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.
84 -d output angles as degrees, minutes, seconds instead of decimal
86 degrees.
87 -: like -d, except use : as a separator instead of the d, ', and "
89 delimiters.
90 -w toggle the longitude first flag (it starts off); if the flag is on,
92 then on input and output, longitude precedes latitude (except that,
93 on input, this can be overridden by a hemisphere designator, N, S,
94 E, W).
95 -b report the back azimuth at point 2 instead of the forward azimuth.
97 -f full output; each line of output consists of 12 quantities: lat1
99 lon1 azi1 lat2 lon2 azi2 s12 a12 m12 M12 M21 S12. a12 is described
100 in "AUXILIARY SPHERE". The four quantities m12, M12, M21, and S12
101 are described in "ADDITIONAL QUANTITIES".
102 -p prec
104 set the output precision to prec (default 3); prec is the precision
105 relative to 1 m. See "PRECISION".
106 -E use "exact" algorithms (based on elliptic integrals) for the
108 geodesic calculations. These are more accurate than the (default)
109 series expansions for |f| > 0.02.
110 --comment-delimiter commentdelim
112 set the comment delimiter to commentdelim (e.g., "#" or "//"). If
113 set, the input lines will be scanned for this delimiter and, if
114 found, the delimiter and the rest of the line will be removed prior
115 to processing and subsequently appended to the output line
116 (separated by a space).
117 --version
119 print version and exit.
120 -h print usage and exit.
122 --help
124 print full documentation and exit.
125 --input-file infile
127 read input from the file infile instead of from standard input; a
128 file name of "-" stands for standard input.
129 --input-string instring
131 read input from the string instring instead of from standard input.
132 All occurrences of the line separator character (default is a
133 semicolon) in instring are converted to newlines before the reading
134 begins.
135 --line-separator linesep
137 set the line separator character to linesep. By default this is a
138 semicolon.
139 --output-file outfile
141 write output to the file outfile instead of to standard output; a
142 file name of "-" stands for standard output.
143
145 GeodSolve measures all angles in degrees and all lengths (s12) in
146 meters, and all areas (S12) in meters^2. On input angles (latitude,
147 longitude, azimuth, arc length) can be as decimal degrees or degrees,
148 minutes, seconds. For example, "40d30", "40d30'", "40:30", "40.5d",
149 and 40.5 are all equivalent. By default, latitude precedes longitude
150 for each point (the -w flag switches this convention); however on input
151 either may be given first by appending (or prepending) N or S to the
152 latitude and E or W to the longitude. Azimuths are measured clockwise
153 from north; however this may be overridden with E or W.
154 For details on the allowed formats for angles, see the "GEOGRAPHIC
156 COORDINATES" section of GeoConvert(1).
157
159 Geodesics on the ellipsoid can be transferred to the auxiliary sphere
160 on which the distance is measured in terms of the arc length a12
161 (measured in degrees) instead of s12. In terms of a12, 180 degrees is
162 the distance from one equator crossing to the next or from the minimum
163 latitude to the maximum latitude. Geodesics with a12 > 180 degrees do
164 not correspond to shortest paths. With the -a flag, s12 (on both input
165 and output) is replaced by a12. The -a flag does not affect the full
166 output given by the -f flag (which always includes both s12 and a12).
167
169 The -f flag reports four additional quantities.
170 The reduced length of the geodesic, m12, is defined such that if the
172 initial azimuth is perturbed by dazi1 (radians) then the second point
173 is displaced by m12 dazi1 in the direction perpendicular to the
174 geodesic. m12 is given in meters. On a curved surface the reduced
175 length obeys a symmetry relation, m12 + m21 = 0. On a flat surface, we
176 have m12 = s12.
177 M12 and M21 are geodesic scales. If two geodesics are parallel at
179 point 1 and separated by a small distance dt, then they are separated
180 by a distance M12 dt at point 2. M21 is defined similarly (with the
181 geodesics being parallel to one another at point 2). M12 and M21 are
182 dimensionless quantities. On a flat surface, we have M12 = M21 = 1.
183 If points 1, 2, and 3 lie on a single geodesic, then the following
185 addition rules hold:
186 s13 = s12 + s23,
188 a13 = a12 + a23,
189 S13 = S12 + S23,
190 m13 = m12 M23 + m23 M21,
191 M13 = M12 M23 - (1 - M12 M21) m23 / m12,
192 M31 = M32 M21 - (1 - M23 M32) m12 / m23.
193 Finally, S12 is the area between the geodesic from point 1 to point 2
195 and the equator; i.e., it is the area, measured counter-clockwise, of
196 the geodesic quadrilateral with corners (lat1,lon1), (0,lon1),
197 (0,lon2), and (lat2,lon2). It is given in meters^2.
198
200 prec gives precision of the output with prec = 0 giving 1 m precision,
201 prec = 3 giving 1 mm precision, etc. prec is the number of digits
202 after the decimal point for lengths. For decimal degrees, the number
203 of digits after the decimal point is prec + 5. For DMS (degree,
204 minute, seconds) output, the number of digits after the decimal point
205 in the seconds component is prec + 1. The minimum value of prec is 0
206 and the maximum is 10.
207
209 An illegal line of input will print an error message to standard output
210 beginning with "ERROR:" and causes GeodSolve to return an exit code of
211 1. However, an error does not cause GeodSolve to terminate; following
212 lines will be converted.
213
215 Using the (default) series solution, GeodSolve is accurate to about 15
216 nm (15 nanometers) for the WGS84 ellipsoid. The approximate maximum
217 error (expressed as a distance) for an ellipsoid with the same
218 equatorial radius as the WGS84 ellipsoid and different values of the
219 flattening is
220 |f| error
222 0.01 25 nm
223 0.02 30 nm
224 0.05 10 um
225 0.1 1.5 mm
226 0.2 300 mm
227 If -E is specified, GeodSolve is accurate to about 40 nm (40
229 nanometers) for the WGS84 ellipsoid. The approximate maximum error
230 (expressed as a distance) for an ellipsoid with a quarter meridian of
231 10000 km and different values of the a/b = 1 - f is
232 1-f error (nm)
234 1/128 387
235 1/64 345
236 1/32 269
237 1/16 210
238 1/8 115
239 1/4 69
240 1/2 36
241 1 15
242 2 25
243 4 96
244 8 318
245 16 985
246 32 2352
247 64 6008
248 128 19024
249
251 The shortest distance returned for the inverse problem is (obviously)
252 uniquely defined. However, in a few special cases there are multiple
253 azimuths which yield the same shortest distance. Here is a catalog of
254 those cases:
255 lat1 = -lat2 (with neither point at a pole)
257 If azi1 = azi2, the geodesic is unique. Otherwise there are two
258 geodesics and the second one is obtained by setting [azi1,azi2] =
259 [azi2,azi1], [M12,M21] = [M21,M12], S12 = -S12. (This occurs when
260 the longitude difference is near +/-180 for oblate ellipsoids.)
261 lon2 = lon1 +/- 180 (with neither point at a pole)
263 If azi1 = 0 or +/-180, the geodesic is unique. Otherwise there are
264 two geodesics and the second one is obtained by setting [azi1,azi2]
265 = [-azi1,-azi2], S12 = -S12. (This occurs when lat2 is near -lat1
266 for prolate ellipsoids.)
267 Points 1 and 2 at opposite poles
269 There are infinitely many geodesics which can be generated by
270 setting [azi1,azi2] = [azi1,azi2] + [d,-d], for arbitrary d. (For
271 spheres, this prescription applies when points 1 and 2 are
272 antipodal.)
273 s12 = 0 (coincident points)
275 There are infinitely many geodesics which can be generated by
276 setting [azi1,azi2] = [azi1,azi2] + [d,d], for arbitrary d.
277
279 Route from JFK Airport to Singapore Changi Airport:
280 echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E |
282 GeodSolve -i -: -p 0
283 003:18:29.9 177:29:09.2 15347628
285 Equally spaced waypoints on the route:
287 for ((i = 0; i <= 10; ++i)); do echo ${i}e-1; done |
289 GeodSolve -I 40:38:23N 073:46:44W 01:21:33N 103:59:22E -F -: -p 0
290 40:38:23.0N 073:46:44.0W 003:18:29.9
292 54:24:51.3N 072:25:39.6W 004:18:44.1
293 68:07:37.7N 069:40:42.9W 006:44:25.4
294 81:38:00.4N 058:37:53.9W 017:28:52.7
295 83:43:26.0N 080:37:16.9E 156:26:00.4
296 70:20:29.2N 097:01:29.4E 172:31:56.4
297 56:38:36.0N 100:14:47.6E 175:26:10.5
298 42:52:37.1N 101:43:37.2E 176:34:28.6
299 29:03:57.0N 102:39:34.8E 177:07:35.2
300 15:13:18.6N 103:22:08.0E 177:23:44.7
301 01:21:33.0N 103:59:22.0E 177:29:09.2
302
304 GeoConvert(1).
305 An online version of this utility is availbable at
307 <https://geographiclib.sourceforge.io/cgi-bin/GeodSolve>.
308 The algorithms are described in C. F. F. Karney, Algorithms for
310 geodesics, J. Geodesy 87, 43-55 (2013); DOI:
311 <https://doi.org/10.1007/s00190-012-0578-z>; addenda:
312 <https://geographiclib.sourceforge.io/geod-addenda.html>.
313 The Wikipedia page, Geodesics on an ellipsoid,
315 <https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.
316
318 GeodSolve was written by Charles Karney.
319
321 GeodSolve was added to GeographicLib,
322 <https://geographiclib.sourceforge.io>, in 2009-03. Prior to version
323 1.30, it was called Geod. (The name was changed to avoid a conflict
324 with the geod utility in proj.4.)
325GeographicLib 1.49 2017-10-05 GEODSOLVE(1)