GeographicLib  2.1.1
GeodesicLineExact.cpp
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1 /**
2  * \file GeodesicLineExact.cpp
3  * \brief Implementation for GeographicLib::GeodesicLineExact class
4  *
5  * Copyright (c) Charles Karney (2012-2022) <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 
36 namespace GeographicLib {
37 
38  using namespace std;
39 
40  void GeodesicLineExact::LineInit(const GeodesicExact& 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  _e2 = g._e2;
56  _nC4 = g._nC4;
57  // Always allow latitude and azimuth and unrolling of longitude
58  _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
59 
60  real cbet1, sbet1;
61  Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
62  // Ensure cbet1 = +epsilon at poles
63  Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
64  _dn1 = (_f >= 0 ? sqrt(1 + g._ep2 * Math::sq(sbet1)) :
65  sqrt(1 - _e2 * Math::sq(cbet1)) / _f1);
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 = 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  // Without normalization we have schi1 = somg1.
84  _cchi1 = _f1 * _dn1 * _comg1;
85  Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
86  // Math::norm(_somg1, _comg1); -- don't need to normalize!
87  // Math::norm(_schi1, _cchi1); -- don't need to normalize!
88 
89  _k2 = Math::sq(_calp0) * g._ep2;
90  _eE.Reset(-_k2, -g._ep2, 1 + _k2, 1 + g._ep2);
91 
92  if (_caps & CAP_E) {
93  _eE0 = _eE.E() / (Math::pi() / 2);
94  _eE1 = _eE.deltaE(_ssig1, _csig1, _dn1);
95  real s = sin(_eE1), c = cos(_eE1);
96  // tau1 = sig1 + B11
97  _stau1 = _ssig1 * c + _csig1 * s;
98  _ctau1 = _csig1 * c - _ssig1 * s;
99  // Not necessary because Einv inverts E
100  // _eE1 = -_eE.deltaEinv(_stau1, _ctau1);
101  }
102 
103  if (_caps & CAP_D) {
104  _dD0 = _eE.D() / (Math::pi() / 2);
105  _dD1 = _eE.deltaD(_ssig1, _csig1, _dn1);
106  }
107 
108  if (_caps & CAP_H) {
109  _hH0 = _eE.H() / (Math::pi() / 2);
110  _hH1 = _eE.deltaH(_ssig1, _csig1, _dn1);
111  }
112 
113  if (_caps & CAP_C4) {
114  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
115  _aA4 = Math::sq(_a) * _calp0 * _salp0 * _e2;
116  if (_aA4 == 0)
117  _bB41 = 0;
118  else {
119  GeodesicExact::I4Integrand i4(g._ep2, _k2);
120  _cC4a.resize(_nC4);
121  g._fft.transform(i4, _cC4a.data());
122  _bB41 = DST::integral(_ssig1, _csig1, _cC4a.data(), _nC4);
123  }
124  }
125 
126  _a13 = _s13 = Math::NaN();
127  }
128 
130  real lat1, real lon1, real azi1,
131  unsigned caps) {
132  azi1 = Math::AngNormalize(azi1);
133  real salp1, calp1;
134  // Guard against underflow in salp0. Also -0 is converted to +0.
135  Math::sincosd(Math::AngRound(azi1), salp1, calp1);
136  LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
137  }
138 
140  real lat1, real lon1,
141  real azi1, real salp1, real calp1,
142  unsigned caps,
143  bool arcmode, real s13_a13) {
144  LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
145  GenSetDistance(arcmode, s13_a13);
146  }
147 
148  Math::real GeodesicLineExact::GenPosition(bool arcmode, real s12_a12,
149  unsigned outmask,
150  real& lat2, real& lon2, real& azi2,
151  real& s12, real& m12,
152  real& M12, real& M21,
153  real& S12) const {
154  outmask &= _caps & OUT_MASK;
155  if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
156  // Uninitialized or impossible distance calculation requested
157  return Math::NaN();
158 
159  // Avoid warning about uninitialized B12.
160  real sig12, ssig12, csig12, E2 = 0, AB1 = 0;
161  if (arcmode) {
162  // Interpret s12_a12 as spherical arc length
163  sig12 = s12_a12 * Math::degree();
164  Math::sincosd(s12_a12, ssig12, csig12);
165  } else {
166  // Interpret s12_a12 as distance
167  real
168  tau12 = s12_a12 / (_b * _eE0),
169  s = sin(tau12),
170  c = cos(tau12);
171  // tau2 = tau1 + tau12
172  E2 = - _eE.deltaEinv(_stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s);
173  sig12 = tau12 - (E2 - _eE1);
174  ssig12 = sin(sig12);
175  csig12 = cos(sig12);
176  }
177 
178  real ssig2, csig2, sbet2, cbet2, salp2, calp2;
179  // sig2 = sig1 + sig12
180  ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
181  csig2 = _csig1 * csig12 - _ssig1 * ssig12;
182  real dn2 = _eE.Delta(ssig2, csig2);
183  if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
184  if (arcmode) {
185  E2 = _eE.deltaE(ssig2, csig2, dn2);
186  }
187  AB1 = _eE0 * (E2 - _eE1);
188  }
189  // sin(bet2) = cos(alp0) * sin(sig2)
190  sbet2 = _calp0 * ssig2;
191  // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
192  cbet2 = hypot(_salp0, _calp0 * csig2);
193  if (cbet2 == 0)
194  // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
195  cbet2 = csig2 = tiny_;
196  // tan(alp0) = cos(sig2)*tan(alp2)
197  salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
198 
199  if (outmask & DISTANCE)
200  s12 = arcmode ? _b * (_eE0 * sig12 + AB1) : s12_a12;
201 
202  if (outmask & LONGITUDE) {
203  real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
204  E = copysign(real(1), _salp0); // east-going?
205  // Without normalization we have schi2 = somg2.
206  real cchi2 = _f1 * dn2 * comg2;
207  real chi12 = outmask & LONG_UNROLL
208  ? E * (sig12
209  - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
210  + (atan2(E * somg2, cchi2) - atan2(E * _somg1, _cchi1)))
211  : atan2(somg2 * _cchi1 - cchi2 * _somg1,
212  cchi2 * _cchi1 + somg2 * _somg1);
213  real lam12 = chi12 -
214  _e2/_f1 * _salp0 * _hH0 *
215  (sig12 + (_eE.deltaH(ssig2, csig2, dn2) - _hH1));
216  real lon12 = lam12 / Math::degree();
217  lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
219  Math::AngNormalize(lon12));
220  }
221 
222  if (outmask & LATITUDE)
223  lat2 = Math::atan2d(sbet2, _f1 * cbet2);
224 
225  if (outmask & AZIMUTH)
226  azi2 = Math::atan2d(salp2, calp2);
227 
228  if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
229  real J12 = _k2 * _dD0 * (sig12 + (_eE.deltaD(ssig2, csig2, dn2) - _dD1));
230  if (outmask & REDUCEDLENGTH)
231  // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
232  // accurate cancellation in the case of coincident points.
233  m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
234  - _csig1 * csig2 * J12);
235  if (outmask & GEODESICSCALE) {
236  real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
237  M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
238  M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
239  }
240  }
241 
242  if (outmask & AREA) {
243  real B42 = _aA4 == 0 ? 0 :
244  DST::integral(ssig2, csig2, _cC4a.data(), _nC4);
245  real salp12, calp12;
246  if (_calp0 == 0 || _salp0 == 0) {
247  // alp12 = alp2 - alp1, used in atan2 so no need to normalize
248  salp12 = salp2 * _calp1 - calp2 * _salp1;
249  calp12 = calp2 * _calp1 + salp2 * _salp1;
250  // We used to include here some patch up code that purported to deal
251  // with nearly meridional geodesics properly. However, this turned out
252  // to be wrong once _salp1 = -0 was allowed (via
253  // GeodesicExact::InverseLine). In fact, the calculation of {s,c}alp12
254  // was already correct (following the IEEE rules for handling signed
255  // zeros). So the patch up code was unnecessary (as well as
256  // dangerous).
257  } else {
258  // tan(alp) = tan(alp0) * sec(sig)
259  // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
260  // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
261  // If csig12 > 0, write
262  // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
263  // else
264  // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
265  // No need to normalize
266  salp12 = _calp0 * _salp0 *
267  (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
268  ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
269  calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
270  }
271  S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41);
272  }
273 
274  return arcmode ? s12_a12 : sig12 / Math::degree();
275  }
276 
278  _s13 = s13;
279  real t;
280  // This will set _a13 to NaN if the GeodesicLineExact doesn't have the
281  // DISTANCE_IN capability.
282  _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
283  }
284 
285  void GeodesicLineExact::SetArc(real a13) {
286  _a13 = a13;
287  // In case the GeodesicLineExact doesn't have the DISTANCE capability.
288  _s13 = Math::NaN();
289  real t;
290  GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
291  }
292 
293  void GeodesicLineExact::GenSetDistance(bool arcmode, real s13_a13) {
294  arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
295  }
296 
297 } // namespace GeographicLib
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::GeodesicLineExact class.
static real integral(real sinx, real cosx, const real F[], int N)
Definition: DST.cpp:110
Exact geodesic calculations.
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
void GenSetDistance(bool arcmode, real s13_a13)
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 pi()
Definition: Math.hpp:190
static T NaN()
Definition: Math.cpp:250
Namespace for GeographicLib.
Definition: Accumulator.cpp:12