-long2
clon = cos(dlon)
slon = sin(dlon)
d = r*atan2(sqrt((c2*slon)**2+(c1*s2-s1*c2*clon)**2), &
(s1*s2+c1*c2*clon))
end function gcdist_vincenty
Test cases
Using a radius of the Earth of 6,378,137 meters, here are a few test cases (using double precision (real64) arithmetic):
Close points
The distance between the following very close points \(([\phi_1,\lambda_1], [\phi_2,\lambda_2]) =([0, 0.000001], [0.0, 0.0])\) (radians) is:
| Algorithm |
Procedure |
Result (meters) |
| Law of Cosines |
gcdist |
6.3784205037462689 |
| Haversine |
gcdist_haversine |
6.3781369999999997 |
| Vincenty |
gcdist_vincenty |
6.3781369999999997 |
Houston to New York
The distance between the following two points \(([\phi_1,\lambda_1], [\phi_2,\lambda_2]) =([29.97, -95.35], [40.77, -73.98])\) (deg) is:
| Algorithm |
Procedure |
Result (meters) |
| Law of Cosines |
gcdist |
2272779.3057236285 |
| Haversine |
gcdist_haversine |
2272779.3057236294 |
| Vincenty |
gcdist_vincenty |
2272779.3057236290 |
Antipodal points
The distance between the following two antipodal points \(([\phi_1,\lambda_1], [\phi_2,\lambda_2]) = ([0, 0], [0.0, \pi/2])\) (radians) is:
| Algorithm |
Procedure |
Result (meters) |
| Law of Cosines |
gcdist |
20037508.342789244 |
| Haversine |
gcdist_haversine |
20037508.342789244 |
| Vincenty |
gcdist_vincenty |
20037508.342789244 |
Nearly antipodal points
The distance between the following two nearly antipodal points \(([\phi_1,\lambda_1], [\phi_2,\lambda_2]) = ([10^{-8}, 10^{-8}], [0.0, \pi/2])\) (radians) is:
| Algorithm |
Procedure |
Result (meters) |
| Law of Cosines |
gcdist |
20037508.342789244 |
| Haversine |
gcdist_haversine |
20037508.342789244 |
| Vincenty |
gcdist_vincenty |
20037508.252588764 |
Other methods
For a more accurate result, using an oblate-spheroid model of the Earth, the Vincenty inverse geodetic algorithm can be used (the equation above is just a special case of the more general algorithm when the ellipsoidal major and minor axes are equal). For the Houston to New York case, the difference between the spherical and oblate-spheroid equations is 282 meters.
See also
Thaddeus Vincenty (1920--2002) Image source: IAG Newsletter
There are two standard problems in geodesy:
- Direct geodetic problem -- Given a point on the Earth (latitude and longitude) and the direction (azimuth) and distance from that point to a second point, determine the latitude and longitude of that second point.
- Inverse geodetic problem -- Given two points on the Earth, determine the azimuth and distance between them.
Assuming the Earth is an oblate spheroid, both of these can be solved using the iterative algorithms by Thaddeus Vincenty [1]. Modern Fortran versions of Vincenty's algorithms can be found in the geodesy module of the Fortran Astrodynamics Toolkit [2].
The direct algorithm was listed in an earlier post. If we assume a WGS84 Earth ellipsoid (a=6378.137 km, f=1/298.257223563), an example direct calculation is:
$$
\begin{array}{rcll}
\phi_1 &= & 29.97^{\circ} & \mathrm{(Latitude ~of ~ first ~ point)}\\
\lambda_1 &= & -95.35^{\circ} & \mathrm{(Longitude ~of ~ first ~point)} \\
Az &= & 20^{\circ} & \mathrm{(Azimuth)}\\
s &= & 50~\mathrm{km} & \mathrm{(Distance)}
\end{array}
$$
The latitude and longitude of the second point is:
$$
\begin{array}{rcl}
\phi_2 &= & 30.393716^{\circ} \\
\lambda_2 &= & -95.172057^{\circ} \\
\end{array}
$$
The more complicated inverse algorithm is listed here (based on the USNGS code from [1]):
subroutine inverse(a, rf, b1, l1, b2, l2, faz, baz, s, it, sig, lam, kind)
use, intrinsic :: iso_fortran_env, wp => real64
implicit none
real(wp), intent(in) :: a !! Equatorial semimajor axis
real(wp), intent(in) :: rf !! reciprocal flattening (1/f)
real(wp), intent(in) :: b1 !! latitude of point 1 (rad, positive north)
real(wp), intent(in) :: l1 !! longitude of point 1 (rad, positive east)
real(wp), intent(in) :: b2 !! latitude of point 2 (rad, positive north)
real(wp), intent(in) :: l2 !! longitude of point 2 (rad, positive east)
real(wp), intent(out) :: faz !! Forward azimuth (rad, clockwise from north)
real(wp), intent(out) :: baz !! Back azimuth (rad, clockwise from north)
real(wp), intent(out) :: s !! Ellipsoidal distance
integer, intent(out) :: it !! iteration count
real(wp), intent(out) :: sig !! spherical distance on auxiliary sphere
real(wp), intent(out) :: lam !! longitude difference on auxiliary sphere
integer, intent(out) :: kind !! solution flag: kind=1, long-line; kind=2, antipodal
real(wp) :: beta1, beta2, biga, bigb, bige, bigf, boa, &
c, cosal2, coslam, cossig, costm, costm2, &
cosu1, cosu2, d, dsig, ep2, l, prev, sinal, &
sinlam, sinsig, sinu1, sinu2, tem1, tem2, &
temp, test, z
real(wp), parameter :: pi = acos(-1.0_wp)
real(wp), parameter :: two_pi = 2.0_wp*pi
real(wp), parameter :: tol = 1.0e-14_wp !! convergence tolerance
real(wp), parameter :: eps = 1.0e-15_wp !! tolerance for zero
boa = 1.0_wp - 1.0_wp/rf
beta1 = atan(boa*tan(b1))
sinu1 = sin(beta1)
cosu1 = cos(beta1)
beta2 = atan(boa*tan(b2))
sinu2 = sin(beta2)
cosu2 = cos(beta2)
l = l2 - l1
if (l > pi) l = l - two_pi
if (l < -pi) l = l + two_pi
prev = l
test = l
it = 0
kind = 1
lam = l
longline: do ! long-line loop (kind=1)
sinlam = sin(lam)
coslam = cos(lam)
temp = cosu1*sinu2 - sinu1*cosu2*coslam
sinsig = sqrt((cosu2*sinlam)**2 + temp**2)
cossig = sinu1*sinu2 + cosu1*cosu2*coslam
sig = atan2(sinsig, cossig)
if (abs(sinsig) < eps) then
sinal = cosu1*cosu2*sinlam/sign(eps, sinsig)
else
sinal = cosu1*cosu2*sinlam/sinsig
end if
cosal2 = -sinal**2 + 1.0_wp
if (abs(cosal2) < eps) then
costm = -2.0_wp*(sinu1*sinu2/sign(eps, &
cosal2)) + cossig
else
costm = -2.0_wp*(sinu1*sinu2/cosal2) + &
cossig
end if
costm2 = costm*costm
c = ((-3.0_wp*cosal2 + 4.0_wp)/rf + 4.0_wp)* &
cosal2/rf/16.0_wp
antipodal: do ! antipodal loop (kind=2)
it = it + 1
d = (((2.0_wp*costm2 - 1.0_wp)*cossig*c + &
costm)*sinsig*c + sig)*(1.0_wp - c)/rf
if (kind == 1) then
lam = l + d*sinal
if (abs(lam - test) >= tol) then
if (abs(lam) > pi) then
kind = 2
lam = pi
if (l < 0.0_wp) lam = -lam
sinal = 0.0_wp
cosal2 = 1.0_wp
test = 2.0_wp
prev = test
sig = pi - abs(atan(sinu1/cosu1) + &
atan(sinu2/cosu2))
sinsig = sin(sig)
cossig = cos(sig)
c = ((-3.0_wp*cosal2 + 4.0_wp)/rf + &
4.0_wp)*cosal2/rf/16.0_wp
if (abs(sinal - prev) < tol) exit longline
if (abs(cosal2) < eps) then
costm = -2.0_wp*(sinu1*sinu2/ &
sign(eps, cosal2)) + cossig
else
costm = -2.0_wp*(sinu1*sinu2/cosal2) + &
cossig
end if
costm2 = costm*costm
cycle antipodal
end if
if (((lam - test)*(test - prev)) < 0.0_wp .and. it > 5) &
lam = (2.0_wp*lam + 3.0_wp*test + prev)/6.0_wp
prev = test
test = lam
cycle longline
end if
else
sinal = (lam - l)/d
if (((sinal - test)*(test - prev)) < 0.0_wp .and. it > 5) &
sinal = (2.0_wp*sinal + 3.0_wp*test + prev)/6.0_wp
prev = test
test = sinal
cosal2 = -sinal**2 + 1.0_wp
sinlam = sinal*sinsig/(cosu1*cosu2)
coslam = -sqrt(abs(-sinlam**2 + 1.0_wp))
lam = atan2(sinlam, coslam)
temp = cosu1*sinu2 - sinu1*cosu2*coslam
sinsig = sqrt((cosu2*sinlam)**2 + temp**2)
cossig = sinu1*sinu2 + cosu1*cosu2*coslam
sig = atan2(sinsig, cossig)
c = ((-3.0_wp*cosal2 + 4.0_wp)/rf + 4.0_wp)* &
cosal2/rf/16.0_wp
if (abs(sinal - prev) >= tol) then
if (abs(cosal2) < eps) then
costm = -2.0_wp*(sinu1*sinu2/ &
sign(eps, cosal2)) + cossig
else
costm = -2.0_wp*(sinu1*sinu2/cosal2) + &
cossig
end if
costm2 = costm*costm
cycle antipodal
end if
end if
exit longline
end do antipodal
end do longline
if (kind == 2) then ! antipodal
faz = sinal/cosu1
baz = sqrt(-faz**2 + 1.0_wp)
if (temp < 0.0_wp) baz = -baz
faz = atan2(faz, baz)
tem1 = -sinal
tem2 = sinu1*sinsig - cosu1*cossig*baz
baz = atan2(tem1, tem2)
else ! long-line
tem1 = cosu2*sinlam
tem2 = cosu1*sinu2 - sinu1*cosu2*coslam
faz = atan2(tem1, tem2)
tem1 = -cosu1*sinlam
tem2 = sinu1*cosu2 - cosu1*sinu2*coslam
baz = atan2(tem1, tem2)
end if
if (faz < 0.0_wp) faz = faz + two_pi
if (baz < 0.0_wp) baz = baz + two_pi
ep2 = 1.0_wp/(boa*boa) - 1.0_wp
bige = sqrt(1.0_wp + ep2*cosal2)
bigf = (bige - 1.0_wp)/(bige + 1.0_wp)
biga = (1.0_wp + bigf*bigf/4.0_wp)/(1.0_wp - bigf)
bigb = bigf*(1.0_wp - 0.375_wp*bigf*bigf)
z = bigb/6.0_wp*costm*(-3.0_wp + 4.0_wp* &
sinsig**2)*(-3.0_wp + 4.0_wp*costm2)
dsig = bigb*sinsig*(costm + bigb/4.0_wp* &
(cossig*(-1.0_wp + 2.0_wp*costm2) - z))
s = (boa*a)*biga*(sig - dsig)
end subroutine inverse
For the same ellipsoid, an example inverse calculation is:
$$
\begin{array}{rcll}
\phi_1 &= & 29.97^{\circ} & \mathrm{(Latitude ~of ~ first ~ point)}\\
\lambda_1 &= & -95.35^{\circ} & \mathrm{(Longitude ~of ~ first ~point)} \\
\phi_2 &= & 40.77^{\circ} & \mathrm{(Latitude ~of ~ second ~ point)}\\
\lambda_2 &= & -73.98^{\circ} & \mathrm{(Longitude ~of ~ second ~point)}
\end{array}
$$
The azimuth and distance from the first to the second point is:
$$
\begin{array}{rcl}
Az &= & 52.400056^{\circ} & \mathrm{(Azimuth)}\\
s &= & 2272.497~\mathrm{km} & \mathrm{(Distance)}
\end{array}
$$
References
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
- 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].