Formula for coordinates (lat, lon), azimuth and distance.

Question

I would like to know how to find a set of coordinates on a small circle some distance from another point.

For example, lets say A = (39.73, -104.98) #Denver

B = (39.83, -106.06) #point approximately 50 nautical miles away from A along the great circle track to C

C = (40.75, -111.88) #Salt Lake City, approximately 323 nautical miles away from A

How do I get the set of orange coordinates when I don't know the angle but I do know the cross track distance?

For concreteness, lets say the outer points near B have a cross track distance of +- 30 nautical miles and the inner points have a cross track of +- 15nm.

And the points near C have a cross track of +- 25 nm.

Also I have an initial bearing from A to C of -1.34 #radians

Answer 1

There is no one formula to solve this problem on a spheroid (since it's a partial differential equation which can only be solved by iterative means), but the solution is a trivial implementation of the "direct" problem (given a point, a bearing, and a distance, locate a point). The US National Geodetic Survey has a web site with FORTRAN source (I converted an earlier version of this code to 'C' and Java without difficulty).

You'd need to determine if the cross-track distance should be a chord or an arc when calculating the change in bearing (inverse sine of distance/radius vs. a fraction of the 2*pi*radius perimeter).

EXAMPLE:

Assuming arc distance, the bearing changes 17.18873 degrees for 15nm, 34.37746 degrees for 30nm, and 4.43466 degrees for 25nm, so the problem becomes:

name    lat     lon     bearing distance
B+30    39.73   -104.98 -42.40  50
B+15    39.73   -104.98 -59.59  50
B       39.73   -104.98 -76.78  50
B-15    39.73   -104.98 -93.97  50
B-30    39.73   -104.98 -111.15 50
C+25    39.73   -104.98 -72.34  323
C       39.73   -104.98 -76.78  323
C-25    39.73   -104.98 -81.21  323

which solved to:

name    lat     lon 
B+30    40.344  -105.715
B+15    40.148  -105.917
B       39.916  -106.034
B-15    39.667  -106.057
B-30    39.425  -105.983
C+25    41.167  -111.778
C       40.758  -111.883
C-25    40.345  -111.945

which looks right in this plot:

Answer 2

If the unknown points are on the specified circles, and if you assume a simple circular reference surface (a sphere, not an ellipsoid), then it is easy to get the angle at A from AC to the unknown points:

angle = ctd / scd, where

ctd is your so-called cross-track distance (what I'd call arc distance), and

scd is your small-circle distance away from A

Then you "just" calculate the new coordinates of unknown point, U, say:

coordsU = traverse (coordsA, azimAC + angle, scd), where

coordsA are coordinates at A,

azimAC is azimuth AC, (conveniently, you already have this in radians),

angle and scd are as above,

traverse is the appropriate function for the so-called "direct" problem of spherical trigonometry -- for now, left as a separate exercise for you to research, or already available from some library.

Answer 3

Geodesic calculations like these can be done in python using http://pypi.python.org/pypi/geographiclib. Assuming you have this installed, the following python computes the answers you want

import sys
sys.path.append("/usr/local/lib/python/site-packages")
from geographiclib.geodesic import Geodesic,Math
NM=1852
MASK=Geodesic.DISTANCE|Geodesic.AZIMUTH|Geodesic.REDUCEDLENGTH
lata=39.73; lona=-104.98
latb=39.83; lonb=-106.06
latc=40.75; lonc=-111.88

gab=Geodesic.WGS84.Inverse(lata,lona,latb,lonb,MASK)
print 'Points around B'
for dist in [-30,-15,0,15,30]:
    g=Geodesic.WGS84.Direct(lata,lona,
                            gab['azi1']+dist*NM/gab['m12']/Math.degree,
                            gab['s12'])
    print g['lat2'],g['lon2']

gac=Geodesic.WGS84.Inverse(lata,lona,latc,lonc,MASK)
print 'Points around C'
for dist in [-25,0,25]:
    g=Geodesic.WGS84.Direct(lata,lona,
                            gac['azi1']+dist*NM/gac['m12']/Math.degree,
                            gac['s12'])
    print g['lat2'],g['lon2']

Running this I get

Points around B
39.3452857009 -105.943379739
39.5808782474 -106.048518139
39.83 -106.06
40.0707028272 -105.975822882
40.2815616574 -105.802535085
Points around C
40.3357882212 -111.941310951
40.75 -111.88
41.1592654552 -111.775837558

Note that the cross track distance should be converted to a change in azimuth using the reduced length, m12, instead of the distance, s12. It doesn't make much difference in this case; but if A and C were separated by a greater distance it would begin to matter.