GeographicLib 2.1.2
GeodesicLine.cpp
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1/**
2 * \file GeodesicLine.cpp
3 * \brief Implementation for GeographicLib::GeodesicLine class
4 *
5 * Copyright (c) Charles Karney (2009-2022) <charles@karney.com> 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
30
31#if defined(_MSC_VER)
32// Squelch warnings about mixing enums
33# pragma warning (disable: 5054)
34#endif
35
36namespace GeographicLib {
37
38 using namespace std;
39
40 void GeodesicLine::LineInit(const Geodesic& g,
41 real lat1, real lon1,
42 real azi1, real salp1, real calp1,
43 unsigned caps) {
44 tiny_ = g.tiny_;
45 _lat1 = Math::LatFix(lat1);
46 _lon1 = lon1;
47 _azi1 = azi1;
48 _salp1 = salp1;
49 _calp1 = calp1;
50 _a = g._a;
51 _f = g._f;
52 _b = g._b;
53 _c2 = g._c2;
54 _f1 = g._f1;
55 // Always allow latitude and azimuth and unrolling of longitude
56 _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
57
58 real cbet1, sbet1;
59 Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
60 // Ensure cbet1 = +epsilon at poles
61 Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
62 _dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1));
63
64 // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
65 _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
66 // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
67 // is slightly better (consider the case salp1 = 0).
68 _calp0 = hypot(_calp1, _salp1 * sbet1);
69 // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
70 // sig = 0 is nearest northward crossing of equator.
71 // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
72 // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
73 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
74 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
75 // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
76 // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
77 // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
78 _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
79 _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
80 Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
81 // Math::norm(_somg1, _comg1); -- don't need to normalize!
82
83 _k2 = Math::sq(_calp0) * g._ep2;
84 real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
85
86 if (_caps & CAP_C1) {
87 _aA1m1 = Geodesic::A1m1f(eps);
88 Geodesic::C1f(eps, _cC1a);
89 _bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_);
90 real s = sin(_bB11), c = cos(_bB11);
91 // tau1 = sig1 + B11
92 _stau1 = _ssig1 * c + _csig1 * s;
93 _ctau1 = _csig1 * c - _ssig1 * s;
94 // Not necessary because C1pa reverts C1a
95 // _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_);
96 }
97
98 if (_caps & CAP_C1p)
99 Geodesic::C1pf(eps, _cC1pa);
100
101 if (_caps & CAP_C2) {
102 _aA2m1 = Geodesic::A2m1f(eps);
103 Geodesic::C2f(eps, _cC2a);
104 _bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_);
105 }
106
107 if (_caps & CAP_C3) {
108 g.C3f(eps, _cC3a);
109 _aA3c = -_f * _salp0 * g.A3f(eps);
110 _bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1);
111 }
112
113 if (_caps & CAP_C4) {
114 g.C4f(eps, _cC4a);
115 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
116 _aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
117 _bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_);
118 }
119
120 _a13 = _s13 = Math::NaN();
121 }
122
124 real lat1, real lon1, real azi1,
125 unsigned caps) {
126 azi1 = Math::AngNormalize(azi1);
127 real salp1, calp1;
128 // Guard against underflow in salp0. Also -0 is converted to +0.
129 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
130 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
131 }
132
134 real lat1, real lon1,
135 real azi1, real salp1, real calp1,
136 unsigned caps, bool arcmode, real s13_a13) {
137 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
138 GenSetDistance(arcmode, s13_a13);
139 }
140
141 Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12,
142 unsigned outmask,
143 real& lat2, real& lon2, real& azi2,
144 real& s12, real& m12,
145 real& M12, real& M21,
146 real& S12) const {
147 outmask &= _caps & OUT_MASK;
148 if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
149 // Uninitialized or impossible distance calculation requested
150 return Math::NaN();
151
152 // Avoid warning about uninitialized B12.
153 real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
154 if (arcmode) {
155 // Interpret s12_a12 as spherical arc length
156 sig12 = s12_a12 * Math::degree();
157 Math::sincosd(s12_a12, ssig12, csig12);
158 } else {
159 // Interpret s12_a12 as distance
160 real
161 tau12 = s12_a12 / (_b * (1 + _aA1m1)),
162 s = sin(tau12),
163 c = cos(tau12);
164 // tau2 = tau1 + tau12
165 B12 = - Geodesic::SinCosSeries(true,
166 _stau1 * c + _ctau1 * s,
167 _ctau1 * c - _stau1 * s,
168 _cC1pa, nC1p_);
169 sig12 = tau12 - (B12 - _bB11);
170 ssig12 = sin(sig12); csig12 = cos(sig12);
171 if (fabs(_f) > 0.01) {
172 // Reverted distance series is inaccurate for |f| > 1/100, so correct
173 // sig12 with 1 Newton iteration. The following table shows the
174 // approximate maximum error for a = WGS_a() and various f relative to
175 // GeodesicExact.
176 // erri = the error in the inverse solution (nm)
177 // errd = the error in the direct solution (series only) (nm)
178 // errda = the error in the direct solution
179 // (series + 1 Newton) (nm)
180 //
181 // f erri errd errda
182 // -1/5 12e6 1.2e9 69e6
183 // -1/10 123e3 12e6 765e3
184 // -1/20 1110 108e3 7155
185 // -1/50 18.63 200.9 27.12
186 // -1/100 18.63 23.78 23.37
187 // -1/150 18.63 21.05 20.26
188 // 1/150 22.35 24.73 25.83
189 // 1/100 22.35 25.03 25.31
190 // 1/50 29.80 231.9 30.44
191 // 1/20 5376 146e3 10e3
192 // 1/10 829e3 22e6 1.5e6
193 // 1/5 157e6 3.8e9 280e6
194 real
195 ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
196 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
197 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
198 real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b;
199 sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
200 ssig12 = sin(sig12); csig12 = cos(sig12);
201 // Update B12 below
202 }
203 }
204
205 real ssig2, csig2, sbet2, cbet2, salp2, calp2;
206 // sig2 = sig1 + sig12
207 ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
208 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
209 real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
210 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
211 if (arcmode || fabs(_f) > 0.01)
212 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
213 AB1 = (1 + _aA1m1) * (B12 - _bB11);
214 }
215 // sin(bet2) = cos(alp0) * sin(sig2)
216 sbet2 = _calp0 * ssig2;
217 // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
218 cbet2 = hypot(_salp0, _calp0 * csig2);
219 if (cbet2 == 0)
220 // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
221 cbet2 = csig2 = tiny_;
222 // tan(alp0) = cos(sig2)*tan(alp2)
223 salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
224
225 if (outmask & DISTANCE)
226 s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12;
227
228 if (outmask & LONGITUDE) {
229 // tan(omg2) = sin(alp0) * tan(sig2)
230 real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
231 E = copysign(real(1), _salp0); // east-going?
232 // omg12 = omg2 - omg1
233 real omg12 = outmask & LONG_UNROLL
234 ? E * (sig12
235 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
236 + (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1)))
237 : atan2(somg2 * _comg1 - comg2 * _somg1,
238 comg2 * _comg1 + somg2 * _somg1);
239 real lam12 = omg12 + _aA3c *
240 ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1)
241 - _bB31));
242 real lon12 = lam12 / Math::degree();
243 lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
245 Math::AngNormalize(lon12));
246 }
247
248 if (outmask & LATITUDE)
249 lat2 = Math::atan2d(sbet2, _f1 * cbet2);
250
251 if (outmask & AZIMUTH)
252 azi2 = Math::atan2d(salp2, calp2);
253
254 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
255 real
256 B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_),
257 AB2 = (1 + _aA2m1) * (B22 - _bB21),
258 J12 = (_aA1m1 - _aA2m1) * sig12 + (AB1 - AB2);
259 if (outmask & REDUCEDLENGTH)
260 // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
261 // accurate cancellation in the case of coincident points.
262 m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
263 - _csig1 * csig2 * J12);
264 if (outmask & GEODESICSCALE) {
265 real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
266 M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
267 M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
268 }
269 }
270
271 if (outmask & AREA) {
272 real
273 B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, nC4_);
274 real salp12, calp12;
275 if (_calp0 == 0 || _salp0 == 0) {
276 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
277 salp12 = salp2 * _calp1 - calp2 * _salp1;
278 calp12 = calp2 * _calp1 + salp2 * _salp1;
279 // We used to include here some patch up code that purported to deal
280 // with nearly meridional geodesics properly. However, this turned out
281 // to be wrong once _salp1 = -0 was allowed (via
282 // Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
283 // was already correct (following the IEEE rules for handling signed
284 // zeros). So the patch up code was unnecessary (as well as
285 // dangerous).
286 } else {
287 // tan(alp) = tan(alp0) * sec(sig)
288 // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
289 // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
290 // If csig12 > 0, write
291 // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
292 // else
293 // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
294 // No need to normalize
295 salp12 = _calp0 * _salp0 *
296 (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
297 ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
298 calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
299 }
300 S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41);
301 }
302
303 return arcmode ? s12_a12 : sig12 / Math::degree();
304 }
305
307 _s13 = s13;
308 real t;
309 // This will set _a13 to NaN if the GeodesicLine doesn't have the
310 // DISTANCE_IN capability.
311 _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
312 }
313
314 void GeodesicLine::SetArc(real a13) {
315 _a13 = a13;
316 // In case the GeodesicLine doesn't have the DISTANCE capability.
317 _s13 = Math::NaN();
318 real t;
319 GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
320 }
321
322 void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) {
323 arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
324 }
325
326} // namespace GeographicLib
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::GeodesicLine class.
void GenSetDistance(bool arcmode, real s13_a13)
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
Geodesic calculations
Definition: Geodesic.hpp:172
static T degree()
Definition: Math.hpp:200
static T LatFix(T x)
Definition: Math.hpp:300
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:106
static T atan2d(T y, T x)
Definition: Math.cpp:183
static void norm(T &x, T &y)
Definition: Math.hpp:222
static T AngRound(T x)
Definition: Math.cpp:97
static T sq(T x)
Definition: Math.hpp:212
static T AngNormalize(T x)
Definition: Math.cpp:71
static T NaN()
Definition: Math.cpp:250
Namespace for GeographicLib.
Definition: Accumulator.cpp:12