GeographicLib  1.44
GeodesicLineExact.cpp
Go to the documentation of this file.
1 /**
2  * \file GeodesicLineExact.cpp
3  * \brief Implementation for GeographicLib::GeodesicLineExact class
4  *
5  * Copyright (c) Charles Karney (2012-2015) <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  , _lat1(Math::LatFix(lat1))
40  , _lon1(lon1)
41  , _azi1(Math::AngNormalize(azi1))
42  , _a(g._a)
43  , _f(g._f)
44  , _b(g._b)
45  , _c2(g._c2)
46  , _f1(g._f1)
47  , _e2(g._e2)
48  , _E(0, 0)
49  // Always allow latitude and azimuth and unrolling of longitude
50  , _caps(caps | LATITUDE | AZIMUTH | LONG_UNROLL)
51  {
52  // Guard against underflow in salp0
53  Math::sincosd(Math::AngRound(_azi1), _salp1, _calp1);
54  real cbet1, sbet1;
55  Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
56  // Ensure cbet1 = +epsilon at poles
57  Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
58  _dn1 = (_f >= 0 ? sqrt(1 + g._ep2 * Math::sq(sbet1)) :
59  sqrt(1 - _e2 * Math::sq(cbet1)) / _f1);
60 
61  // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
62  _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
63  // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
64  // is slightly better (consider the case salp1 = 0).
65  _calp0 = Math::hypot(_calp1, _salp1 * sbet1);
66  // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
67  // sig = 0 is nearest northward crossing of equator.
68  // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
69  // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
70  // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
71  // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
72  // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
73  // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
74  // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
75  _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
76  _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
77  // Without normalization we have schi1 = somg1.
78  _cchi1 = _f1 * _dn1 * _comg1;
79  Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
80  // Math::norm(_somg1, _comg1); -- don't need to normalize!
81  // Math::norm(_schi1, _cchi1); -- don't need to normalize!
82 
83  _k2 = Math::sq(_calp0) * g._ep2;
84  _E.Reset(-_k2, -g._ep2, 1 + _k2, 1 + g._ep2);
85 
86  if (_caps & CAP_E) {
87  _E0 = _E.E() / (Math::pi() / 2);
88  _E1 = _E.deltaE(_ssig1, _csig1, _dn1);
89  real s = sin(_E1), c = cos(_E1);
90  // tau1 = sig1 + B11
91  _stau1 = _ssig1 * c + _csig1 * s;
92  _ctau1 = _csig1 * c - _ssig1 * s;
93  // Not necessary because Einv inverts E
94  // _E1 = -_E.deltaEinv(_stau1, _ctau1);
95  }
96 
97  if (_caps & CAP_D) {
98  _D0 = _E.D() / (Math::pi() / 2);
99  _D1 = _E.deltaD(_ssig1, _csig1, _dn1);
100  }
101 
102  if (_caps & CAP_H) {
103  _H0 = _E.H() / (Math::pi() / 2);
104  _H1 = _E.deltaH(_ssig1, _csig1, _dn1);
105  }
106 
107  if (_caps & CAP_C4) {
108  real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
109  g.C4f(eps, _C4a);
110  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
111  _A4 = Math::sq(_a) * _calp0 * _salp0 * _e2;
112  _B41 = GeodesicExact::CosSeries(_ssig1, _csig1, _C4a, nC4_);
113  }
114  }
115 
116  Math::real GeodesicLineExact::GenPosition(bool arcmode, real s12_a12,
117  unsigned outmask,
118  real& lat2, real& lon2, real& azi2,
119  real& s12, real& m12,
120  real& M12, real& M21,
121  real& S12)
122  const {
123  outmask &= _caps & OUT_MASK;
124  if (!( Init() && (arcmode || (_caps & DISTANCE_IN & OUT_MASK)) ))
125  // Uninitialized or impossible distance calculation requested
126  return Math::NaN();
127 
128  // Avoid warning about uninitialized B12.
129  real sig12, ssig12, csig12, E2 = 0, AB1 = 0;
130  if (arcmode) {
131  // Interpret s12_a12 as spherical arc length
132  sig12 = s12_a12 * Math::degree();
133  real s12a = abs(s12_a12);
134  s12a -= 180 * floor(s12a / 180);
135  ssig12 = s12a == 0 ? 0 : sin(sig12);
136  csig12 = s12a == 90 ? 0 : cos(sig12);
137  } else {
138  // Interpret s12_a12 as distance
139  real
140  tau12 = s12_a12 / (_b * _E0),
141  s = sin(tau12),
142  c = cos(tau12);
143  // tau2 = tau1 + tau12
144  E2 = - _E.deltaEinv(_stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s);
145  sig12 = tau12 - (E2 - _E1);
146  ssig12 = sin(sig12);
147  csig12 = cos(sig12);
148  }
149 
150  real ssig2, csig2, sbet2, cbet2, salp2, calp2;
151  // sig2 = sig1 + sig12
152  ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
153  csig2 = _csig1 * csig12 - _ssig1 * ssig12;
154  real dn2 = _E.Delta(ssig2, csig2);
155  if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
156  if (arcmode) {
157  E2 = _E.deltaE(ssig2, csig2, dn2);
158  }
159  AB1 = _E0 * (E2 - _E1);
160  }
161  // sin(bet2) = cos(alp0) * sin(sig2)
162  sbet2 = _calp0 * ssig2;
163  // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
164  cbet2 = Math::hypot(_salp0, _calp0 * csig2);
165  if (cbet2 == 0)
166  // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
167  cbet2 = csig2 = tiny_;
168  // tan(alp0) = cos(sig2)*tan(alp2)
169  salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
170 
171  if (outmask & DISTANCE)
172  s12 = arcmode ? _b * (_E0 * sig12 + AB1) : s12_a12;
173 
174  if (outmask & LONGITUDE) {
175  real somg2 = _salp0 * ssig2, comg2 = csig2; // No need to normalize
176  int E = _salp0 < 0 ? -1 : 1; // east-going?
177  // Without normalization we have schi2 = somg2.
178  real cchi2 = _f1 * dn2 * comg2;
179  real chi12 = outmask & LONG_UNROLL
180  ? E * (sig12
181  - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
182  + (atan2(E * somg2, cchi2) - atan2(E * _somg1, _cchi1)))
183  : atan2(somg2 * _cchi1 - cchi2 * _somg1,
184  cchi2 * _cchi1 + somg2 * _somg1);
185  real lam12 = chi12 -
186  _e2/_f1 * _salp0 * _H0 * (sig12 + (_E.deltaH(ssig2, csig2, dn2) - _H1));
187  real lon12 = lam12 / Math::degree();
188  lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
190  Math::AngNormalize(lon12));
191  }
192 
193  if (outmask & LATITUDE)
194  lat2 = Math::atan2d(sbet2, _f1 * cbet2);
195 
196  if (outmask & AZIMUTH)
197  azi2 = Math::atan2d(salp2, calp2);
198 
199  if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
200  real J12 = _k2 * _D0 * (sig12 + (_E.deltaD(ssig2, csig2, dn2) - _D1));
201  if (outmask & REDUCEDLENGTH)
202  // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
203  // accurate cancellation in the case of coincident points.
204  m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
205  - _csig1 * csig2 * J12);
206  if (outmask & GEODESICSCALE) {
207  real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
208  M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
209  M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
210  }
211  }
212 
213  if (outmask & AREA) {
214  real
215  B42 = GeodesicExact::CosSeries(ssig2, csig2, _C4a, nC4_);
216  real salp12, calp12;
217  if (_calp0 == 0 || _salp0 == 0) {
218  // alp12 = alp2 - alp1, used in atan2 so no need to normalize
219  salp12 = salp2 * _calp1 - calp2 * _salp1;
220  calp12 = calp2 * _calp1 + salp2 * _salp1;
221  // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
222  // salp12 = -0 and alp12 = -180. However this depends on the sign being
223  // attached to 0 correctly. The following ensures the correct behavior.
224  if (salp12 == 0 && calp12 < 0) {
225  salp12 = tiny_ * _calp1;
226  calp12 = -1;
227  }
228  } else {
229  // tan(alp) = tan(alp0) * sec(sig)
230  // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
231  // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
232  // If csig12 > 0, write
233  // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
234  // else
235  // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
236  // No need to normalize
237  salp12 = _calp0 * _salp0 *
238  (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
239  ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
240  calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
241  }
242  S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
243  }
244 
245  return arcmode ? s12_a12 : sig12 / Math::degree();
246  }
247 
248 } // namespace GeographicLib
static T AngNormalize(T x)
Definition: Math.hpp:451
static T NaN()
Definition: Math.hpp:783
Math::real deltaEinv(real stau, real ctau) const
static T pi()
Definition: Math.hpp:216
void Reset(real k2=0, real alpha2=0)
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:102
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.hpp:559
Math::real Delta(real sn, real cn) const
static void norm(T &x, T &y)
Definition: Math.hpp:398
static T hypot(T x, T y)
Definition: Math.hpp:257
static T sq(T x)
Definition: Math.hpp:246
Header for GeographicLib::GeodesicLineExact class.
static T atan2d(T y, T x)
Definition: Math.hpp:676
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
static T degree()
Definition: Math.hpp:230
Exact geodesic calculations.
Math::real deltaE(real sn, real cn, real dn) const
Math::real deltaH(real sn, real cn, real dn) const
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
static T AngRound(T x)
Definition: Math.hpp:530
Math::real deltaD(real sn, real cn, real dn) const