1RHUMBSOLVE(1) GeographicLib Utilities RHUMBSOLVE(1)
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6 RhumbSolve -- perform rhumb line calculations
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9 RhumbSolve [ -i | -L lat1 lon1 azi12 ] [ -e a f ] [ -d | -: ] [ -w ] [
10 -p prec ] [ -s ] [ --comment-delimiter commentdelim ] [ --version | -h
11 | --help ] [ --input-file infile | --input-string instring ] [
12 --line-separator linesep ] [ --output-file outfile ]
13
15 The path with constant heading between two points on the ellipsoid at
16 (lat1, lon1) and (lat2, lon2) is called the rhumb line or loxodrome.
17 Its length is s12 and the rhumb line has a forward azimuth azi12 along
18 its length. Also computed is S12 is the area between the rhumb line
19 from point 1 to point 2 and the equator; i.e., it is the area, measured
20 counter-clockwise, of the geodesic quadrilateral with corners
21 (lat1,lon1), (0,lon1), (0,lon2), and (lat2,lon2). A point at a pole is
22 treated as a point a tiny distance away from the pole on the given line
23 of longitude. The longitude becomes indeterminate when a rhumb line
24 passes through a pole, and RhumbSolve reports NaNs for the longitude
25 and the area in this case.
26 NOTE: the rhumb line is not the shortest path between two points; that
28 is the geodesic and it is calculated by GeodSolve(1).
29 RhumbSolve operates in one of three modes:
31 1. By default, RhumbSolve accepts lines on the standard input
33 containing lat1 lon1 azi12 s12 and prints lat2 lon2 S12 on standard
34 output. This is the direct calculation.
35 2. With the -i command line argument, RhumbSolve performs the inverse
37 calculation. It reads lines containing lat1 lon1 lat2 lon2 and
38 prints the values of azi12 s12 S12 for the corresponding shortest
39 rhumb lines. If the end points are on opposite meridians, there
40 are two shortest rhumb lines and the east-going one is chosen.
41 3. Command line arguments -L lat1 lon1 azi12 specify a rhumb line.
43 RhumbSolve then accepts a sequence of s12 values (one per line) on
44 standard input and prints lat2 lon2 S12 for each. This generates a
45 sequence of points on a rhumb line.
46
48 -i perform an inverse calculation (see 2 above).
49 -L lat1 lon1 azi12
51 line mode (see 3 above); generate a sequence of points along the
52 rhumb line specified by lat1 lon1 azi12. The -w flag can be used
53 to swap the default order of the 2 geographic coordinates, provided
54 that it appears before -L. (-l is an alternative, deprecated,
55 spelling of this flag.)
56 -e a f
58 specify the ellipsoid via the equatorial radius, a and the
59 flattening, f. Setting f = 0 results in a sphere. Specify f < 0
60 for a prolate ellipsoid. A simple fraction, e.g., 1/297, is
61 allowed for f. By default, the WGS84 ellipsoid is used, a =
62 6378137 m, f = 1/298.257223563.
63 -d output angles as degrees, minutes, seconds instead of decimal
65 degrees.
66 -: like -d, except use : as a separator instead of the d, ', and "
68 delimiters.
69 -w on input and output, longitude precedes latitude (except that on
71 input this can be overridden by a hemisphere designator, N, S, E,
72 W).
73 -p prec
75 set the output precision to prec (default 3); prec is the precision
76 relative to 1 m. See "PRECISION".
77 -s By default, the rhumb line calculations are carried out exactly in
79 terms of elliptic integrals. This includes the use of the addition
80 theorem for elliptic integrals to compute the divided difference of
81 the isometric and rectifying latitudes. If -s is supplied this
82 divided difference is computed using Krueger series for the
83 transverse Mercator projection which is only accurate for |f| <
84 0.01. See "ACCURACY".
85 --comment-delimiter commentdelim
87 set the comment delimiter to commentdelim (e.g., "#" or "//"). If
88 set, the input lines will be scanned for this delimiter and, if
89 found, the delimiter and the rest of the line will be removed prior
90 to processing and subsequently appended to the output line
91 (separated by a space).
92 --version
94 print version and exit.
95 -h print usage and exit.
97 --help
99 print full documentation and exit.
100 --input-file infile
102 read input from the file infile instead of from standard input; a
103 file name of "-" stands for standard input.
104 --input-string instring
106 read input from the string instring instead of from standard input.
107 All occurrences of the line separator character (default is a
108 semicolon) in instring are converted to newlines before the reading
109 begins.
110 --line-separator linesep
112 set the line separator character to linesep. By default this is a
113 semicolon.
114 --output-file outfile
116 write output to the file outfile instead of to standard output; a
117 file name of "-" stands for standard output.
118
120 RhumbSolve measures all angles in degrees, all lengths (s12) in meters,
121 and all areas (S12) in meters^2. On input angles (latitude, longitude,
122 azimuth, arc length) can be as decimal degrees or degrees, minutes,
123 seconds. For example, "40d30", "40d30'", "40:30", "40.5d", and 40.5
124 are all equivalent. By default, latitude precedes longitude for each
125 point (the -w flag switches this convention); however on input either
126 may be given first by appending (or prepending) N or S to the latitude
127 and E or W to the longitude. Azimuths are measured clockwise from
128 north; however this may be overridden with E or W.
129 For details on the allowed formats for angles, see the "GEOGRAPHIC
131 COORDINATES" section of GeoConvert(1).
132
134 prec gives precision of the output with prec = 0 giving 1 m precision,
135 prec = 3 giving 1 mm precision, etc. prec is the number of digits
136 after the decimal point for lengths. For decimal degrees, the number
137 of digits after the decimal point is prec + 5. For DMS (degree,
138 minute, seconds) output, the number of digits after the decimal point
139 in the seconds component is prec + 1. The minimum value of prec is 0
140 and the maximum is 10.
141
143 An illegal line of input will print an error message to standard output
144 beginning with "ERROR:" and causes RhumbSolve to return an exit code of
145 1. However, an error does not cause RhumbSolve to terminate; following
146 lines will be converted.
147
149 The algorithm used by RhumbSolve uses exact formulas for converting
150 between the latitude, rectifying latitude (mu), and isometric latitude
151 (psi). These formulas are accurate for any value of the flattening.
152 The computation of rhumb lines involves the ratio (psi1 - psi2) / (mu1
153 - mu2) and this is subject to large round-off errors if lat1 is close
154 to lat2. So this ratio is computed using divided differences using one
155 of two methods: by default, this uses the addition theorem for elliptic
156 integrals (accurate for all values of f); however, with the -s options,
157 it is computed using the series expansions used by
158 TransverseMercatorProj(1) for the conversions between rectifying and
159 conformal latitudes (accurate for |f| < 0.01). For the WGS84
160 ellipsoid, the error is about 10 nanometers using either method.
161
163 Route from JFK Airport to Singapore Changi Airport:
164 echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E |
166 RhumbSolve -i -: -p 0
167 103:34:58.2 18523563
169 N.B. This is not the route typically taken by aircraft because it's
171 considerably longer than the geodesic given by GeodSolve(1).
172 Waypoints on the route at intervals of 2000km:
174 for ((i = 0; i <= 20; i += 2)); do echo ${i}000000;done |
176 RhumbSolve -L 40:38:23N 073:46:44W 103:34:58.2 -: -p 0
177 40:38:23.0N 073:46:44.0W 0
179 36:24:30.3N 051:28:26.4W 9817078307821
180 32:10:26.8N 030:20:57.3W 18224745682005
181 27:56:13.2N 010:10:54.2W 25358020327741
182 23:41:50.1N 009:12:45.5E 31321269267102
183 19:27:18.7N 027:59:22.1E 36195163180159
184 15:12:40.2N 046:17:01.1E 40041499143669
185 10:57:55.9N 064:12:52.8E 42906570007050
186 06:43:07.3N 081:53:28.8E 44823504180200
187 02:28:16.2N 099:24:54.5E 45813843358737
188 01:46:36.0S 116:52:59.7E 45888525219677
189
191 GeoConvert(1), GeodSolve(1), TransverseMercatorProj(1).
192 An online version of this utility is availbable at
194 <https://geographiclib.sourceforge.io/cgi-bin/RhumbSolve>.
195 The Wikipedia page, Rhumb line,
197 <https://en.wikipedia.org/wiki/Rhumb_line>.
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200 RhumbSolve was written by Charles Karney.
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203 RhumbSolve was added to GeographicLib,
204 <https://geographiclib.sourceforge.io>, in version 1.37.
205GeographicLib 1.51 2020-11-22 RHUMBSOLVE(1)