Geodesic and GeodesicLine provide accurate solutions to the direct and inverse geodesic problems. The GeodSolve utility provides an interface to these classes. AzimuthalEquidistant implements the azimuthal equidistant projection in terms of geodesics. CassiniSoldner implements a transverse cylindrical equidistant projection in terms of geodesics. The GeodesicProj utility provides an interface to these projections.
The algorithms used by Geodesic and GeodesicLine are based on a Taylor expansion of the geodesic integrals valid when the flattening f is small. GeodesicExact and GeodesicLineExact evaluate the integrals exactly (in terms of incomplete elliptic integrals). For the WGS84 ellipsoid, the series solutions are about 2--3 times faster and 2--3 times more accurate (because it's easier to control round-off errors with series solutions); thus Geodesic and GeodesicLine are recommended for most geodetic applications. However, in applications where the absolute value of f is greater than about 0.02, the exact classes should be used.
Go to
For some background information on geodesics on triaxial ellipsoids, see Geodesics on a triaxial ellipsoid.
References:
A test set a geodesics is available at
This is about 39 MB (compressed). This consists of a set of geodesics for the WGS84 ellipsoid. A subset of this (consisting of 1/50 of the members — about 690 kB, compressed) is available at
Each line of the test set gives 10 space delimited numbers
These are computed using as direct geodesic calculations with the given lat1, lon1, azi1, and s12. The distance s12 always corresponds to an arc length a12 180, so the given geodesics give the shortest paths from point 1 to point 2. For simplicity and without loss of generality, lat1 is chosen in [0, 90], lon1 is taken to be zero, azi1 is chosen in [0, 180]. Furthermore, lat1 and azi1 are taken to be multiples of 1012 deg and s12 is a multiple of 0.1 m in [0 m, 20003931.4586254 m]. This results lon2 in [0, 180] and azi2 in [0, 180].
The direct calculation uses an expansion of the geodesic equations accurate to f30 (approximately 1 part in 1050) and is computed with with Maxima's bfloats and fpprec set to 100 (so the errors in the data are probably 1/2 of the values quoted above).
The contents of the file are as follows:
(a total of 500000 entries). The values for s12 for the geodesics running between vertices are truncated to a multiple of 0.1 pm and this is used to determine point 2.
This data can be fed to the GeodSolve utility as follows
gunzip -c GeodTest.dat.gz | cut -d' ' -f1,2,3,7 | ./GeodSolve
gunzip -c GeodTest.dat.gz | cut -d' ' -f4,5,6,7 | sed "s/ \([^ ]*$\)/ -\1/" | ./GeodSolve
gunzip -c GeodTest.dat.gz | cut -d' ' -f1,2,4,5 | ./GeodSolve -i
Add, e.g., "-p 6", to the call to GeodSolve to change the precision of the output. Adding "-f" causes GeodSolve to print 12 fields specifying the geodesic; these include the 10 fields in the test set plus the geodesic scales M12 and M21 which are inserted between m12 and S12.
Code for computing arbitrarily accurate geodesics in maxima is available in geodesic.mac (this depends on ellint.mac and uses the series computed by geod.mac). This solve both the direct and inverse geodesic problems and offers the ability to solve the problems either using series expansions (similar to Geodesic) or in terms of elliptic integrals (similar to GeodesicExact).
We give here the series expansions for the various geodesic integrals valid to order f10. In this release of the code, we use a 6th-order expansions. This is sufficient to maintain accuracy for doubles for the SRMmax ellipsoid (a = 6400 km, f = 1/150). However, the preprocessor macro GEOGRAPHICLIB_GEODESIC_ORDER can be used to select any order up to 8. (If using long doubles, with a 64-bit fraction, the default order is 7.) The series expanded to order f30 are given in geodseries30.html.
In the formulas below ^ indicates exponentiation (f^3 = f3) and / indicates real division (3/5 = 0.6). The equations need to be converted to Horner form, but are here left in expanded form so that they can be easily truncated to lower order. These expansions were obtained using the Maxima code, geod.mac.
In the expansions below, we have
The formula for distance is
where
and
A1 = (1 + 1/4 * eps^2 + 1/64 * eps^4 + 1/256 * eps^6 + 25/16384 * eps^8 + 49/65536 * eps^10) / (1 - eps);
C1[1] = - 1/2 * eps + 3/16 * eps^3 - 1/32 * eps^5 + 19/2048 * eps^7 - 3/4096 * eps^9; C1[2] = - 1/16 * eps^2 + 1/32 * eps^4 - 9/2048 * eps^6 + 7/4096 * eps^8 + 1/65536 * eps^10; C1[3] = - 1/48 * eps^3 + 3/256 * eps^5 - 3/2048 * eps^7 + 17/24576 * eps^9; C1[4] = - 5/512 * eps^4 + 3/512 * eps^6 - 11/16384 * eps^8 + 3/8192 * eps^10; C1[5] = - 7/1280 * eps^5 + 7/2048 * eps^7 - 3/8192 * eps^9; C1[6] = - 7/2048 * eps^6 + 9/4096 * eps^8 - 117/524288 * eps^10; C1[7] = - 33/14336 * eps^7 + 99/65536 * eps^9; C1[8] = - 429/262144 * eps^8 + 143/131072 * eps^10; C1[9] = - 715/589824 * eps^9; C1[10] = - 2431/2621440 * eps^10;
The function may be inverted by series reversion giving
where
C1'[1] = + 1/2 * eps - 9/32 * eps^3 + 205/1536 * eps^5 - 4879/73728 * eps^7 + 9039/327680 * eps^9; C1'[2] = + 5/16 * eps^2 - 37/96 * eps^4 + 1335/4096 * eps^6 - 86171/368640 * eps^8 + 4119073/28311552 * eps^10; C1'[3] = + 29/96 * eps^3 - 75/128 * eps^5 + 2901/4096 * eps^7 - 443327/655360 * eps^9; C1'[4] = + 539/1536 * eps^4 - 2391/2560 * eps^6 + 1082857/737280 * eps^8 - 2722891/1548288 * eps^10; C1'[5] = + 3467/7680 * eps^5 - 28223/18432 * eps^7 + 1361343/458752 * eps^9; C1'[6] = + 38081/61440 * eps^6 - 733437/286720 * eps^8 + 10820079/1835008 * eps^10; C1'[7] = + 459485/516096 * eps^7 - 709743/163840 * eps^9; C1'[8] = + 109167851/82575360 * eps^8 - 550835669/74317824 * eps^10; C1'[9] = + 83141299/41287680 * eps^9; C1'[10] = + 9303339907/2972712960 * eps^10;
The reduced length is given by
where
A2 = (1 + 1/4 * eps^2 + 9/64 * eps^4 + 25/256 * eps^6 + 1225/16384 * eps^8 + 3969/65536 * eps^10) * (1 - eps);
C2[1] = + 1/2 * eps + 1/16 * eps^3 + 1/32 * eps^5 + 41/2048 * eps^7 + 59/4096 * eps^9; C2[2] = + 3/16 * eps^2 + 1/32 * eps^4 + 35/2048 * eps^6 + 47/4096 * eps^8 + 557/65536 * eps^10; C2[3] = + 5/48 * eps^3 + 5/256 * eps^5 + 23/2048 * eps^7 + 191/24576 * eps^9; C2[4] = + 35/512 * eps^4 + 7/512 * eps^6 + 133/16384 * eps^8 + 47/8192 * eps^10; C2[5] = + 63/1280 * eps^5 + 21/2048 * eps^7 + 51/8192 * eps^9; C2[6] = + 77/2048 * eps^6 + 33/4096 * eps^8 + 2607/524288 * eps^10; C2[7] = + 429/14336 * eps^7 + 429/65536 * eps^9; C2[8] = + 6435/262144 * eps^8 + 715/131072 * eps^10; C2[9] = + 12155/589824 * eps^9; C2[10] = + 46189/2621440 * eps^10;
The longitude is given in terms of the spherical longitude by
where
and
A3 = 1 - (1/2 - 1/2*n) * eps - (1/4 + 1/8*n - 3/8*n^2) * eps^2 - (1/16 + 3/16*n + 1/16*n^2 - 5/16*n^3) * eps^3 - (3/64 + 1/32*n + 5/32*n^2 + 5/128*n^3 - 35/128*n^4) * eps^4 - (3/128 + 5/128*n + 5/256*n^2 + 35/256*n^3 + 7/256*n^4) * eps^5 - (5/256 + 15/1024*n + 35/1024*n^2 + 7/512*n^3) * eps^6 - (25/2048 + 35/2048*n + 21/2048*n^2) * eps^7 - (175/16384 + 35/4096*n) * eps^8 - 245/32768 * eps^9;
C3[1] = + (1/4 - 1/4*n) * eps + (1/8 - 1/8*n^2) * eps^2 + (3/64 + 3/64*n - 1/64*n^2 - 5/64*n^3) * eps^3 + (5/128 + 1/64*n + 1/64*n^2 - 1/64*n^3 - 7/128*n^4) * eps^4 + (3/128 + 11/512*n + 3/512*n^2 + 1/256*n^3 - 7/512*n^4) * eps^5 + (21/1024 + 5/512*n + 13/1024*n^2 + 1/512*n^3) * eps^6 + (243/16384 + 189/16384*n + 83/16384*n^2) * eps^7 + (435/32768 + 109/16384*n) * eps^8 + 345/32768 * eps^9; C3[2] = + (1/16 - 3/32*n + 1/32*n^2) * eps^2 + (3/64 - 1/32*n - 3/64*n^2 + 1/32*n^3) * eps^3 + (3/128 + 1/128*n - 9/256*n^2 - 3/128*n^3 + 7/256*n^4) * eps^4 + (5/256 + 1/256*n - 1/128*n^2 - 7/256*n^3 - 3/256*n^4) * eps^5 + (27/2048 + 69/8192*n - 39/8192*n^2 - 47/4096*n^3) * eps^6 + (187/16384 + 39/8192*n + 31/16384*n^2) * eps^7 + (287/32768 + 47/8192*n) * eps^8 + 255/32768 * eps^9; C3[3] = + (5/192 - 3/64*n + 5/192*n^2 - 1/192*n^3) * eps^3 + (3/128 - 5/192*n - 1/64*n^2 + 5/192*n^3 - 1/128*n^4) * eps^4 + (7/512 - 1/384*n - 77/3072*n^2 + 5/3072*n^3 + 65/3072*n^4) * eps^5 + (3/256 - 1/1024*n - 71/6144*n^2 - 47/3072*n^3) * eps^6 + (139/16384 + 143/49152*n - 383/49152*n^2) * eps^7 + (243/32768 + 95/49152*n) * eps^8 + 581/98304 * eps^9; C3[4] = + (7/512 - 7/256*n + 5/256*n^2 - 7/1024*n^3 + 1/1024*n^4) * eps^4 + (7/512 - 5/256*n - 7/2048*n^2 + 9/512*n^3 - 21/2048*n^4) * eps^5 + (9/1024 - 43/8192*n - 129/8192*n^2 + 39/4096*n^3) * eps^6 + (127/16384 - 23/8192*n - 165/16384*n^2) * eps^7 + (193/32768 + 3/8192*n) * eps^8 + 171/32768 * eps^9; C3[5] = + (21/2560 - 9/512*n + 15/1024*n^2 - 7/1024*n^3 + 9/5120*n^4) * eps^5 + (9/1024 - 15/1024*n + 3/2048*n^2 + 57/5120*n^3) * eps^6 + (99/16384 - 91/16384*n - 781/81920*n^2) * eps^7 + (179/32768 - 55/16384*n) * eps^8 + 141/32768 * eps^9; C3[6] = + (11/2048 - 99/8192*n + 275/24576*n^2 - 77/12288*n^3) * eps^6 + (99/16384 - 275/24576*n + 55/16384*n^2) * eps^7 + (143/32768 - 253/49152*n) * eps^8 + 33/8192 * eps^9; C3[7] = + (429/114688 - 143/16384*n + 143/16384*n^2) * eps^7 + (143/32768 - 143/16384*n) * eps^8 + 429/131072 * eps^9; C3[8] = + (715/262144 - 429/65536*n) * eps^8 + 429/131072 * eps^9; C3[9] = + 2431/1179648 * eps^9;
The formula for area between the geodesic and the equator is given in Sec. 6 of Algorithms for geodesics in terms of S,
where
In the paper, this was expanded in and
. However, the series converges faster for eccentric ellipsoids if the expansion is in
and
. The series to order
becomes
C4[0] = + (2/3 - 4/15*n + 8/105*n^2 + 4/315*n^3 + 16/3465*n^4 + 20/9009*n^5 + 8/6435*n^6 + 28/36465*n^7 + 32/62985*n^8 + 4/11305*n^9) - (1/5 - 16/35*n + 32/105*n^2 - 16/385*n^3 - 64/15015*n^4 - 16/15015*n^5 - 32/85085*n^6 - 112/692835*n^7 - 128/1616615*n^8) * eps - (2/105 + 32/315*n - 1088/3465*n^2 + 1184/5005*n^3 - 128/3465*n^4 - 3232/765765*n^5 - 1856/1616615*n^6 - 6304/14549535*n^7) * eps^2 + (11/315 - 368/3465*n - 32/6435*n^2 + 976/4095*n^3 - 154048/765765*n^4 + 368/11115*n^5 + 5216/1322685*n^6) * eps^3 + (4/1155 + 1088/45045*n - 128/1287*n^2 + 64/3927*n^3 + 2877184/14549535*n^4 - 370112/2078505*n^5) * eps^4 + (97/15015 - 464/45045*n + 4192/153153*n^2 - 88240/969969*n^3 + 31168/1322685*n^4) * eps^5 + (10/9009 + 4192/765765*n - 188096/14549535*n^2 + 23392/855855*n^3) * eps^6 + (193/85085 - 6832/2078505*n + 106976/14549535*n^2) * eps^7 + (632/1322685 + 3456/1616615*n) * eps^8 + 107/101745 * eps^9; C4[1] = + (1/45 - 16/315*n + 32/945*n^2 - 16/3465*n^3 - 64/135135*n^4 - 16/135135*n^5 - 32/765765*n^6 - 112/6235515*n^7 - 128/14549535*n^8) * eps - (2/105 - 64/945*n + 128/1485*n^2 - 1984/45045*n^3 + 256/45045*n^4 + 64/109395*n^5 + 128/855855*n^6 + 2368/43648605*n^7) * eps^2 - (1/105 - 16/2079*n - 5792/135135*n^2 + 3568/45045*n^3 - 103744/2297295*n^4 + 264464/43648605*n^5 + 544/855855*n^6) * eps^3 + (4/1155 - 2944/135135*n + 256/9009*n^2 + 17536/765765*n^3 - 3053056/43648605*n^4 + 1923968/43648605*n^5) * eps^4 + (1/9009 + 16/19305*n - 2656/153153*n^2 + 65072/2078505*n^3 + 526912/43648605*n^4) * eps^5 + (10/9009 - 1472/459459*n + 106112/43648605*n^2 - 204352/14549535*n^3) * eps^6 + (349/2297295 + 28144/43648605*n - 32288/8729721*n^2) * eps^7 + (632/1322685 - 44288/43648605*n) * eps^8 + 43/479655 * eps^9; C4[2] = + (4/525 - 32/1575*n + 64/3465*n^2 - 32/5005*n^3 + 128/225225*n^4 + 32/765765*n^5 + 64/8083075*n^6 + 32/14549535*n^7) * eps^2 - (8/1575 - 128/5775*n + 256/6825*n^2 - 6784/225225*n^3 + 4608/425425*n^4 - 128/124355*n^5 - 5888/72747675*n^6) * eps^3 - (8/1925 - 1856/225225*n - 128/17325*n^2 + 42176/1276275*n^3 - 2434816/72747675*n^4 + 195136/14549535*n^5) * eps^4 + (8/10725 - 128/17325*n + 64256/3828825*n^2 - 128/25935*n^3 - 266752/10392525*n^4) * eps^5 - (4/25025 + 928/3828825*n + 292288/72747675*n^2 - 106528/6613425*n^3) * eps^6 + (464/1276275 - 17152/10392525*n + 83456/72747675*n^2) * eps^7 + (1168/72747675 + 128/1865325*n) * eps^8 + 208/1119195 * eps^9; C4[3] = + (8/2205 - 256/24255*n + 512/45045*n^2 - 256/45045*n^3 + 1024/765765*n^4 - 256/2909907*n^5 - 512/101846745*n^6) * eps^3 - (16/8085 - 1024/105105*n + 2048/105105*n^2 - 1024/51051*n^3 + 4096/373065*n^4 - 1024/357357*n^5) * eps^4 - (136/63063 - 256/45045*n + 512/1072071*n^2 + 494336/33948915*n^3 - 44032/1996995*n^4) * eps^5 + (64/315315 - 16384/5360355*n + 966656/101846745*n^2 - 868352/101846745*n^3) * eps^6 - (16/97461 + 14848/101846745*n + 74752/101846745*n^2) * eps^7 + (5024/33948915 - 96256/101846745*n) * eps^8 - 1744/101846745 * eps^9; C4[4] = + (64/31185 - 512/81081*n + 1024/135135*n^2 - 512/109395*n^3 + 2048/1247103*n^4 - 2560/8729721*n^5) * eps^4 - (128/135135 - 2048/405405*n + 77824/6891885*n^2 - 198656/14549535*n^3 + 8192/855855*n^4) * eps^5 - (512/405405 - 2048/530145*n + 299008/130945815*n^2 + 280576/43648605*n^3) * eps^6 + (128/2297295 - 2048/1438965*n + 241664/43648605*n^2) * eps^7 - (17536/130945815 + 1024/43648605*n) * eps^8 + 2944/43648605 * eps^9; C4[5] = + (128/99099 - 2048/495495*n + 4096/765765*n^2 - 6144/1616615*n^3 + 8192/4849845*n^4) * eps^5 - (256/495495 - 8192/2807805*n + 376832/53348295*n^2 - 8192/855855*n^3) * eps^6 - (6784/8423415 - 432128/160044885*n + 397312/160044885*n^2) * eps^7 + (512/53348295 - 16384/22863555*n) * eps^8 - 16768/160044885 * eps^9; C4[6] = + (512/585585 - 4096/1422135*n + 8192/2078505*n^2 - 4096/1322685*n^3) * eps^6 - (1024/3318315 - 16384/9006855*n + 98304/21015995*n^2) * eps^7 - (103424/189143955 - 8192/4203199*n) * eps^8 - 1024/189143955 * eps^9; C4[7] = + (1024/1640925 - 65536/31177575*n + 131072/43648605*n^2) * eps^7 - (2048/10392525 - 262144/218243025*n) * eps^8 - 84992/218243025 * eps^9; C4[8] = + (16384/35334585 - 131072/82447365*n) * eps^8 - 32768/247342095 * eps^9; C4[9] = + 32768/92147055 * eps^9;
GeodesicExact and GeodesicLineExact solve the geodesic problem using elliptic integrals. The formulation of geodesic in terms of incomplete elliptic integrals is given in
It is most convenient to use the form derived for a prolate ellipsoid in Appendix D. For an oblate ellipsoid this results in elliptic integrals with an imaginary modulus. However, the integrals themselves are real and the algorithms used to compute the elliptic integrals handles the case of an imaginary modulus using real arithmetic.
The key relations used by GeographicLib are
where is a modified spherical longitude given by
and
and ,
,
, and
, are incomplete elliptic integrals (see http://dlmf.nist.gov/19.2.ii). The formula for
and the first expression for
are given by Legendre (1811) and are the most common representation of geodesics in terms of elliptic integrals. The second (equivalent) expression for
, which was given by Cayley (1870), is useful in that the elliptic integral is relegated to a small correction term. This form allows the longitude to be computed more accurately and is used in GeographicLib. (The equivalence of the two expressions for
follows from http://dlmf.nist.gov/19.7.E8.)
Nominally, GeodesicExact and GeodesicLineExact will give "exact" results for any value of the flattening. However, the geographic latitude is a distorted measure of distance from the equator with very eccentric ellipsoids and this introducing an irreducible representational error in the algorithms in this case. It is therefore recommended to restrict the use of these classes to b/a [0.01, 100] or f [-99, 0.99]. Note that GeodesicExact still uses a series expansion for the area S12. However the series is taken out to 30th order and gives accurate results for b/a [1/2, 2]; the accuracy is about 8 decimal digits for b/a [1/4, 4]. Additional work planned for this aspect of the geodesic problem:
Thomas (1952) and Rollins (2010) use a different independent variable for geodesics, instead of
, where
. The corresponding expressions for
and
are given here for completeness:
where . The expression for
can be written in terms of elliptic integrals of the second kind and Cayley's technique can be used to subtract out the leading order behavior of
to give
where
The tangents of the three "longitude-like" angles are in geometric progression, .
The formulas for given in the previous section are the same as those for the distance along a meridian for an ellipsoid with equatorial radius
and polar semi-axis
. Here is a list of possible ways of expressing the meridian distance in terms of elliptic integrals using the notation:
The eccentricities are real (resp. imaginary) for oblate (resp. prolate) ellipsoids. The elliptic variables
and
are defined by
and are linearly related by
The cartesian coordinates for the meridian are given by
The distance along the meridian can be expressed variously as
with the quarter meridian distance given by
(Here are elliptic integrals defined in http://dlmf.nist.gov/19.2.ii.
are Jacobi elliptic functions defined in http://dlmf.nist.gov/22.2 and http://dlmf.nist.gov/22.16.)
There are several considerations in the choice of independent variable for evaluate the meridian distance
GeographicLib uses the geodesic generalization of , namely
. See Geodesics in terms of elliptic integrals.
Comparison of meridian measures
Here we describe Bowring's method for solving the inverse geodesic problem in the limit of short geodesics and contrast it with the great circle solution using Bessel's auxiliary sphere. References:
Bowring considers the conformal mapping of the ellipsoid to a sphere of radius such that circles of latitude and meridians are preserved (and hence the azimuth of a line is preserved). Let
and
be the latitude and longitude on the ellipsoid and sphere respectively. Define isometric latitudes for the sphere and the ellipsoid as
The most general conformal mapping satisfying Bowring's conditions is
where and
are constants. (In fact a constant can be added to the equation for
, but this does affect the analysis.) The scale of this mapping is
where is the transverse radius of curvature. (Note that in Bowring's Eq. (10),
should be replaced by
.) The mapping from the ellipsoid to the sphere depends on three parameters
. These will be selected to satisfy certain conditions at some representative latitude
. Two possible choices are given below.
Bowring (1981) requires that
i.e, in the vicinity of
. This gives
where is the second eccentricity,
, and
is defined implicitly by the equation for
. The radius
is the (Gaussian) mean radius of curvature of the ellipsoid at
(so near
the ellipsoid can be deformed to fit the sphere snugly). The third derivative of
is given by
The method for solving the inverse problem between two nearby points and
is as follows: Set
. Compute
, and hence find
and
. Finally, solve for the great circle on a sphere of radius
; the resulting distance and azimuths are good approximations for the corresponding quantities for the ellipsoidal geodesic.
Consistent with the accuracy of this method, we can compute and
using a Taylor expansion about
. This also avoids numerical errors that arise from subtracting nearly equal quantities when using the equation for
directly. Write
and
; then we have
where the error is . This is essentially Bowring's method. Significant differences between this result, "Bowring (improved)", compared to Bowring's paper, "Bowring
(original)", are:
In fact, arguably, the highest order terms should be dropped altogether. Their inclusion does result in a better estimate for the distance. However, if your goal is to generate both accurate distances and accurate azimuths, then
needs to be restricted sufficiently to allow these terms to be dropped to give the "Bowring (truncated)" method.
With highly eccentric ellipsoids, the parametric latitude is a better behaved independent variable to use. In this case,
is naturally defined using
and in terms of
, we have
where , and the error once again is
. This is the "Bowring (using
)" method.
GeographicLib's uses the auxiliary sphere method of Legendre, Bessel, and Helmert. For short geodesics, this is equivalent to picking so that
Bowring's requirement that the second derivative of vanish has been replaced by the last relation which states that
, the parametric latitude corresponding to
. This gives
Similar to Bowring's method, we can compute and
using a Taylor expansion about
. This results in the simple expression
where the error is .
In assessing the accuracy of these methods we use two metrics:
(The second metric is much more stringent.) We may now compare the methods by asking for a bound to the length of a geodesic which ensures that the one or other of the errors fall below 1 mm (an "engineering" definition of accurate) or 1 nm (1 nanometer, about the round-off limit).
method | ||||
---|---|---|---|---|
1 mm error | 1 nm error | 1 mm error | 1 nm error | |
Bowring (original) | ||||
Bowring (improved) | ||||
Bowring (truncated) | ||||
Bowring (using | ||||
Bessel's aux. sphere |
For example, if you're only interested in measuring distances and an accuracy of 1 mm is sufficient, then Bowring's improved method can be used for distances up to 180 km. On the other hand, GeographicLib uses Bessel's auxiliary sphere and we require both the distance and the azimuth to be accurate, so the great circle approximation can only be used for distances less than 1.7 m. The reason that GeographicLib does not use Bowring's method is that the information necessary for auxiliary sphere method is already available as part of the general solution and, as much as possible, we allow all geodesics to be computed by the general method.