1GEODSOLVE(1) GeographicLib Utilities GEODSOLVE(1)
2
6 GeodSolve -- perform geodesic calculations
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9 GeodSolve [ -i | -L lat1 lon1 azi1 | -D lat1 lon1 azi1 s13 | -I lat1
10 lon1 lat3 lon3 ] [ -a ] [ -e a f ] [ -u ] [ -F ] [ -d | -: ] [ -w ] [
11 -b ] [ -f ] [ -p prec ] [ -E ] [ --comment-delimiter commentdelim ] [
12 --version | -h | --help ] [ --input-file infile | --input-string
13 instring ] [ --line-separator linesep ] [ --output-file outfile ]
14
16 The shortest path between two points on the ellipsoid at (lat1, lon1)
17 and (lat2, lon2) is called the geodesic. Its length is s12 and the
18 geodesic from point 1 to point 2 has forward azimuths azi1 and azi2 at
19 the two end points.
20 GeodSolve operates in one of three modes:
22 1. By default, GeodSolve accepts lines on the standard input
24 containing lat1 lon1 azi1 s12 and prints lat2 lon2 azi2 on standard
25 output. This is the direct geodesic calculation.
26 2. With the -i command line argument, GeodSolve performs the inverse
28 geodesic calculation. It reads lines containing lat1 lon1 lat2
29 lon2 and prints the corresponding values of azi1 azi2 s12.
30 3. Command line arguments -L lat1 lon1 azi1 specify a geodesic line.
32 GeodSolve then accepts a sequence of s12 values (one per line) on
33 standard input and prints lat2 lon2 azi2 for each. This generates
34 a sequence of points on a single geodesic. Command line arguments
35 -D and -I work similarly with the geodesic line defined in terms of
36 a direct or inverse geodesic calculation, respectively.
37
39 -i perform an inverse geodesic calculation (see 2 above).
40 -L lat1 lon1 azi1
42 line mode (see 3 above); generate a sequence of points along the
43 geodesic specified by lat1 lon1 azi1. The -w flag can be used to
44 swap the default order of the 2 geographic coordinates, provided
45 that it appears before -L. (-l is an alternative, deprecated,
46 spelling of this flag.)
47 -D lat1 lon1 azi1 s13
49 line mode (see 3 above); generate a sequence of points along the
50 geodesic specified by lat1 lon1 azi1 s13. The -w flag can be used
51 to swap the default order of the 2 geographic coordinates, provided
52 that it appears before -D. Similarly, the -a flag can be used to
53 change the interpretation of s13 to a13, provided that it appears
54 before -D.
55 -I lat1 lon1 lat3 lon3
57 line mode (see 3 above); generate a sequence of points along the
58 geodesic specified by lat1 lon1 lat3 lon3. The -w flag can be used
59 to swap the default order of the 2 geographic coordinates, provided
60 that it appears before -I.
61 -a toggle the arc mode flag (it starts off); if this flag is on, then
63 on input and output s12 is replaced by a12 the arc length (in
64 degrees) on the auxiliary sphere. See "AUXILIARY SPHERE".
65 -e a f
67 specify the ellipsoid via the equatorial radius, a and the
68 flattening, f. Setting f = 0 results in a sphere. Specify f < 0
69 for a prolate ellipsoid. A simple fraction, e.g., 1/297, is
70 allowed for f. By default, the WGS84 ellipsoid is used, a =
71 6378137 m, f = 1/298.257223563.
72 -u unroll the longitude. Normally, on output longitudes are reduced
74 to lie in [-180deg,180deg). However with this option, the returned
75 longitude lon2 is "unrolled" so that lon2 - lon1 indicates how
76 often and in what sense the geodesic has encircled the earth. Use
77 the -f option, to get both longitudes printed.
78 -F fractional mode. This only has any effect with the -D and -I
80 options (and is otherwise ignored). The values read on standard
81 input are interpreted as fractional distances to point 3, i.e., as
82 s12/s13 instead of s12. If arc mode is in effect, then the values
83 denote fractional arc length, i.e., a12/a13. The fractional
84 distances can be entered as a simple fraction, e.g., 3/4.
85 -d output angles as degrees, minutes, seconds instead of decimal
87 degrees.
88 -: like -d, except use : as a separator instead of the d, ', and "
90 delimiters.
91 -w toggle the longitude first flag (it starts off); if the flag is on,
93 then on input and output, longitude precedes latitude (except that,
94 on input, this can be overridden by a hemisphere designator, N, S,
95 E, W).
96 -b report the back azimuth at point 2 instead of the forward azimuth.
98 -f full output; each line of output consists of 12 quantities: lat1
100 lon1 azi1 lat2 lon2 azi2 s12 a12 m12 M12 M21 S12. a12 is described
101 in "AUXILIARY SPHERE". The four quantities m12, M12, M21, and S12
102 are described in "ADDITIONAL QUANTITIES".
103 -p prec
105 set the output precision to prec (default 3); prec is the precision
106 relative to 1 m. See "PRECISION".
107 -E use "exact" algorithms (based on elliptic integrals) for the
109 geodesic calculations. These are more accurate than the (default)
110 series expansions for |f| > 0.02.
111 --comment-delimiter commentdelim
113 set the comment delimiter to commentdelim (e.g., "#" or "//"). If
114 set, the input lines will be scanned for this delimiter and, if
115 found, the delimiter and the rest of the line will be removed prior
116 to processing and subsequently appended to the output line
117 (separated by a space).
118 --version
120 print version and exit.
121 -h print usage and exit.
123 --help
125 print full documentation and exit.
126 --input-file infile
128 read input from the file infile instead of from standard input; a
129 file name of "-" stands for standard input.
130 --input-string instring
132 read input from the string instring instead of from standard input.
133 All occurrences of the line separator character (default is a
134 semicolon) in instring are converted to newlines before the reading
135 begins.
136 --line-separator linesep
138 set the line separator character to linesep. By default this is a
139 semicolon.
140 --output-file outfile
142 write output to the file outfile instead of to standard output; a
143 file name of "-" stands for standard output.
144
146 GeodSolve measures all angles in degrees and all lengths (s12) in
147 meters, and all areas (S12) in meters^2. On input angles (latitude,
148 longitude, azimuth, arc length) can be as decimal degrees or degrees,
149 minutes, seconds. For example, "40d30", "40d30'", "40:30", "40.5d",
150 and 40.5 are all equivalent. By default, latitude precedes longitude
151 for each point (the -w flag switches this convention); however on input
152 either may be given first by appending (or prepending) N or S to the
153 latitude and E or W to the longitude. Azimuths are measured clockwise
154 from north; however this may be overridden with E or W.
155 For details on the allowed formats for angles, see the "GEOGRAPHIC
157 COORDINATES" section of GeoConvert(1).
158
160 Geodesics on the ellipsoid can be transferred to the auxiliary sphere
161 on which the distance is measured in terms of the arc length a12
162 (measured in degrees) instead of s12. In terms of a12, 180 degrees is
163 the distance from one equator crossing to the next or from the minimum
164 latitude to the maximum latitude. Geodesics with a12 > 180 degrees do
165 not correspond to shortest paths. With the -a flag, s12 (on both input
166 and output) is replaced by a12. The -a flag does not affect the full
167 output given by the -f flag (which always includes both s12 and a12).
168
170 The -f flag reports four additional quantities.
171 The reduced length of the geodesic, m12, is defined such that if the
173 initial azimuth is perturbed by dazi1 (radians) then the second point
174 is displaced by m12 dazi1 in the direction perpendicular to the
175 geodesic. m12 is given in meters. On a curved surface the reduced
176 length obeys a symmetry relation, m12 + m21 = 0. On a flat surface, we
177 have m12 = s12.
178 M12 and M21 are geodesic scales. If two geodesics are parallel at
180 point 1 and separated by a small distance dt, then they are separated
181 by a distance M12 dt at point 2. M21 is defined similarly (with the
182 geodesics being parallel to one another at point 2). M12 and M21 are
183 dimensionless quantities. On a flat surface, we have M12 = M21 = 1.
184 If points 1, 2, and 3 lie on a single geodesic, then the following
186 addition rules hold:
187 s13 = s12 + s23,
189 a13 = a12 + a23,
190 S13 = S12 + S23,
191 m13 = m12 M23 + m23 M21,
192 M13 = M12 M23 - (1 - M12 M21) m23 / m12,
193 M31 = M32 M21 - (1 - M23 M32) m12 / m23.
194 Finally, S12 is the area between the geodesic from point 1 to point 2
196 and the equator; i.e., it is the area, measured counter-clockwise, of
197 the geodesic quadrilateral with corners (lat1,lon1), (0,lon1),
198 (0,lon2), and (lat2,lon2). It is given in meters^2.
199
201 prec gives precision of the output with prec = 0 giving 1 m precision,
202 prec = 3 giving 1 mm precision, etc. prec is the number of digits
203 after the decimal point for lengths. For decimal degrees, the number
204 of digits after the decimal point is prec + 5. For DMS (degree,
205 minute, seconds) output, the number of digits after the decimal point
206 in the seconds component is prec + 1. The minimum value of prec is 0
207 and the maximum is 10.
208
210 An illegal line of input will print an error message to standard output
211 beginning with "ERROR:" and causes GeodSolve to return an exit code of
212 1. However, an error does not cause GeodSolve to terminate; following
213 lines will be converted.
214
216 Using the (default) series solution, GeodSolve is accurate to about 15
217 nm (15 nanometers) for the WGS84 ellipsoid. The approximate maximum
218 error (expressed as a distance) for an ellipsoid with the same
219 equatorial radius as the WGS84 ellipsoid and different values of the
220 flattening is
221 |f| error
223 0.01 25 nm
224 0.02 30 nm
225 0.05 10 um
226 0.1 1.5 mm
227 0.2 300 mm
228 If -E is specified, GeodSolve is accurate to about 40 nm (40
230 nanometers) for the WGS84 ellipsoid. The approximate maximum error
231 (expressed as a distance) for an ellipsoid with a quarter meridian of
232 10000 km and different values of the a/b = 1 - f is
233 1-f error (nm)
235 1/128 387
236 1/64 345
237 1/32 269
238 1/16 210
239 1/8 115
240 1/4 69
241 1/2 36
242 1 15
243 2 25
244 4 96
245 8 318
246 16 985
247 32 2352
248 64 6008
249 128 19024
250
252 The shortest distance returned for the inverse problem is (obviously)
253 uniquely defined. However, in a few special cases there are multiple
254 azimuths which yield the same shortest distance. Here is a catalog of
255 those cases:
256 lat1 = -lat2 (with neither point at a pole)
258 If azi1 = azi2, the geodesic is unique. Otherwise there are two
259 geodesics and the second one is obtained by setting [azi1,azi2] =
260 [azi2,azi1], [M12,M21] = [M21,M12], S12 = -S12. (This occurs when
261 the longitude difference is near +/-180 for oblate ellipsoids.)
262 lon2 = lon1 +/- 180 (with neither point at a pole)
264 If azi1 = 0 or +/-180, the geodesic is unique. Otherwise there are
265 two geodesics and the second one is obtained by setting [azi1,azi2]
266 = [-azi1,-azi2], S12 = -S12. (This occurs when lat2 is near -lat1
267 for prolate ellipsoids.)
268 Points 1 and 2 at opposite poles
270 There are infinitely many geodesics which can be generated by
271 setting [azi1,azi2] = [azi1,azi2] + [d,-d], for arbitrary d. (For
272 spheres, this prescription applies when points 1 and 2 are
273 antipodal.)
274 s12 = 0 (coincident points)
276 There are infinitely many geodesics which can be generated by
277 setting [azi1,azi2] = [azi1,azi2] + [d,d], for arbitrary d.
278
280 Route from JFK Airport to Singapore Changi Airport:
281 echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E |
283 GeodSolve -i -: -p 0
284 003:18:29.9 177:29:09.2 15347628
286 Equally spaced waypoints on the route:
288 for ((i = 0; i <= 10; ++i)); do echo $i/10; done |
290 GeodSolve -I 40:38:23N 073:46:44W 01:21:33N 103:59:22E -F -: -p 0
291 40:38:23.0N 073:46:44.0W 003:18:29.9
293 54:24:51.3N 072:25:39.6W 004:18:44.1
294 68:07:37.7N 069:40:42.9W 006:44:25.4
295 81:38:00.4N 058:37:53.9W 017:28:52.7
296 83:43:26.0N 080:37:16.9E 156:26:00.4
297 70:20:29.2N 097:01:29.4E 172:31:56.4
298 56:38:36.0N 100:14:47.6E 175:26:10.5
299 42:52:37.1N 101:43:37.2E 176:34:28.6
300 29:03:57.0N 102:39:34.8E 177:07:35.2
301 15:13:18.6N 103:22:08.0E 177:23:44.7
302 01:21:33.0N 103:59:22.0E 177:29:09.2
303
305 GeoConvert(1).
306 An online version of this utility is availbable at
308 <https://geographiclib.sourceforge.io/cgi-bin/GeodSolve>.
309 The algorithms are described in C. F. F. Karney, Algorithms for
311 geodesics, J. Geodesy 87, 43-55 (2013); DOI:
312 <https://doi.org/10.1007/s00190-012-0578-z>; addenda:
313 <https://geographiclib.sourceforge.io/geod-addenda.html>.
314 The Wikipedia page, Geodesics on an ellipsoid,
316 <https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.
317
319 GeodSolve was written by Charles Karney.
320
322 GeodSolve was added to GeographicLib,
323 <https://geographiclib.sourceforge.io>, in 2009-03. Prior to version
324 1.30, it was called Geod. (The name was changed to avoid a conflict
325 with the geod utility in proj.4.)
326GeographicLib 1.52 2022-02-24 GEODSOLVE(1)