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GeodesicExact.cpp
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/**
* \file GeodesicExact.cpp
* \brief Implementation for GeographicLib::GeodesicExact class
*
* Copyright (c) Charles Karney (2012-2025) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
*
* This is a reformulation of the geodesic problem. The notation is as
* follows:
* - at a general point (no suffix or 1 or 2 as suffix)
* - phi = latitude
* - beta = latitude on auxiliary sphere
* - omega = longitude on auxiliary sphere
* - lambda = longitude
* - alpha = azimuth of great circle
* - sigma = arc length along great circle
* - s = distance
* - tau = scaled distance (= sigma at multiples of pi/2)
* - at northwards equator crossing
* - beta = phi = 0
* - omega = lambda = 0
* - alpha = alpha0
* - sigma = s = 0
* - a 12 suffix means a difference, e.g., s12 = s2 - s1.
* - s and c prefixes mean sin and cos
**********************************************************************/
#include <GeographicLib/GeodesicExact.hpp>
#include <GeographicLib/GeodesicLineExact.hpp>
#include <vector>
#if defined(_MSC_VER)
// Squelch warnings about potentially uninitialized local variables
# pragma warning (disable: 4701)
#endif
namespace GeographicLib {
using namespace std;
GeodesicExact::GeodesicExact(real a, real f)
: maxit2_(maxit1_ + Math::digits() + 10)
// Underflow guard. We require
// tiny_ * epsilon() > 0
// tiny_ + epsilon() == epsilon()
, tiny_(sqrt(numeric_limits<real>::min()))
, tol0_(numeric_limits<real>::epsilon())
// Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
// case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
// which otherwise failed for Visual Studio 10 (Release and Debug)
, tol1_(200 * tol0_)
, tol2_(sqrt(tol0_))
, tolb_(tol0_) // Check on bisection interval
, xthresh_(1000 * tol2_)
, _a(a)
, _f(f)
, _f1(1 - _f)
, _e2(_f * (2 - _f))
, _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
, _n(_f / ( 2 - _f))
, _b(_a * _f1)
// The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) in
// the definition of _c2. The latter is more accurate for very oblate
// ellipsoids (which the Geodesic class does not attempt to handle).
, _c2((Math::sq(_a) + Math::sq(_b) *
(_f == 0 ? 1 :
(_f > 0 ? asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) /
sqrt(fabs(_e2))))/2) // authalic radius squared
// The sig12 threshold for "really short". Using the auxiliary sphere
// solution with dnm computed at (bet1 + bet2) / 2, the relative error in
// the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
// (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a
// given f and sig12, the max error occurs for lines near the pole. If
// the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
// increases by a factor of 2.) Setting this equal to epsilon gives
// sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100)
// and max(0.001, abs(f)) stops etol2 getting too large in the nearly
// spherical case.
, _etol2(real(0.1) * tol2_ /
sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 ))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_b) && _b > 0))
throw GeographicErr("Polar semiaxis is not positive");
// Required number of terms in DST for full accuracy for all precisions as
// a function of n in [-0.99, 0.99]. Values determined by running
// develop/AreaEst compiled with GEOGRAPHICLIB_PRECISION = 5. For
// precision 4 and 5, GEOGRAPHICLIB_DIGITS was set to, resp., 384 and 768.
// The error criterion is relative error less than or equal to epsilon/2 =
// 0.5^digits, with digits = 24, 53, 64, 113, 256. The first 4 are the the
// "standard" values for float, double, long double, and float128; the last
// is the default for GeographicLib + mpfr. Also listed is the value of
// alp0 resulting in the most error for the given N.
//
// float double long double quad mpfr
// n N alp0 N alp0 N alp0 N alp0 N alp0
// -0.99 1024 0.09 3072 0.05 4096 0.04 8192 43.50 16384 22.01
// -0.98 512 0.18 1536 0.10 2048 0.09 4096 0.06 8192 0.04
// -0.97 384 0.25 1024 0.16 1536 0.13 3072 0.09 6144 0.06
// -0.96 256 0.36 768 0.21 1024 0.18 2048 0.13 4096 0.09
// -0.95 192 0.47 768 0.23 768 0.23 1536 0.17 4096 0.10
// -0.94 192 0.51 512 0.31 768 0.26 1536 0.18 3072 0.13
// -0.93 192 0.55 384 0.39 512 0.34 1024 0.24 3072 0.14
// -0.92 128 0.73 384 0.42 512 0.37 1024 0.26 2048 0.18
// -0.91 128 0.77 384 0.45 384 0.45 768 0.32 2048 0.19
// -0.90 96 0.94 256 0.58 384 0.47 768 0.34 2048 0.21
// -0.89 96 0.99 256 0.61 384 0.50 768 0.35 1536 0.25
// -0.88 96 1.04 256 0.64 384 0.52 768 0.37 1536 0.26
// -0.87 96 1.09 192 0.77 256 0.67 512 0.47 1536 0.27
// -0.86 64 1.38 192 0.80 256 0.69 512 0.49 1536 0.28
// -0.85 64 1.43 192 0.83 256 0.72 512 0.51 1024 0.36
// -0.84 64 1.49 192 0.86 256 0.75 384 0.61 1024 0.37
// -0.83 64 1.54 192 0.89 192 0.89 384 0.63 1024 0.39
// -0.82 48 1.82 192 0.92 192 0.92 384 0.65 1024 0.40
// -0.81 48 1.88 128 1.16 192 0.95 384 0.67 1024 0.41
// -0.80 48 1.94 128 1.19 192 0.97 384 0.69 768 0.49
// -0.79 48 1.99 128 1.23 192 1.00 384 0.71 768 0.50
// -0.78 48 2.04 128 1.26 192 1.03 384 0.73 768 0.51
// -0.77 48 2.10 128 1.29 192 1.05 256 0.91 768 0.53
// -0.76 48 2.15 128 1.32 128 1.32 256 0.93 768 0.54
// -0.75 48 2.20 96 1.56 128 1.35 256 0.96 768 0.55
// -0.74 32 2.74 96 1.60 128 1.38 256 0.98 768 0.57
// -0.73 32 2.81 96 1.63 128 1.41 256 1.00 768 0.58
// -0.72 32 2.87 96 1.67 128 1.44 256 1.02 512 0.72
// -0.71 32 2.93 96 1.70 128 1.47 192 1.20 512 0.74
// -0.70 32 2.99 96 1.73 96 1.73 192 1.23 512 0.75
// -0.69 32 3.05 96 1.77 96 1.77 192 1.25 512 0.77
// -0.68 32 3.11 96 1.80 96 1.80 192 1.28 512 0.78
// -0.67 24 3.64 96 1.84 96 1.84 192 1.30 512 0.80
// -0.66 24 3.71 96 1.87 96 1.87 192 1.32 512 0.81
// -0.65 24 3.77 64 2.33 96 1.90 192 1.35 384 0.95
// -0.64 24 3.84 64 2.37 96 1.93 192 1.37 384 0.97
// -0.63 24 3.90 64 2.41 96 1.97 192 1.39 384 0.98
// -0.62 24 3.97 64 2.45 96 2.00 192 1.42 384 1.00
// -0.61 24 4.04 64 2.49 96 2.03 192 1.44 384 1.02
// -0.60 24 4.10 64 2.53 96 2.06 192 1.46 384 1.03
// -0.59 24 4.16 64 2.57 64 2.57 128 1.82 384 1.05
// -0.58 24 4.23 64 2.60 64 2.60 128 1.84 384 1.07
// -0.57 24 4.29 48 3.05 64 2.64 128 1.87 384 1.08
// -0.56 24 4.36 48 3.10 64 2.68 128 1.90 384 1.10
// -0.55 16 5.38 48 3.14 64 2.72 128 1.93 384 1.11
// -0.54 16 5.46 48 3.19 64 2.76 128 1.96 384 1.13
// -0.53 16 5.54 48 3.23 64 2.80 128 1.98 256 1.40
// -0.52 16 5.61 48 3.27 64 2.84 128 2.01 256 1.42
// -0.51 16 5.69 48 3.32 64 2.88 128 2.04 256 1.44
// -0.50 16 5.77 48 3.36 64 2.92 96 2.38 256 1.46
// -0.49 16 5.85 48 3.41 48 3.41 96 2.42 256 1.48
// -0.48 16 5.92 48 3.45 48 3.45 96 2.45 256 1.50
// -0.47 16 6.00 48 3.50 48 3.50 96 2.48 256 1.52
// -0.46 16 6.08 48 3.54 48 3.54 96 2.51 256 1.54
// -0.45 12 7.06 48 3.59 48 3.59 96 2.54 256 1.56
// -0.44 12 7.15 48 3.63 48 3.63 96 2.57 256 1.58
// -0.43 12 7.24 32 4.49 48 3.68 96 2.61 256 1.60
// -0.42 12 7.33 32 4.55 48 3.72 96 2.64 192 1.87
// -0.41 12 7.42 32 4.60 48 3.77 96 2.67 192 1.89
// -0.40 12 7.51 32 4.66 48 3.81 96 2.70 192 1.91
// -0.39 12 7.60 32 4.71 48 3.86 96 2.73 192 1.94
// -0.38 12 7.69 32 4.77 48 3.90 96 2.77 192 1.96
// -0.37 12 7.77 32 4.82 48 3.95 96 2.80 192 1.98
// -0.36 12 7.86 32 4.88 48 3.99 96 2.83 192 2.00
// -0.35 12 7.95 32 4.94 32 4.94 64 3.50 192 2.03
// -0.34 12 8.04 32 4.99 32 4.99 64 3.54 192 2.05
// -0.33 12 8.13 24 5.81 32 5.05 64 3.58 192 2.07
// -0.32 12 8.22 24 5.88 32 5.10 64 3.62 192 2.10
// -0.31 12 8.31 24 5.94 32 5.16 64 3.66 192 2.12
// -0.30 8 10.16 24 6.01 32 5.22 64 3.70 192 2.14
// -0.29 8 10.27 24 6.07 32 5.27 64 3.74 192 2.17
// -0.28 8 10.38 24 6.14 32 5.33 64 3.78 128 2.68
// -0.27 8 10.49 24 6.20 32 5.39 64 3.82 128 2.71
// -0.26 8 10.60 24 6.27 32 5.45 64 3.87 128 2.74
// -0.25 8 10.72 24 6.34 32 5.50 48 4.51 128 2.77
// -0.24 8 10.83 24 6.40 24 6.40 48 4.55 128 2.80
// -0.23 8 10.94 24 6.47 24 6.47 48 4.60 128 2.83
// -0.22 8 11.05 24 6.54 24 6.54 48 4.65 128 2.86
// -0.21 6 12.72 24 6.60 24 6.60 48 4.70 128 2.89
// -0.20 6 12.85 24 6.67 24 6.67 48 4.75 128 2.92
// -0.19 6 12.97 16 8.21 24 6.74 48 4.80 128 2.95
// -0.18 6 13.10 16 8.29 24 6.81 48 4.85 96 3.44
// -0.17 6 13.23 16 8.37 24 6.88 48 4.89 96 3.47
// -0.16 6 13.36 16 8.46 24 6.95 48 4.94 96 3.51
// -0.15 6 13.49 16 8.54 24 7.02 48 5.00 96 3.54
// -0.14 6 13.62 16 8.62 24 7.09 48 5.05 96 3.58
// -0.13 6 13.75 16 8.71 24 7.16 48 5.10 96 3.62
// -0.12 6 13.88 16 8.80 16 8.80 32 6.28 96 3.65
// -0.11 6 14.01 12 10.19 16 8.88 32 6.35 96 3.69
// -0.10 4 16.85 12 10.28 16 8.97 32 6.41 96 3.73
// -0.09 4 17.01 12 10.38 16 9.06 32 6.47 96 3.76
// -0.08 4 17.17 12 10.48 16 9.14 32 6.54 96 3.80
// -0.07 4 17.32 12 10.58 16 9.23 32 6.60 64 4.69
// -0.06 4 17.48 12 10.69 12 10.69 24 7.67 64 4.74
// -0.05 4 17.64 12 10.79 12 10.79 24 7.75 64 4.79
// -0.04 4 17.80 12 10.89 12 10.89 24 7.82 64 4.84
// -0.03 4 17.96 8 13.26 12 10.99 24 7.90 48 5.63
// -0.02 4 18.11 8 13.38 12 11.10 24 7.97 48 5.68
// -0.01 4 18.27 6 15.36 8 13.51 16 9.78 48 5.74
// 0.00 4 1.00 4 1.00 4 1.00 4 1.00 4 1.00
// 0.01 4 18.57 6 15.62 8 13.75 16 9.96 48 5.85
// 0.02 4 18.70 8 13.86 12 11.51 24 8.28 48 5.91
// 0.03 4 18.83 8 13.97 12 11.61 24 8.36 48 5.97
// 0.04 4 18.96 12 11.71 12 11.71 24 8.44 64 5.23
// 0.05 4 19.09 12 11.81 12 11.81 24 8.52 64 5.28
// 0.06 4 19.22 12 11.92 12 11.92 24 8.60 64 5.33
// 0.07 4 19.36 12 12.02 16 10.52 32 7.55 64 5.39
// 0.08 4 19.49 12 12.13 16 10.61 32 7.63 64 5.44
// 0.09 4 19.62 12 12.23 16 10.71 32 7.70 96 4.50
// 0.10 4 19.76 12 12.34 16 10.80 32 7.77 96 4.54
// 0.11 4 19.89 12 12.45 16 10.90 32 7.85 96 4.59
// 0.12 6 17.01 16 11.00 16 11.00 32 7.92 96 4.63
// 0.13 6 17.14 16 11.10 16 11.10 32 8.00 96 4.68
// 0.14 6 17.27 16 11.20 24 9.26 48 6.64 96 4.73
// 0.15 6 17.40 16 11.30 24 9.35 48 6.70 96 4.77
// 0.16 6 17.53 16 11.40 24 9.44 48 6.77 96 4.82
// 0.17 6 17.67 16 11.51 24 9.53 48 6.84 96 4.87
// 0.18 6 17.80 16 11.61 24 9.62 48 6.90 96 4.92
// 0.19 6 17.94 16 11.72 24 9.71 48 6.97 128 4.31
// 0.20 6 18.07 16 11.83 24 9.80 48 7.04 128 4.36
// 0.21 6 18.21 24 9.90 24 9.90 48 7.11 128 4.40
// 0.22 6 18.35 24 9.99 24 9.99 48 7.18 128 4.45
// 0.23 6 18.49 24 10.09 24 10.09 48 7.26 128 4.49
// 0.24 6 18.63 24 10.19 24 10.19 48 7.33 128 4.54
// 0.25 6 18.77 24 10.28 24 10.28 48 7.41 128 4.59
// 0.26 6 18.92 24 10.39 24 10.39 48 7.48 128 4.64
// 0.27 8 17.00 24 10.49 32 9.17 64 6.58 128 4.69
// 0.28 8 17.14 24 10.59 32 9.26 64 6.65 128 4.74
// 0.29 8 17.28 24 10.69 32 9.35 64 6.72 192 3.92
// 0.30 8 17.43 24 10.80 32 9.45 64 6.79 192 3.96
// 0.31 8 17.57 24 10.91 32 9.54 64 6.86 192 4.01
// 0.32 8 17.72 24 11.02 32 9.64 64 6.93 192 4.05
// 0.33 8 17.87 24 11.13 32 9.74 64 7.01 192 4.09
// 0.34 8 18.02 24 11.24 32 9.84 64 7.08 192 4.14
// 0.35 8 18.17 24 11.36 32 9.95 64 7.16 192 4.19
// 0.36 8 18.32 24 11.47 32 10.05 64 7.24 192 4.23
// 0.37 8 18.48 32 10.16 32 10.16 64 7.32 192 4.28
// 0.38 8 18.63 32 10.27 32 10.27 96 6.08 192 4.33
// 0.39 8 18.79 32 10.38 48 8.58 96 6.15 192 4.38
// 0.40 8 18.95 32 10.49 48 8.68 96 6.22 192 4.43
// 0.41 8 19.11 32 10.60 48 8.78 96 6.30 192 4.49
// 0.42 12 16.45 32 10.72 48 8.88 96 6.37 192 4.54
// 0.43 12 16.61 32 10.84 48 8.98 96 6.45 192 4.59
// 0.44 12 16.77 32 10.96 48 9.08 96 6.52 256 4.04
// 0.45 12 16.93 32 11.09 48 9.19 96 6.60 256 4.09
// 0.46 12 17.10 32 11.21 48 9.30 96 6.68 256 4.14
// 0.47 12 17.26 32 11.34 48 9.41 96 6.77 256 4.19
// 0.48 12 17.44 32 11.47 48 9.52 96 6.85 256 4.24
// 0.49 12 17.61 48 9.64 48 9.64 96 6.94 256 4.30
// 0.50 12 17.79 48 9.76 48 9.76 96 7.03 256 4.35
// 0.51 12 17.97 48 9.88 48 9.88 96 7.12 256 4.41
// 0.52 12 18.15 48 10.00 48 10.00 96 7.21 256 4.47
// 0.53 12 18.34 48 10.13 48 10.13 128 6.36 256 4.53
// 0.54 12 18.53 48 10.26 48 10.26 128 6.45 256 4.59
// 0.55 12 18.72 48 10.40 64 9.10 128 6.53 384 3.82
// 0.56 12 18.92 48 10.53 64 9.22 128 6.63 384 3.87
// 0.57 12 19.12 48 10.68 64 9.35 128 6.72 384 3.93
// 0.58 12 19.33 48 10.82 64 9.48 128 6.82 384 3.98
// 0.59 12 19.54 48 10.97 64 9.61 128 6.92 384 4.05
// 0.60 12 19.75 48 11.13 64 9.75 128 7.02 384 4.11
// 0.61 12 19.97 48 11.28 64 9.89 128 7.13 384 4.17
// 0.62 12 20.20 48 11.45 64 10.04 128 7.24 384 4.24
// 0.63 16 18.39 48 11.62 64 10.19 192 6.05 384 4.31
// 0.64 16 18.62 64 10.35 64 10.35 192 6.15 384 4.38
// 0.65 16 18.86 64 10.51 96 8.71 192 6.25 384 4.45
// 0.66 16 19.10 64 10.68 96 8.85 192 6.36 384 4.53
// 0.67 16 19.35 64 10.86 96 9.00 192 6.47 512 4.00
// 0.68 16 19.60 64 11.04 96 9.16 192 6.58 512 4.08
// 0.69 16 19.87 64 11.23 96 9.32 192 6.70 512 4.15
// 0.70 16 20.14 64 11.43 96 9.49 192 6.83 512 4.23
// 0.71 16 20.42 64 11.63 96 9.67 192 6.96 512 4.32
// 0.72 16 20.71 64 11.85 96 9.85 192 7.10 512 4.40
// 0.73 16 21.01 96 10.04 96 10.04 192 7.25 512 4.50
// 0.74 16 21.32 96 10.25 96 10.25 256 6.44 768 3.76
// 0.75 16 21.65 96 10.46 96 10.46 256 6.58 768 3.84
// 0.76 16 21.99 96 10.68 128 9.36 256 6.73 768 3.93
// 0.77 24 19.41 96 10.92 128 9.57 256 6.89 768 4.03
// 0.78 24 19.76 96 11.17 128 9.79 256 7.06 768 4.13
// 0.79 24 20.13 96 11.44 128 10.03 256 7.24 768 4.24
// 0.80 24 20.51 96 11.72 128 10.29 384 6.11 768 4.35
// 0.81 24 20.92 96 12.02 128 10.56 384 6.28 768 4.48
// 0.82 24 21.35 96 12.34 192 8.99 384 6.46 1024 4.00
// 0.83 24 21.81 128 11.16 192 9.26 384 6.66 1024 4.13
// 0.84 24 22.29 128 11.50 192 9.55 384 6.88 1024 4.26
// 0.85 24 22.82 128 11.86 192 9.87 384 7.11 1024 4.41
// 0.86 24 23.38 128 12.26 192 10.21 384 7.37 1024 4.58
// 0.87 24 24.00 128 12.70 192 10.59 512 6.67 1536 3.90
// 0.88 24 24.67 192 11.01 192 11.01 512 6.95 1536 4.06
// 0.89 24 25.41 192 11.48 256 10.07 512 7.26 1536 4.25
// 0.90 24 26.24 192 12.00 256 10.54 768 6.27 1536 4.47
// 0.91 24 27.17 192 12.61 256 11.09 768 6.62 2048 4.10
// 0.92 24 28.23 192 13.30 384 9.74 768 7.02 2048 4.35
// 0.93 24 29.45 256 12.46 384 10.38 768 7.50 3072 3.82
// 0.94 24 30.86 256 13.36 384 11.16 1024 7.05 3072 4.13
// 0.95 24 32.53 384 12.14 512 10.67 1024 7.72 3072 4.53
// 0.96 24 34.51 384 13.42 512 11.83 1536 7.09 4096 4.40
// 0.97 24 36.88 512 13.45 768 11.24 2048 7.11 6144 4.16
// 0.98 16 41.78 768 13.48 1024 11.88 3072 7.12 8192 4.42
// 0.99 8 44.82 1024 16.00 1536 13.51 6144 7.14 16384 4.43
static const int ndiv = 100;
// Encode N as small integer: 2,3,4,6,8,12... -> 0,1,2,3,4,5...
// using this awk script
//
// {
// n = $1;
// if (n % 3 == 0) {
// s = 1;
// n = n/3;
// } else {
// s = 0;
// n = n/2;
// }
// p = int( log(n)/log(2)+0.5 );
// printf "%d\n", 2*p+s;
// }
//
// A couple of changes have been made: (1) the decrease in N for float and
// n > 0.97 has been removed; (2) entrys of n=+/-1 have been included
// (incrementing the previous code value by 1).
#if GEOGRAPHICLIB_PRECISION == 1
static const unsigned char narr[2*ndiv+1] = {
19,18,16,15,14,13,13,13,12,12,11,11,11,11,10,10,10,10,9,9,9,9,9,9,9,9,8,
8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5,
5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,
2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,
4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,
6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8
};
#elif GEOGRAPHICLIB_PRECISION == 2
static const unsigned char narr[2*ndiv+1] = {
22,21,19,18,17,17,16,15,15,15,14,14,14,13,13,13,13,13,13,12,12,12,12,12,
12,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10,10,9,9,9,9,9,9,9,9,
9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,
6,6,5,5,5,5,5,5,5,5,4,4,3,2,3,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,
7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,
9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,
12,12,12,13,13,13,13,13,14,14,15,15,16,17,18,19
};
#elif GEOGRAPHICLIB_PRECISION == 3
static const unsigned char narr[2*ndiv+1] = {
23,22,20,19,18,17,17,16,16,15,15,15,15,14,14,14,14,13,13,13,13,13,13,13,
12,12,12,12,12,12,11,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10,
10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,
7,7,7,7,7,6,6,6,6,6,6,5,5,5,5,5,4,2,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,
7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,12,12,12,
12,12,12,13,13,13,13,13,13,13,14,14,14,15,15,15,16,16,17,18,19,20
};
#elif GEOGRAPHICLIB_PRECISION == 4
static const unsigned char narr[2*ndiv+1] = {
25,24,22,21,20,19,19,18,18,17,17,17,17,16,16,16,15,15,15,15,15,15,15,14,
14,14,14,14,14,13,13,13,13,13,13,13,13,13,13,13,13,12,12,12,12,12,12,12,
12,12,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10,
10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,7,7,7,7,7,6,2,6,7,7,7,7,7,
8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,
11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,
12,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,
15,16,16,16,17,17,17,17,18,18,19,20,21,23,24
};
#elif GEOGRAPHICLIB_PRECISION >= 5
static const unsigned char narr[2*ndiv+1] = {
27,26,24,23,22,22,21,21,20,20,20,19,19,19,19,18,18,18,18,18,17,17,17,17,
17,17,17,17,16,16,16,16,16,16,16,15,15,15,15,15,15,15,15,15,15,15,15,14,
14,14,14,14,14,14,14,14,14,14,13,13,13,13,13,13,13,13,13,13,13,13,13,13,
12,12,12,12,12,12,12,12,12,12,11,11,11,11,11,11,11,11,11,11,11,10,10,10,
10,9,9,9,2,9,9,9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,12,
12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,
14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16,
16,16,16,16,17,17,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,20,20,21,
21,21,22,23,24,26,27
};
#else
#error "Bad value for GEOGRAPHICLIB_PRECISION"
#endif
real n = ndiv * _n; // n in [-ndiv, ndiv]
int j = ndiv + int(n < 0 ? floor(n) : ceil(n)); // j in [0, 2*ndiv]
int N = int(narr[j]);
// Decode 0,1,2,3,4,5... -> 2,3,4,6,8,12...
N = (N % 2 == 0 ? 2 : 3) * (1 << (N/2));
#if GEOGRAPHICLIB_PRECISION == 5
if (Math::digits() > 256) {
// Scale up N by the number of digits in the precision relative to
// the number used for the test = 256.
int M = (Math::digits() * N) / 256;
while (N < M) N = N % 3 == 0 ? 4*N/3 : 3*N/2;
}
#endif
_fft.reset(N);
_nC4 = N;
}
const GeodesicExact& GeodesicExact::WGS84() {
static const GeodesicExact wgs84(Constants::WGS84_a(),
Constants::WGS84_f());
return wgs84;
}
GeodesicLineExact GeodesicExact::Line(real lat1, real lon1, real azi1,
unsigned caps) const {
return GeodesicLineExact(*this, lat1, lon1, azi1, caps);
}
Math::real GeodesicExact::GenDirect(real lat1, real lon1, real azi1,
bool arcmode, real s12_a12,
unsigned outmask,
real& lat2, real& lon2, real& azi2,
real& s12, real& m12,
real& M12, real& M21,
real& S12) const {
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) outmask |= DISTANCE_IN;
return GeodesicLineExact(*this, lat1, lon1, azi1, outmask)
. // Note the dot!
GenPosition(arcmode, s12_a12, outmask,
lat2, lon2, azi2, s12, m12, M12, M21, S12);
}
GeodesicLineExact GeodesicExact::GenDirectLine(real lat1, real lon1,
real azi1,
bool arcmode, real s12_a12,
unsigned caps) const {
azi1 = Math::AngNormalize(azi1);
real salp1, calp1;
// Guard against underflow in salp0. Also -0 is converted to +0.
Math::sincosd(Math::AngRound(azi1), salp1, calp1);
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) caps |= DISTANCE_IN;
return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1,
caps, arcmode, s12_a12);
}
GeodesicLineExact GeodesicExact::DirectLine(real lat1, real lon1,
real azi1, real s12,
unsigned caps) const {
return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
}
GeodesicLineExact GeodesicExact::ArcDirectLine(real lat1, real lon1,
real azi1, real a12,
unsigned caps) const {
return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
}
Math::real GeodesicExact::GenInverse(real lat1, real lon1,
real lat2, real lon2,
unsigned outmask, real& s12,
real& salp1, real& calp1,
real& salp2, real& calp2,
real& m12, real& M12, real& M21,
real& S12) const {
// Compute longitude difference (AngDiff does this carefully). Result is
// in [-180, 180] but -180 is only for west-going geodesics. 180 is for
// east-going and meridional geodesics.
real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
// Make longitude difference positive.
int lonsign = signbit(lon12) ? -1 : 1;
lon12 *= lonsign; lon12s *= lonsign;
real
lam12 = lon12 * Math::degree(),
slam12, clam12;
// Calculate sincos of lon12 + error (this applies AngRound internally).
Math::sincosde(lon12, lon12s, slam12, clam12);
// the supplementary longitude difference
lon12s = (Math::hd - lon12) - lon12s;
// If really close to the equator, treat as on equator.
lat1 = Math::AngRound(Math::LatFix(lat1));
lat2 = Math::AngRound(Math::LatFix(lat2));
// Swap points so that point with higher (abs) latitude is point 1
// If one latitude is a nan, then it becomes lat1.
int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1;
if (swapp < 0) {
lonsign *= -1;
swap(lat1, lat2);
}
// Make lat1 <= -0
int latsign = signbit(lat1) ? 1 : -1;
lat1 *= latsign;
lat2 *= latsign;
// Now we have
//
// 0 <= lon12 <= 180
// -90 <= lat1 <= -0
// lat1 <= lat2 <= -lat1
//
// longsign, swapp, latsign register the transformation to bring the
// coordinates to this canonical form. In all cases, 1 means no change was
// made. We make these transformations so that there are few cases to
// check, e.g., on verifying quadrants in atan2. In addition, this
// enforces some symmetries in the results returned.
real sbet1, cbet1, sbet2, cbet2, s12x, m12x = Math::NaN();
// Initialize for the meridian. No longitude calculation is done in this
// case to let the parameter default to 0.
EllipticFunction E(-_ep2);
Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
// Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
// will be <= 2*tiny for two points at the same pole.
Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
// Ensure cbet2 = +epsilon at poles
Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2);
// If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
// |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
// a better measure. This logic is used in assigning calp2 in Lambda12.
// Sometimes these quantities vanish and in that case we force bet2 = +/-
// bet1 exactly. An example where is is necessary is the inverse problem
// 48.522876735459 0 -48.52287673545898293 179.599720456223079643
// which failed with Visual Studio 10 (Release and Debug)
if (cbet1 < -sbet1) {
if (cbet2 == cbet1)
sbet2 = copysign(sbet1, sbet2);
} else {
if (fabs(sbet2) == -sbet1)
cbet2 = cbet1;
}
real
dn1 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet1)) :
sqrt(1 - _e2 * Math::sq(cbet1)) / _f1),
dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) :
sqrt(1 - _e2 * Math::sq(cbet2)) / _f1);
real a12, sig12;
bool meridian = lat1 == -Math::qd || slam12 == 0;
if (meridian) {
// Endpoints are on a single full meridian, so the geodesic might lie on
// a meridian.
calp1 = clam12; salp1 = slam12; // Head to the target longitude
calp2 = 1; salp2 = 0; // At the target we're heading north
real
// tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2;
// sig12 = sig2 - sig1
sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2);
{
real dummy;
Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, outmask | REDUCEDLENGTH,
s12x, m12x, dummy, M12, M21);
}
// Add the check for sig12 since zero length geodesics might yield m12 <
// 0. Test case was
//
// echo 20.001 0 20.001 0 | GeodSolve -i
if (sig12 < tol2_ || m12x >= 0) {
// Need at least 2, to handle 90 0 90 180
if (sig12 < 3 * tiny_ ||
// Prevent negative s12 or m12 for short lines
(sig12 < tol0_ && (s12x < 0 || m12x < 0)))
sig12 = m12x = s12x = 0;
m12x *= _b;
s12x *= _b;
a12 = sig12 / Math::degree();
} else
// m12 < 0, i.e., prolate and too close to anti-podal
meridian = false;
}
// somg12 == 2 marks that it needs to be calculated
real omg12 = 0, somg12 = 2, comg12 = 0;
if (!meridian &&
sbet1 == 0 && // and sbet2 == 0
(_f <= 0 || lon12s >= _f * Math::hd)) {
// Geodesic runs along equator
calp1 = calp2 = 0; salp1 = salp2 = 1;
s12x = _a * lam12;
sig12 = omg12 = lam12 / _f1;
m12x = _b * sin(sig12);
if (outmask & GEODESICSCALE)
M12 = M21 = cos(sig12);
a12 = lon12 / _f1;
} else if (!meridian) {
// Now point1 and point2 belong within a hemisphere bounded by a
// meridian and geodesic is neither meridional or equatorial.
// Figure a starting point for Newton's method
real dnm;
sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12,
salp1, calp1, salp2, calp2, dnm);
if (sig12 >= 0) {
// Short lines (InverseStart sets salp2, calp2, dnm)
s12x = sig12 * _b * dnm;
m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
if (outmask & GEODESICSCALE)
M12 = M21 = cos(sig12 / dnm);
a12 = sig12 / Math::degree();
omg12 = lam12 / (_f1 * dnm);
} else {
// Newton's method. This is a straightforward solution of f(alp1) =
// lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
// root in the interval (0, pi) and its derivative is positive at the
// root. Thus f(alp) is positive for alp > alp1 and negative for alp <
// alp1. During the course of the iteration, a range (alp1a, alp1b) is
// maintained which brackets the root and with each evaluation of
// f(alp) the range is shrunk, if possible. Newton's method is
// restarted whenever the derivative of f is negative (because the new
// value of alp1 is then further from the solution) or if the new
// estimate of alp1 lies outside (0,pi); in this case, the new starting
// guess is taken to be (alp1a + alp1b) / 2.
//
// initial values to suppress warnings (if loop is executed 0 times)
real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, domg12 = 0;
unsigned numit = 0;
// Bracketing range
real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
for (bool tripn = false, tripb = false;; ++numit) {
// 1/4 meridian = 10e6 m and random input. max err is estimated max
// error in nm (checking solution of inverse problem by direct
// solution). iter is mean and sd of number of iterations
//
// max iter
// log2(b/a) err mean sd
// -7 387 5.33 3.68
// -6 345 5.19 3.43
// -5 269 5.00 3.05
// -4 210 4.76 2.44
// -3 115 4.55 1.87
// -2 69 4.35 1.38
// -1 36 4.05 1.03
// 0 15 0.01 0.13
// 1 25 5.10 1.53
// 2 96 5.61 2.09
// 3 318 6.02 2.74
// 4 985 6.24 3.22
// 5 2352 6.32 3.44
// 6 6008 6.30 3.45
// 7 19024 6.19 3.30
real dv;
real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
slam12, clam12,
salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
E, domg12, numit < maxit1_, dv);
if (tripb ||
// Reversed test to allow escape with NaNs
!(fabs(v) >= (tripn ? 8 : 1) * tol0_) ||
// Enough bisections to get accurate result
numit == maxit2_)
break;
// Update bracketing values
if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
{ salp1b = salp1; calp1b = calp1; }
else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
{ salp1a = salp1; calp1a = calp1; }
if (numit < maxit1_ && dv > 0) {
real
dalp1 = -v/dv;
// |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION
// = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages
// (because of the need to do accurate range reduction).
if (fabs(dalp1) < Math::pi()) {
real
sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
if (nsalp1 > 0) {
calp1 = calp1 * cdalp1 - salp1 * sdalp1;
salp1 = nsalp1;
Math::norm(salp1, calp1);
// In some regimes we don't get quadratic convergence because
// slope -> 0. So use convergence conditions based on epsilon
// instead of sqrt(epsilon).
tripn = fabs(v) <= 16 * tol0_;
continue;
}
}
}
// Either dv was not positive or updated value was outside legal
// range. Use the midpoint of the bracket as the next estimate.
// This mechanism is not needed for the WGS84 ellipsoid, but it does
// catch problems with more eccentric ellipsoids. Its efficacy is
// such for the WGS84 test set with the starting guess set to alp1 =
// 90deg:
// the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
// WGS84 and random input: mean = 4.74, sd = 0.99
salp1 = (salp1a + salp1b)/2;
calp1 = (calp1a + calp1b)/2;
Math::norm(salp1, calp1);
tripn = false;
tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
}
{
real dummy;
Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
}
m12x *= _b;
s12x *= _b;
a12 = sig12 / Math::degree();
if (outmask & AREA) {
// omg12 = lam12 - domg12
real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
somg12 = slam12 * cdomg12 - clam12 * sdomg12;
comg12 = clam12 * cdomg12 + slam12 * sdomg12;
}
}
}
if (outmask & DISTANCE)
s12 = real(0) + s12x; // Convert -0 to 0
if (outmask & REDUCEDLENGTH)
m12 = real(0) + m12x; // Convert -0 to 0
if (outmask & AREA) {
real
// From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
real alp12,
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = Math::sq(_a) * calp0 * salp0 * _e2;
if (A4 != 0) {
real
k2 = Math::sq(calp0) * _ep2,
// From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2;
Math::norm(ssig1, csig1);
Math::norm(ssig2, csig2);
I4Integrand i4(_ep2, k2);
vector<real> C4a(_nC4);
_fft.transform(i4, C4a.data());
S12 = A4 * DST::integral(ssig1, csig1, ssig2, csig2, C4a.data(), _nC4);
} else
// Avoid problems with indeterminate sig1, sig2 on equator
S12 = 0;
if (!meridian && somg12 == 2) {
somg12 = sin(omg12); comg12 = cos(omg12);
}
if (!meridian &&
// omg12 < 3/4 * pi
comg12 > -real(0.7071) && // Long difference not too big
sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
// Use tan(Gamma/2) = tan(omg12/2)
// * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
// with tan(x/2) = sin(x)/(1+cos(x))
real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
} else {
// alp12 = alp2 - alp1, used in atan2 so no need to normalize
real
salp12 = salp2 * calp1 - calp2 * salp1,
calp12 = calp2 * calp1 + salp2 * salp1;
// The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
// salp12 = -0 and alp12 = -180. However this depends on the sign
// being attached to 0 correctly. The following ensures the correct
// behavior.
if (salp12 == 0 && calp12 < 0) {
salp12 = tiny_ * calp1;
calp12 = -1;
}
alp12 = atan2(salp12, calp12);
}
S12 += _c2 * alp12;
S12 *= swapp * lonsign * latsign;
// Convert -0 to 0
S12 += 0;
}
// Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if (swapp < 0) {
swap(salp1, salp2);
swap(calp1, calp2);
if (outmask & GEODESICSCALE)
swap(M12, M21);
}
salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
// Returned value in [0, 180]
return a12;
}
Math::real GeodesicExact::GenInverse(real lat1, real lon1,
real lat2, real lon2,
unsigned outmask,
real& s12, real& azi1, real& azi2,
real& m12, real& M12, real& M21,
real& S12) const {
outmask &= OUT_MASK;
real salp1, calp1, salp2, calp2,
a12 = GenInverse(lat1, lon1, lat2, lon2,
outmask, s12, salp1, calp1, salp2, calp2,
m12, M12, M21, S12);
if (outmask & AZIMUTH) {
azi1 = Math::atan2d(salp1, calp1);
azi2 = Math::atan2d(salp2, calp2);
}
return a12;
}
GeodesicLineExact GeodesicExact::InverseLine(real lat1, real lon1,
real lat2, real lon2,
unsigned caps) const {
real t, salp1, calp1, salp2, calp2,
a12 = GenInverse(lat1, lon1, lat2, lon2,
// No need to specify AZIMUTH here
0u, t, salp1, calp1, salp2, calp2,
t, t, t, t),
azi1 = Math::atan2d(salp1, calp1);
// Ensure that a12 can be converted to a distance
if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1, caps,
true, a12);
}
void GeodesicExact::Lengths(const EllipticFunction& E,
real sig12,
real ssig1, real csig1, real dn1,
real ssig2, real csig2, real dn2,
real cbet1, real cbet2, unsigned outmask,
real& s12b, real& m12b, real& m0,
real& M12, real& M21) const {
// Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
// and m0 = coefficient of secular term in expression for reduced length.
outmask &= OUT_ALL;
// outmask & DISTANCE: set s12b
// outmask & REDUCEDLENGTH: set m12b & m0
// outmask & GEODESICSCALE: set M12 & M21
// It's OK to have repeated dummy arguments,
// e.g., s12b = m0 = M12 = M21 = dummy
if (outmask & DISTANCE)
// Missing a factor of _b
s12b = E.E() / (Math::pi() / 2) *
(sig12 + (E.deltaE(ssig2, csig2, dn2) - E.deltaE(ssig1, csig1, dn1)));
if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
real
m0x = - E.k2() * E.D() / (Math::pi() / 2),
J12 = m0x *
(sig12 + (E.deltaD(ssig2, csig2, dn2) - E.deltaD(ssig1, csig1, dn1)));
if (outmask & REDUCEDLENGTH) {
m0 = m0x;
// Missing a factor of _b. Add parens around (csig1 * ssig2) and
// (ssig1 * csig2) to ensure accurate cancellation in the case of
// coincident points.
m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
csig1 * csig2 * J12;
}
if (outmask & GEODESICSCALE) {
real csig12 = csig1 * csig2 + ssig1 * ssig2;
real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
}
}
}
Math::real GeodesicExact::Astroid(real x, real y) {
// Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
// This solution is adapted from Geocentric::Reverse.
real k;
real
p = Math::sq(x),
q = Math::sq(y),
r = (p + q - 1) / 6;
if ( !(q == 0 && r <= 0) ) {
real
// Avoid possible division by zero when r = 0 by multiplying equations
// for s and t by r^3 and r, resp.
S = p * q / 4, // S = r^3 * s
r2 = Math::sq(r),
r3 = r * r2,
// The discriminant of the quadratic equation for T3. This is zero on
// the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3);
real u = r;
if (disc >= 0) {
real T3 = S + r3;
// Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
// of precision due to cancellation. The result is unchanged because
// of the way the T is used in definition of u.
T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
// N.B. cbrt always returns the real root. cbrt(-8) = -2.
real T = cbrt(T3); // T = r * t
// T can be zero; but then r2 / T -> 0.
u += T + (T != 0 ? r2 / T : 0);
} else {
// T is complex, but the way u is defined the result is real.
real ang = atan2(sqrt(-disc), -(S + r3));
// There are three possible cube roots. We choose the root which
// avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * cos(ang / 3);
}
real
v = sqrt(Math::sq(u) + q), // guaranteed positive
// Avoid loss of accuracy when u < 0.
uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
w = (uv - q) / (2 * v); // positive?
// Rearrange expression for k to avoid loss of accuracy due to
// subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
} else { // q == 0 && r <= 0
// y = 0 with |x| <= 1. Handle this case directly.
// for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0;
}
return k;
}
Math::real GeodesicExact::InverseStart(EllipticFunction& E,
real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real lam12, real slam12, real clam12,
real& salp1, real& calp1,
// Only updated if return val >= 0
real& salp2, real& calp2,
// Only updated for short lines
real& dnm) const {
// Return a starting point for Newton's method in salp1 and calp1 (function
// value is -1). If Newton's method doesn't need to be used, return also
// salp2 and calp2 and function value is sig12.
real
sig12 = -1, // Return value
// bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
cbet2 * lam12 < real(0.5);
real somg12, comg12;
if (shortline) {
real sbetm2 = Math::sq(sbet1 + sbet2);
// sin((bet1+bet2)/2)^2
// = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
dnm = sqrt(1 + _ep2 * sbetm2);
real omg12 = lam12 / (_f1 * dnm);
somg12 = sin(omg12); comg12 = cos(omg12);
} else {
somg12 = slam12; comg12 = clam12;
}
salp1 = cbet2 * somg12;
calp1 = comg12 >= 0 ?
sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
real
ssig12 = hypot(salp1, calp1),
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
if (shortline && ssig12 < _etol2) {
// really short lines
salp2 = cbet1 * somg12;
calp2 = sbet12 - cbet1 * sbet2 *
(comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
Math::norm(salp2, calp2);
// Set return value
sig12 = atan2(ssig12, csig12);
} else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric
csig12 >= 0 ||
ssig12 >= 6 * fabs(_n) * Math::pi() * Math::sq(cbet1)) {
// Nothing to do, zeroth order spherical approximation is OK
} else {
// Scale lam12 and bet2 to x, y coordinate system where antipodal point
// is at origin and singular point is at y = 0, x = -1.
real x, y, lamscale, betscale;
real lam12x = atan2(-slam12, -clam12); // lam12 - pi
if (_f >= 0) { // In fact f == 0 does not get here
// x = dlong, y = dlat
{
real k2 = Math::sq(sbet1) * _ep2;
E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
lamscale = _e2/_f1 * cbet1 * 2 * E.H();
}
betscale = lamscale * cbet1;
x = lam12x / lamscale;
y = sbet12a / betscale;
} else { // _f < 0
// x = dlat, y = dlong
real
cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
bet12a = atan2(sbet12a, cbet12a);
real m12b, m0, dummy;
// In the case of lon12 = 180, this repeats a calculation made in
// Inverse.
Lengths(E, Math::pi() + bet12a,
sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy);
x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
betscale = x < -real(0.01) ? sbet12a / x :
-_f * Math::sq(cbet1) * Math::pi();
lamscale = betscale / cbet1;
y = lam12x / lamscale;
}
if (y > -tol1_ && x > -1 - xthresh_) {
// strip near cut
// Need real(x) here to cast away the volatility of x for min/max
if (_f >= 0) {
salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1));
} else {
calp1 = fmax(real(x > -tol1_ ? 0 : -1), x);
salp1 = sqrt(1 - Math::sq(calp1));
}
} else {
// Estimate alp1, by solving the astroid problem.
//
// Could estimate alpha1 = theta + pi/2, directly, i.e.,
// calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
// calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
//
// However, it's better to estimate omg12 from astroid and use
// spherical formula to compute alp1. This reduces the mean number of
// Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
// (min 0 max 5). The changes in the number of iterations are as
// follows:
//
// change percent
// 1 5
// 0 78
// -1 16
// -2 0.6
// -3 0.04
// -4 0.002
//
// The histogram of iterations is (m = number of iterations estimating
// alp1 directly, n = number of iterations estimating via omg12, total
// number of trials = 148605):
//
// iter m n
// 0 148 186
// 1 13046 13845
// 2 93315 102225
// 3 36189 32341
// 4 5396 7
// 5 455 1
// 6 56 0
//
// Because omg12 is near pi, estimate work with omg12a = pi - omg12
real k = Astroid(x, y);
real
omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
somg12 = sin(omg12a); comg12 = -cos(omg12a);
// Update spherical estimate of alp1 using omg12 instead of lam12
salp1 = cbet2 * somg12;
calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
}
}
// Sanity check on starting guess. Backwards check allows NaN through.
if (!(salp1 <= 0))
Math::norm(salp1, calp1);
else {
salp1 = 1; calp1 = 0;
}
return sig12;
}
Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real salp1, real calp1,
real slam120, real clam120,
real& salp2, real& calp2,
real& sig12,
real& ssig1, real& csig1,
real& ssig2, real& csig2,
EllipticFunction& E,
real& domg12,
bool diffp, real& dlam12) const
{
if (sbet1 == 0 && calp1 == 0)
// Break degeneracy of equatorial line. This case has already been
// handled.
calp1 = -tiny_;
real
// sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
real somg1, comg1, somg2, comg2, somg12, comg12, cchi1, cchi2, lam12;
// tan(bet1) = tan(sig1) * cos(alp1)
// tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
ssig1 = sbet1; somg1 = salp0 * sbet1;
csig1 = comg1 = calp1 * cbet1;
// Without normalization we have schi1 = somg1.
cchi1 = _f1 * dn1 * comg1;
Math::norm(ssig1, csig1);
// Math::norm(somg1, comg1); -- don't need to normalize!
// Math::norm(schi1, cchi1); -- don't need to normalize!
// Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
// about this case, since this can yield singularities in the Newton
// iteration.
// sin(alp2) * cos(bet2) = sin(alp0)
salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
// calp2 = sqrt(1 - sq(salp2))
// = sqrt(sq(calp0) - sq(sbet2)) / cbet2
// and subst for calp0 and rearrange to give (choose positive sqrt
// to give alp2 in [0, pi/2]).
calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
sqrt(Math::sq(calp1 * cbet1) +
(cbet1 < -sbet1 ?
(cbet2 - cbet1) * (cbet1 + cbet2) :
(sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
fabs(calp1);
// tan(bet2) = tan(sig2) * cos(alp2)
// tan(omg2) = sin(alp0) * tan(sig2).
ssig2 = sbet2; somg2 = salp0 * sbet2;
csig2 = comg2 = calp2 * cbet2;
// Without normalization we have schi2 = somg2.
cchi2 = _f1 * dn2 * comg2;
Math::norm(ssig2, csig2);
// Math::norm(somg2, comg2); -- don't need to normalize!
// Math::norm(schi2, cchi2); -- don't need to normalize!
// sig12 = sig2 - sig1, limit to [0, pi]
sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2);
// omg12 = omg2 - omg1, limit to [0, pi]
somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2);
comg12 = comg1 * comg2 + somg1 * somg2;
real k2 = Math::sq(calp0) * _ep2;
E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
// chi12 = chi2 - chi1, limit to [0, pi]
real
schi12 = fmax(real(0), cchi1 * somg2 - somg1 * cchi2),
cchi12 = cchi1 * cchi2 + somg1 * somg2;
// eta = chi12 - lam120
real eta = atan2(schi12 * clam120 - cchi12 * slam120,
cchi12 * clam120 + schi12 * slam120);
real deta12 = -_e2/_f1 * salp0 * E.H() / (Math::pi() / 2) *
(sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1)));
lam12 = eta + deta12;
// domg12 = deta12 + chi12 - omg12
domg12 = deta12 + atan2(schi12 * comg12 - cchi12 * somg12,
cchi12 * comg12 + schi12 * somg12);
if (diffp) {
if (calp2 == 0)
dlam12 = - 2 * _f1 * dn1 / sbet1;
else {
real dummy;
Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, REDUCEDLENGTH,
dummy, dlam12, dummy, dummy, dummy);
dlam12 *= _f1 / (calp2 * cbet2);
}
}
return lam12;
}
Math::real GeodesicExact::I4Integrand::asinhsqrt(real x) {
// return asinh(sqrt(x))/sqrt(x)
return x == 0 ? 1 :
(x > 0 ? asinh(sqrt(x))/sqrt(x) :
asin(sqrt(-x))/sqrt(-x)); // NaNs end up here
}
Math::real GeodesicExact::I4Integrand::t(real x) {
// This differs by from t as defined following Eq 61 in Karney (2013) by
// the final subtraction of 1. This changes nothing since Eq 61 uses the
// difference of two evaluations of t and improves the accuracy(?).
// Group terms to minimize roundoff
// with x = ep2, this is the same as
// e2/(1-e2) + (atanh(e)/e - 1)
return x + (sqrt(1 + x) * asinhsqrt(x) - 1);
}
Math::real GeodesicExact::I4Integrand::td(real x) {
// d t(x) / dx
return x == 0 ? 4/real(3) :
// Group terms to minimize roundoff
1 + (1 - asinhsqrt(x) / sqrt(1+x)) / (2*x);
}
// Math::real GeodesicExact::I4Integrand::Dt(real x, real y) {
// // ( t(x) - t(y) ) / (x - y)
// if (x == y) return td(x);
// if (x * y <= 0) return ( t(x) - t(y) ) / (x - y);
// real
// sx = sqrt(fabs(x)), sx1 = sqrt(1 + x),
// sy = sqrt(fabs(y)), sy1 = sqrt(1 + y),
// z = (x - y) / (sx * sy1 + sy * sx1),
// d1 = 2 * sx * sy,
// d2 = 2 * (x * sy * sy1 + y * sx * sx1);
// return x > 0 ?
// ( 1 + (asinh(z)/z) / d1 - (asinh(sx) + asinh(sy)) / d2 ) :
// // NaNs fall through to here
// ( 1 - (asin (z)/z) / d1 - (asin (sx) + asin (sy)) / d2 );
// }
Math::real GeodesicExact::I4Integrand::DtX(real y) const {
// idiot version:
// return ( tX - t(y) ) / (X - y);
if (X == y) return tdX;
if (X * y <= 0) return ( tX - t(y) ) / (X - y);
real
sy = sqrt(fabs(y)), sy1 = sqrt(1 + y),
z = (X - y) / (sX * sy1 + sy * sX1),
d1 = 2 * sX * sy,
d2 = 2 * (X * sy * sy1 + y * sXX1);
return X > 0 ?
( 1 + (asinh(z)/z) / d1 - (asinhsX + asinh(sy)) / d2 ) :
// NaNs fall through to here
( 1 - (asin (z)/z) / d1 - (asinhsX + asin (sy)) / d2 );
}
GeodesicExact::I4Integrand::I4Integrand(real ep2, real k2)
: X( ep2 )
, tX( t(X) )
, tdX( td(X) )
, _k2( k2 )
{
sX = sqrt(fabs(X)); // ep
sX1 = sqrt(1 + X); // 1/(1-f)
sXX1 = sX * sX1;
asinhsX = X > 0 ? asinh(sX) : asin(sX); // atanh(e)
}
Math::real GeodesicExact::I4Integrand::operator()(real sig) const {
real ssig = sin(sig);
return - DtX(_k2 * Math::sq(ssig)) * ssig/2;
}
} // namespace GeographicLib
ang
GeographicLib::Angle ang
Definition Geod3Solve.cpp:26
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeodesicExact.hpp
Header for GeographicLib::GeodesicExact class.
GeodesicLineExact.hpp
Header for GeographicLib::GeodesicLineExact class.
GeographicLib::Constants::WGS84_f
static T WGS84_f()
Definition Constants.hpp:118
GeographicLib::Constants::WGS84_a
static T WGS84_a()
Definition Constants.hpp:112
GeographicLib::DST::integral
static real integral(real sinx, real cosx, const real F[], int N)
Definition DST.cpp:112
GeographicLib::EllipticFunction
Elliptic integrals and functions.
Definition EllipticFunction.hpp:63
GeographicLib::EllipticFunction::deltaE
Math::real deltaE(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:525
GeographicLib::EllipticFunction::E
Math::real E() const
Definition EllipticFunction.hpp:203
GeographicLib::EllipticFunction::k2
Math::real k2() const
Definition EllipticFunction.hpp:158
GeographicLib::EllipticFunction::deltaD
Math::real deltaD(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:537
GeographicLib::EllipticFunction::D
Math::real D() const
Definition EllipticFunction.hpp:216
GeographicLib::GeodesicExact::InverseLine
GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition GeodesicExact.cpp:814
GeographicLib::GeodesicExact::GenDirectLine
GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition GeodesicExact.cpp:417
GeographicLib::GeodesicExact::DirectLine
GeodesicLineExact DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition GeodesicExact.cpp:431
GeographicLib::GeodesicExact::GenDirect
Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
Definition GeodesicExact.cpp:402
GeographicLib::GeodesicExact::GeodesicLineExact
friend class GeodesicLineExact
Definition GeodesicExact.hpp:91
GeographicLib::GeodesicExact::Line
GeodesicLineExact Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition GeodesicExact.cpp:397
GeographicLib::GeodesicExact::DISTANCE
@ DISTANCE
Definition GeodesicExact.hpp:194
GeographicLib::GeodesicExact::AREA
@ AREA
Definition GeodesicExact.hpp:222
GeographicLib::GeodesicExact::DISTANCE_IN
@ DISTANCE_IN
Definition GeodesicExact.hpp:207
GeographicLib::GeodesicExact::AZIMUTH
@ AZIMUTH
Definition GeodesicExact.hpp:189
GeographicLib::GeodesicExact::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition GeodesicExact.hpp:212
GeographicLib::GeodesicExact::GEODESICSCALE
@ GEODESICSCALE
Definition GeodesicExact.hpp:217
GeographicLib::GeodesicExact::WGS84
static const GeodesicExact & WGS84()
Definition GeodesicExact.cpp:391
GeographicLib::GeodesicExact::ArcDirectLine
GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition GeodesicExact.cpp:437
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:344
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition Math.hpp:82
GeographicLib::Math::degree
static T degree()
Definition Math.hpp:197
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition Math.hpp:303
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition Math.cpp:104
GeographicLib::Math::atan2d
static T atan2d(T y, T x)
Definition Math.cpp:212
GeographicLib::Math::norm
static void norm(T &x, T &y)
Definition Math.hpp:219
GeographicLib::Math::AngRound
static T AngRound(T x)
Definition Math.cpp:95
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:142
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::digits
static int digits()
Definition Math.cpp:20
GeographicLib::Math::sincosde
static void sincosde(T x, T t, T &sinx, T &cosx)
Definition Math.cpp:131
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:187
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:301
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::hd
static constexpr int hd
degrees per half turn
Definition Math.hpp:145
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition NearestNeighbor.hpp:819