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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 (2009-2012) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * http://geographiclib.sourceforge.net/
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 namespace GeographicLib {
32 
33  using namespace std;
34 
36  real lat1, real lon1, real azi1,
37  unsigned caps)
38  : tiny_(g.tiny_)
39  , _a(g._a)
40  , _f(g._f)
41  , _b(g._b)
42  , _c2(g._c2)
43  , _f1(g._f1)
44  // Always allow latitude and azimuth
45  , _caps(caps | LATITUDE | AZIMUTH)
46  {
47  // Guard against underflow in salp0
48  azi1 = Geodesic::AngRound(Math::AngNormalize(azi1));
49  lon1 = Math::AngNormalize(lon1);
50  _lat1 = lat1;
51  _lon1 = lon1;
52  _azi1 = azi1;
53  // alp1 is in [0, pi]
54  real alp1 = azi1 * Math::degree();
55  // Enforce sin(pi) == 0 and cos(pi/2) == 0. Better to face the ensuing
56  // problems directly than to skirt them.
57  _salp1 = azi1 == -180 ? 0 : sin(alp1);
58  _calp1 = abs(azi1) == 90 ? 0 : cos(alp1);
59  real cbet1, sbet1, phi;
60  phi = lat1 * Math::degree();
61  // Ensure cbet1 = +epsilon at poles
62  sbet1 = _f1 * sin(phi);
63  cbet1 = abs(lat1) == 90 ? tiny_ : cos(phi);
64  Geodesic::SinCosNorm(sbet1, cbet1);
65  _dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1));
66 
67  // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
68  _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
69  // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
70  // is slightly better (consider the case salp1 = 0).
71  _calp0 = Math::hypot(_calp1, _salp1 * sbet1);
72  // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
73  // sig = 0 is nearest northward crossing of equator.
74  // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
75  // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
76  // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
77  // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
78  // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
79  // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
80  // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
81  _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
82  _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
83  Geodesic::SinCosNorm(_ssig1, _csig1); // sig1 in (-pi, pi]
84  // Geodesic::SinCosNorm(_somg1, _comg1); -- don't need to normalize!
85 
86  _k2 = Math::sq(_calp0) * g._ep2;
87  real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
88 
89  if (_caps & CAP_C1) {
90  _A1m1 = Geodesic::A1m1f(eps);
91  Geodesic::C1f(eps, _C1a);
92  _B11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C1a, nC1_);
93  real s = sin(_B11), c = cos(_B11);
94  // tau1 = sig1 + B11
95  _stau1 = _ssig1 * c + _csig1 * s;
96  _ctau1 = _csig1 * c - _ssig1 * s;
97  // Not necessary because C1pa reverts C1a
98  // _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_);
99  }
100 
101  if (_caps & CAP_C1p)
102  Geodesic::C1pf(eps, _C1pa);
103 
104  if (_caps & CAP_C2) {
105  _A2m1 = Geodesic::A2m1f(eps);
106  Geodesic::C2f(eps, _C2a);
107  _B21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C2a, nC2_);
108  }
109 
110  if (_caps & CAP_C3) {
111  g.C3f(eps, _C3a);
112  _A3c = -_f * _salp0 * g.A3f(eps);
113  _B31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C3a, nC3_-1);
114  }
115 
116  if (_caps & CAP_C4) {
117  g.C4f(eps, _C4a);
118  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
119  _A4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
120  _B41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _C4a, nC4_);
121  }
122  }
123 
124  Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12,
125  unsigned outmask,
126  real& lat2, real& lon2, real& azi2,
127  real& s12, real& m12,
128  real& M12, real& M21,
129  real& S12)
130  const {
131  outmask &= _caps & OUT_ALL;
132  if (!( Init() && (arcmode || (_caps & DISTANCE_IN & OUT_ALL)) ))
133  // Uninitialized or impossible distance calculation requested
134  return Math::NaN();
135 
136  // Avoid warning about uninitialized B12.
137  real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
138  if (arcmode) {
139  // Interpret s12_a12 as spherical arc length
140  sig12 = s12_a12 * Math::degree();
141  real s12a = abs(s12_a12);
142  s12a -= 180 * floor(s12a / 180);
143  ssig12 = s12a == 0 ? 0 : sin(sig12);
144  csig12 = s12a == 90 ? 0 : cos(sig12);
145  } else {
146  // Interpret s12_a12 as distance
147  real
148  tau12 = s12_a12 / (_b * (1 + _A1m1)),
149  s = sin(tau12),
150  c = cos(tau12);
151  // tau2 = tau1 + tau12
152  B12 = - Geodesic::SinCosSeries(true,
153  _stau1 * c + _ctau1 * s,
154  _ctau1 * c - _stau1 * s,
155  _C1pa, nC1p_);
156  sig12 = tau12 - (B12 - _B11);
157  ssig12 = sin(sig12); csig12 = cos(sig12);
158  if (abs(_f) > 0.01) {
159  // Reverted distance series is inaccurate for |f| > 1/100, so correct
160  // sig12 with 1 Newton iteration. The following table shows the
161  // approximate maximum error for a = WGS_a() and various f relative to
162  // GeodesicExact.
163  // erri = the error in the inverse solution (nm)
164  // errd = the error in the direct solution (series only) (nm)
165  // errda = the error in the direct solution (series + 1 Newton) (nm)
166  //
167  // f erri errd errda
168  // -1/5 12e6 1.2e9 69e6
169  // -1/10 123e3 12e6 765e3
170  // -1/20 1110 108e3 7155
171  // -1/50 18.63 200.9 27.12
172  // -1/100 18.63 23.78 23.37
173  // -1/150 18.63 21.05 20.26
174  // 1/150 22.35 24.73 25.83
175  // 1/100 22.35 25.03 25.31
176  // 1/50 29.80 231.9 30.44
177  // 1/20 5376 146e3 10e3
178  // 1/10 829e3 22e6 1.5e6
179  // 1/5 157e6 3.8e9 280e6
180  real
181  ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
182  csig2 = _csig1 * csig12 - _ssig1 * ssig12;
183  B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1_);
184  real serr = (1 + _A1m1) * (sig12 + (B12 - _B11)) - s12_a12 / _b;
185  sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
186  ssig12 = sin(sig12); csig12 = cos(sig12);
187  // Update B12 below
188  }
189  }
190 
191  real omg12, lam12, lon12;
192  real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2;
193  // sig2 = sig1 + sig12
194  ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
195  csig2 = _csig1 * csig12 - _ssig1 * ssig12;
196  real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
197  if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
198  if (arcmode || abs(_f) > 0.01)
199  B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1_);
200  AB1 = (1 + _A1m1) * (B12 - _B11);
201  }
202  // sin(bet2) = cos(alp0) * sin(sig2)
203  sbet2 = _calp0 * ssig2;
204  // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
205  cbet2 = Math::hypot(_salp0, _calp0 * csig2);
206  if (cbet2 == 0)
207  // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
208  cbet2 = csig2 = tiny_;
209  // tan(omg2) = sin(alp0) * tan(sig2)
210  somg2 = _salp0 * ssig2; comg2 = csig2; // No need to normalize
211  // tan(alp0) = cos(sig2)*tan(alp2)
212  salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
213  // omg12 = omg2 - omg1
214  omg12 = atan2(somg2 * _comg1 - comg2 * _somg1,
215  comg2 * _comg1 + somg2 * _somg1);
216 
217  if (outmask & DISTANCE)
218  s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12;
219 
220  if (outmask & LONGITUDE) {
221  lam12 = omg12 + _A3c *
222  ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _C3a, nC3_-1)
223  - _B31));
224  lon12 = lam12 / Math::degree();
225  // Use Math::AngNormalize2 because longitude might have wrapped multiple
226  // times.
227  lon12 = Math::AngNormalize2(lon12);
228  lon2 = Math::AngNormalize(_lon1 + lon12);
229  }
230 
231  if (outmask & LATITUDE)
232  lat2 = atan2(sbet2, _f1 * cbet2) / Math::degree();
233 
234  if (outmask & AZIMUTH)
235  // minus signs give range [-180, 180). 0- converts -0 to +0.
236  azi2 = 0 - atan2(-salp2, calp2) / Math::degree();
237 
238  if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
239  real
240  B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _C2a, nC2_),
241  AB2 = (1 + _A2m1) * (B22 - _B21),
242  J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2);
243  if (outmask & REDUCEDLENGTH)
244  // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
245  // accurate cancellation in the case of coincident points.
246  m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
247  - _csig1 * csig2 * J12);
248  if (outmask & GEODESICSCALE) {
249  real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
250  M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
251  M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
252  }
253  }
254 
255  if (outmask & AREA) {
256  real
257  B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _C4a, nC4_);
258  real salp12, calp12;
259  if (_calp0 == 0 || _salp0 == 0) {
260  // alp12 = alp2 - alp1, used in atan2 so no need to normalized
261  salp12 = salp2 * _calp1 - calp2 * _salp1;
262  calp12 = calp2 * _calp1 + salp2 * _salp1;
263  // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
264  // salp12 = -0 and alp12 = -180. However this depends on the sign being
265  // attached to 0 correctly. The following ensures the correct behavior.
266  if (salp12 == 0 && calp12 < 0) {
267  salp12 = tiny_ * _calp1;
268  calp12 = -1;
269  }
270  } else {
271  // tan(alp) = tan(alp0) * sec(sig)
272  // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
273  // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
274  // If csig12 > 0, write
275  // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
276  // else
277  // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
278  // No need to normalize
279  salp12 = _calp0 * _salp0 *
280  (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
281  ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
282  calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
283  }
284  S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
285  }
286 
287  return arcmode ? s12_a12 : sig12 / Math::degree();
288  }
289 
290 } // namespace GeographicLib