GeographicLib: GeodesicLine.cpp Source File

GeographicLib 2.6

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GeodesicLine.cpp
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1/**
2 * \file GeodesicLine.cpp
3 * \brief Implementation for GeographicLib::GeodesicLine class
4 *
5 * Copyright (c) Charles Karney (2009-2023) <karney@alum.mit.edu> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 *
9 * This is a reformulation of the geodesic problem. The notation is as
10 * follows:
11 * - at a general point (no suffix or 1 or 2 as suffix)
12 * - phi = latitude
13 * - beta = latitude on auxiliary sphere
14 * - omega = longitude on auxiliary sphere
15 * - lambda = longitude
16 * - alpha = azimuth of great circle
17 * - sigma = arc length along great circle
18 * - s = distance
19 * - tau = scaled distance (= sigma at multiples of pi/2)
20 * - at northwards equator crossing
21 * - beta = phi = 0
22 * - omega = lambda = 0
23 * - alpha = alpha0
24 * - sigma = s = 0
25 * - a 12 suffix means a difference, e.g., s12 = s2 - s1.
26 * - s and c prefixes mean sin and cos
27 **********************************************************************/
28
29#include <GeographicLib/GeodesicLine.hpp>
30
31namespace GeographicLib {
32
33 using namespace std;
34
35 void GeodesicLine::LineInit(const Geodesic& g,
36 real lat1, real lon1,
37 real azi1, real salp1, real calp1,
38 unsigned caps) {
39 tiny_ = g.tiny_;
40 _lat1 = Math::LatFix(lat1);
41 _lon1 = lon1;
42 _azi1 = azi1;
43 _salp1 = salp1;
44 _calp1 = calp1;
45 _a = g._a;
46 _f = g._f;
47 _b = g._b;
48 _c2 = g._c2;
49 _f1 = g._f1;
50 // Always allow latitude and azimuth and unrolling of longitude
51 _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
52
53 real cbet1, sbet1;
54 Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
55 // Ensure cbet1 = +epsilon at poles
56 Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
57 _dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1));
58
59 // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
60 _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
61 // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
62 // is slightly better (consider the case salp1 = 0).
63 _calp0 = hypot(_calp1, _salp1 * sbet1);
64 // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
65 // sig = 0 is nearest northward crossing of equator.
66 // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
67 // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
68 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
69 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
70 // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
71 // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
72 // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
73 _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
74 _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
75 Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
76 // Math::norm(_somg1, _comg1); -- don't need to normalize!
77
78 _a13 = _s13 = Math::NaN();
79 _exact = g._exact;
80 if (_exact) {
81 _lineexact.LineInit(g._geodexact, lat1, lon1, azi1, salp1, calp1, caps);
82 return;
83 }
84
85 _k2 = Math::sq(_calp0) * g._ep2;
86 real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
87
88 if (_caps & CAP_C1) {
89 _aA1m1 = Geodesic::A1m1f(eps);
90 Geodesic::C1f(eps, _cC1a);
91 _bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_);
92 real s = sin(_bB11), c = cos(_bB11);
93 // tau1 = sig1 + B11
94 _stau1 = _ssig1 * c + _csig1 * s;
95 _ctau1 = _csig1 * c - _ssig1 * s;
96 // Not necessary because C1pa reverts C1a
97 // _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_);
98 }
99
100 if (_caps & CAP_C1p)
101 Geodesic::C1pf(eps, _cC1pa);
102
103 if (_caps & CAP_C2) {
104 _aA2m1 = Geodesic::A2m1f(eps);
105 Geodesic::C2f(eps, _cC2a);
106 _bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_);
107 }
108
109 if (_caps & CAP_C3) {
110 g.C3f(eps, _cC3a);
111 _aA3c = -_f * _salp0 * g.A3f(eps);
112 _bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1);
113 }
114
115 if (_caps & CAP_C4) {
116 g.C4f(eps, _cC4a);
117 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
118 _aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
119 _bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_);
120 }
121
122 }
123
124 GeodesicLine::GeodesicLine(const Geodesic& g,
125 real lat1, real lon1, real azi1,
126 unsigned caps) {
127 azi1 = Math::AngNormalize(azi1);
128 real salp1, calp1;
129 // Guard against underflow in salp0. Also -0 is converted to +0.
130 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
131 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
132 }
133
134 GeodesicLine::GeodesicLine(const Geodesic& g,
135 real lat1, real lon1,
136 real azi1, real salp1, real calp1,
137 unsigned caps, bool arcmode, real s13_a13) {
138 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
139 GenSetDistance(arcmode, s13_a13);
140 }
141
142 Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12,
143 unsigned outmask,
144 real& lat2, real& lon2, real& azi2,
145 real& s12, real& m12,
146 real& M12, real& M21,
147 real& S12) const {
148 if (_exact)
149 return _lineexact.GenPosition(arcmode, s12_a12, outmask,
150 lat2, lon2, azi2,
151 s12, m12, M12, M21, S12);
152 outmask &= _caps & OUT_MASK;
153 if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
154 // Uninitialized or impossible distance calculation requested
155 return Math::NaN();
156
157 // Avoid warning about uninitialized B12.
158 real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
159 if (arcmode) {
160 // Interpret s12_a12 as spherical arc length
161 sig12 = s12_a12 * Math::degree();
162 Math::sincosd(s12_a12, ssig12, csig12);
163 } else {
164 // Interpret s12_a12 as distance
165 real
166 tau12 = s12_a12 / (_b * (1 + _aA1m1)),
167 s = sin(tau12),
168 c = cos(tau12);
169 // tau2 = tau1 + tau12
170 B12 = - Geodesic::SinCosSeries(true,
171 _stau1 * c + _ctau1 * s,
172 _ctau1 * c - _stau1 * s,
173 _cC1pa, nC1p_);
174 sig12 = tau12 - (B12 - _bB11);
175 ssig12 = sin(sig12); csig12 = cos(sig12);
176 if (fabs(_f) > 0.01) {
177 // Reverted distance series is inaccurate for |f| > 1/100, so correct
178 // sig12 with 1 Newton iteration. The following table shows the
179 // approximate maximum error for a = WGS_a() and various f relative to
180 // GeodesicExact.
181 // erri = the error in the inverse solution (nm)
182 // errd = the error in the direct solution (series only) (nm)
183 // errda = the error in the direct solution
184 // (series + 1 Newton) (nm)
185 //
186 // f erri errd errda
187 // -1/5 12e6 1.2e9 69e6
188 // -1/10 123e3 12e6 765e3
189 // -1/20 1110 108e3 7155
190 // -1/50 18.63 200.9 27.12
191 // -1/100 18.63 23.78 23.37
192 // -1/150 18.63 21.05 20.26
193 // 1/150 22.35 24.73 25.83
194 // 1/100 22.35 25.03 25.31
195 // 1/50 29.80 231.9 30.44
196 // 1/20 5376 146e3 10e3
197 // 1/10 829e3 22e6 1.5e6
198 // 1/5 157e6 3.8e9 280e6
199 real
200 ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
201 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
202 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
203 real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b;
204 sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
205 ssig12 = sin(sig12); csig12 = cos(sig12);
206 // Update B12 below
207 }
208 }
209
210 real ssig2, csig2, sbet2, cbet2, salp2, calp2;
211 // sig2 = sig1 + sig12
212 ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
213 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
214 real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
215 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
216 if (arcmode || fabs(_f) > 0.01)
217 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
218 AB1 = (1 + _aA1m1) * (B12 - _bB11);
219 }
220 // sin(bet2) = cos(alp0) * sin(sig2)
221 sbet2 = _calp0 * ssig2;
222 // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
223 cbet2 = hypot(_salp0, _calp0 * csig2);
224 if (cbet2 == 0)
225 // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
226 cbet2 = csig2 = tiny_;
227 // tan(alp0) = cos(sig2)*tan(alp2)
228 salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
229
230 if (outmask & DISTANCE)
231 s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12;
232
233 if (outmask & LONGITUDE) {
234 // tan(omg2) = sin(alp0) * tan(sig2)
235 real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
236 E = copysign(real(1), _salp0); // east-going?
237 // omg12 = omg2 - omg1
238 real omg12 = outmask & LONG_UNROLL
239 ? E * (sig12
240 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
241 + (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1)))
242 : atan2(somg2 * _comg1 - comg2 * _somg1,
243 comg2 * _comg1 + somg2 * _somg1);
244 real lam12 = omg12 + _aA3c *
245 ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1)
246 - _bB31));
247 real lon12 = lam12 / Math::degree();
248 lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
249 Math::AngNormalize(Math::AngNormalize(_lon1) +
250 Math::AngNormalize(lon12));
251 }
252
253 if (outmask & LATITUDE)
254 lat2 = Math::atan2d(sbet2, _f1 * cbet2);
255
256 if (outmask & AZIMUTH)
257 azi2 = Math::atan2d(salp2, calp2);
258
259 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
260 real
261 B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_),
262 AB2 = (1 + _aA2m1) * (B22 - _bB21),
263 J12 = (_aA1m1 - _aA2m1) * sig12 + (AB1 - AB2);
264 if (outmask & REDUCEDLENGTH)
265 // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
266 // accurate cancellation in the case of coincident points.
267 m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
268 - _csig1 * csig2 * J12);
269 if (outmask & GEODESICSCALE) {
270 real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
271 M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
272 M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
273 }
274 }
275
276 if (outmask & AREA) {
277 real
278 B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, nC4_);
279 real salp12, calp12;
280 if (_calp0 == 0 || _salp0 == 0) {
281 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
282 salp12 = salp2 * _calp1 - calp2 * _salp1;
283 calp12 = calp2 * _calp1 + salp2 * _salp1;
284 // We used to include here some patch up code that purported to deal
285 // with nearly meridional geodesics properly. However, this turned out
286 // to be wrong once _salp1 = -0 was allowed (via
287 // Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
288 // was already correct (following the IEEE rules for handling signed
289 // zeros). So the patch up code was unnecessary (as well as
290 // dangerous).
291 } else {
292 // tan(alp) = tan(alp0) * sec(sig)
293 // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
294 // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
295 // If csig12 > 0, write
296 // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
297 // else
298 // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
299 // No need to normalize
300 salp12 = _calp0 * _salp0 *
301 (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
302 ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
303 calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
304 }
305 S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41);
306 }
307
308 return arcmode ? s12_a12 : sig12 / Math::degree();
309 }
310
311 void GeodesicLine::SetDistance(real s13) {
312 _s13 = s13;
313 real t;
314 // This will set _a13 to NaN if the GeodesicLine doesn't have the
315 // DISTANCE_IN capability.
316 _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
317 }
318
319 void GeodesicLine::SetArc(real a13) {
320 _a13 = a13;
321 // In case the GeodesicLine doesn't have the DISTANCE capability.
322 _s13 = Math::NaN();
323 real t;
324 GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
325 }
326
327 void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) {
328 arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
329 }
330
331} // namespace GeographicLib
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
GeographicLib::GeodesicLine::Geodesic
friend class Geodesic
Definition GeodesicLine.hpp:77
GeographicLib::GeodesicLine::SetArc
void SetArc(real a13)
Definition GeodesicLine.cpp:319
GeographicLib::GeodesicLine::Init
bool Init() const
Definition GeodesicLine.hpp:605
GeographicLib::GeodesicLine::GenSetDistance
void GenSetDistance(bool arcmode, real s13_a13)
Definition GeodesicLine.cpp:327
GeographicLib::GeodesicLine::GenPosition
Math::real GenPosition(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 GeodesicLine.cpp:142
GeographicLib::GeodesicLine::SetDistance
void SetDistance(real s13)
Definition GeodesicLine.cpp:311
GeographicLib::GeodesicLine::LATITUDE
@ LATITUDE
Definition GeodesicLine.hpp:137
GeographicLib::GeodesicLine::AZIMUTH
@ AZIMUTH
Definition GeodesicLine.hpp:149
GeographicLib::GeodesicLine::DISTANCE
@ DISTANCE
Definition GeodesicLine.hpp:154
GeographicLib::GeodesicLine::AREA
@ AREA
Definition GeodesicLine.hpp:181
GeographicLib::GeodesicLine::GEODESICSCALE
@ GEODESICSCALE
Definition GeodesicLine.hpp:176
GeographicLib::GeodesicLine::DISTANCE_IN
@ DISTANCE_IN
Definition GeodesicLine.hpp:166
GeographicLib::GeodesicLine::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition GeodesicLine.hpp:171
GeographicLib::GeodesicLine::LONGITUDE
@ LONGITUDE
Definition GeodesicLine.hpp:142
GeographicLib::GeodesicLine::LONG_UNROLL
@ LONG_UNROLL
Definition GeodesicLine.hpp:186
GeographicLib::GeodesicLine::GeodesicLine
GeodesicLine()
Definition GeodesicLine.hpp:251
GeographicLib::Geodesic
Geodesic calculations
Definition Geodesic.hpp:175
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::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:301
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12