GeographicLib 1.52
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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 (2009-2020) <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
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 = max(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 _k2 = Math::sq(_calp0) * g._ep2;
79 real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
80
81 if (_caps & CAP_C1) {
82 _A1m1 = Geodesic::A1m1f(eps);
83 Geodesic::C1f(eps, _C1a);
84 _B11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C1a, nC1_);
85 real s = sin(_B11), c = cos(_B11);
86 // tau1 = sig1 + B11
87 _stau1 = _ssig1 * c + _csig1 * s;
88 _ctau1 = _csig1 * c - _ssig1 * s;
89 // Not necessary because C1pa reverts C1a
90 // _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_);
91 }
92
93 if (_caps & CAP_C1p)
94 Geodesic::C1pf(eps, _C1pa);
95
96 if (_caps & CAP_C2) {
97 _A2m1 = Geodesic::A2m1f(eps);
98 Geodesic::C2f(eps, _C2a);
99 _B21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C2a, nC2_);
100 }
101
102 if (_caps & CAP_C3) {
103 g.C3f(eps, _C3a);
104 _A3c = -_f * _salp0 * g.A3f(eps);
105 _B31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C3a, nC3_-1);
106 }
107
108 if (_caps & CAP_C4) {
109 g.C4f(eps, _C4a);
110 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
111 _A4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
112 _B41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _C4a, nC4_);
113 }
114
115 _a13 = _s13 = Math::NaN();
116 }
117
119 real lat1, real lon1, real azi1,
120 unsigned caps) {
121 azi1 = Math::AngNormalize(azi1);
122 real salp1, calp1;
123 // Guard against underflow in salp0. Also -0 is converted to +0.
124 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
125 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
126 }
127
129 real lat1, real lon1,
130 real azi1, real salp1, real calp1,
131 unsigned caps, bool arcmode, real s13_a13) {
132 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
133 GenSetDistance(arcmode, s13_a13);
134 }
135
136 Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12,
137 unsigned outmask,
138 real& lat2, real& lon2, real& azi2,
139 real& s12, real& m12,
140 real& M12, real& M21,
141 real& S12) const {
142 outmask &= _caps & OUT_MASK;
143 if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
144 // Uninitialized or impossible distance calculation requested
145 return Math::NaN();
146
147 // Avoid warning about uninitialized B12.
148 real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
149 if (arcmode) {
150 // Interpret s12_a12 as spherical arc length
151 sig12 = s12_a12 * Math::degree();
152 Math::sincosd(s12_a12, ssig12, csig12);
153 } else {
154 // Interpret s12_a12 as distance
155 real
156 tau12 = s12_a12 / (_b * (1 + _A1m1)),
157 s = sin(tau12),
158 c = cos(tau12);
159 // tau2 = tau1 + tau12
160 B12 = - Geodesic::SinCosSeries(true,
161 _stau1 * c + _ctau1 * s,
162 _ctau1 * c - _stau1 * s,
163 _C1pa, nC1p_);
164 sig12 = tau12 - (B12 - _B11);
165 ssig12 = sin(sig12); csig12 = cos(sig12);
166 if (abs(_f) > 0.01) {
167 // Reverted distance series is inaccurate for |f| > 1/100, so correct
168 // sig12 with 1 Newton iteration. The following table shows the
169 // approximate maximum error for a = WGS_a() and various f relative to
170 // GeodesicExact.
171 // erri = the error in the inverse solution (nm)
172 // errd = the error in the direct solution (series only) (nm)
173 // errda = the error in the direct solution
174 // (series + 1 Newton) (nm)
175 //
176 // f erri errd errda
177 // -1/5 12e6 1.2e9 69e6
178 // -1/10 123e3 12e6 765e3
179 // -1/20 1110 108e3 7155
180 // -1/50 18.63 200.9 27.12
181 // -1/100 18.63 23.78 23.37
182 // -1/150 18.63 21.05 20.26
183 // 1/150 22.35 24.73 25.83
184 // 1/100 22.35 25.03 25.31
185 // 1/50 29.80 231.9 30.44
186 // 1/20 5376 146e3 10e3
187 // 1/10 829e3 22e6 1.5e6
188 // 1/5 157e6 3.8e9 280e6
189 real
190 ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
191 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
192 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1_);
193 real serr = (1 + _A1m1) * (sig12 + (B12 - _B11)) - s12_a12 / _b;
194 sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
195 ssig12 = sin(sig12); csig12 = cos(sig12);
196 // Update B12 below
197 }
198 }
199
200 real ssig2, csig2, sbet2, cbet2, salp2, calp2;
201 // sig2 = sig1 + sig12
202 ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
203 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
204 real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
205 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
206 if (arcmode || abs(_f) > 0.01)
207 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1_);
208 AB1 = (1 + _A1m1) * (B12 - _B11);
209 }
210 // sin(bet2) = cos(alp0) * sin(sig2)
211 sbet2 = _calp0 * ssig2;
212 // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
213 cbet2 = hypot(_salp0, _calp0 * csig2);
214 if (cbet2 == 0)
215 // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
216 cbet2 = csig2 = tiny_;
217 // tan(alp0) = cos(sig2)*tan(alp2)
218 salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
219
220 if (outmask & DISTANCE)
221 s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12;
222
223 if (outmask & LONGITUDE) {
224 // tan(omg2) = sin(alp0) * tan(sig2)
225 real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
226 E = copysign(real(1), _salp0); // east-going?
227 // omg12 = omg2 - omg1
228 real omg12 = outmask & LONG_UNROLL
229 ? E * (sig12
230 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
231 + (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1)))
232 : atan2(somg2 * _comg1 - comg2 * _somg1,
233 comg2 * _comg1 + somg2 * _somg1);
234 real lam12 = omg12 + _A3c *
235 ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _C3a, nC3_-1)
236 - _B31));
237 real lon12 = lam12 / Math::degree();
238 lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
240 Math::AngNormalize(lon12));
241 }
242
243 if (outmask & LATITUDE)
244 lat2 = Math::atan2d(sbet2, _f1 * cbet2);
245
246 if (outmask & AZIMUTH)
247 azi2 = Math::atan2d(salp2, calp2);
248
249 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
250 real
251 B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _C2a, nC2_),
252 AB2 = (1 + _A2m1) * (B22 - _B21),
253 J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2);
254 if (outmask & REDUCEDLENGTH)
255 // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
256 // accurate cancellation in the case of coincident points.
257 m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
258 - _csig1 * csig2 * J12);
259 if (outmask & GEODESICSCALE) {
260 real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
261 M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
262 M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
263 }
264 }
265
266 if (outmask & AREA) {
267 real
268 B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _C4a, nC4_);
269 real salp12, calp12;
270 if (_calp0 == 0 || _salp0 == 0) {
271 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
272 salp12 = salp2 * _calp1 - calp2 * _salp1;
273 calp12 = calp2 * _calp1 + salp2 * _salp1;
274 // We used to include here some patch up code that purported to deal
275 // with nearly meridional geodesics properly. However, this turned out
276 // to be wrong once _salp1 = -0 was allowed (via
277 // Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
278 // was already correct (following the IEEE rules for handling signed
279 // zeros). So the patch up code was unnecessary (as well as
280 // dangerous).
281 } else {
282 // tan(alp) = tan(alp0) * sec(sig)
283 // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
284 // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
285 // If csig12 > 0, write
286 // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
287 // else
288 // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
289 // No need to normalize
290 salp12 = _calp0 * _salp0 *
291 (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
292 ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
293 calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
294 }
295 S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
296 }
297
298 return arcmode ? s12_a12 : sig12 / Math::degree();
299 }
300
302 _s13 = s13;
303 real t;
304 // This will set _a13 to NaN if the GeodesicLine doesn't have the
305 // DISTANCE_IN capability.
306 _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
307 }
308
309 void GeodesicLine::SetArc(real a13) {
310 _a13 = a13;
311 // In case the GeodesicLine doesn't have the DISTANCE capability.
312 _s13 = Math::NaN();
313 real t;
314 GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
315 }
316
317 void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) {
318 arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
319 }
320
321} // 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 AngNormalize(T x)
Definition: Math.hpp:420
static T degree()
Definition: Math.hpp:159
static T LatFix(T x)
Definition: Math.hpp:433
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T atan2d(T y, T x)
Definition: Math.cpp:180
static void norm(T &x, T &y)
Definition: Math.hpp:355
static T AngRound(T x)
Definition: Math.cpp:117
static T sq(T x)
Definition: Math.hpp:171
static T NaN()
Definition: Math.cpp:260
Namespace for GeographicLib.
Definition: Accumulator.cpp:12