GeographicLib  1.46
GeodesicExact.cpp
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1 /**
2  * \file GeodesicExact.cpp
3  * \brief Implementation for GeographicLib::GeodesicExact class
4  *
5  * Copyright (c) Charles Karney (2012-2016) <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 
31 
32 #if defined(_MSC_VER)
33 // Squelch warnings about potentially uninitialized local variables and
34 // constant conditional expressions
35 # pragma warning (disable: 4701 4127)
36 #endif
37 
38 namespace GeographicLib {
39 
40  using namespace std;
41 
43  : maxit2_(maxit1_ + Math::digits() + 10)
44  // Underflow guard. We require
45  // tiny_ * epsilon() > 0
46  // tiny_ + epsilon() == epsilon()
47  , tiny_(sqrt(numeric_limits<real>::min()))
48  , tol0_(numeric_limits<real>::epsilon())
49  // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
50  // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
51  // which otherwise failed for Visual Studio 10 (Release and Debug)
52  , tol1_(200 * tol0_)
53  , tol2_(sqrt(tol0_))
54  , tolb_(tol0_ * tol2_) // Check on bisection interval
55  , xthresh_(1000 * tol2_)
56  , _a(a)
57  , _f(f)
58  , _f1(1 - _f)
59  , _e2(_f * (2 - _f))
60  , _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
61  , _n(_f / ( 2 - _f))
62  , _b(_a * _f1)
63  // The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) in
64  // the definition of _c2. The latter is more accurate for very oblate
65  // ellipsoids (which the Geodesic class does not attempt to handle). Of
66  // course, the area calculation in GeodesicExact is still based on a
67  // series and so only holds for moderately oblate (or prolate)
68  // ellipsoids.
69  , _c2((Math::sq(_a) + Math::sq(_b) *
70  (_f == 0 ? 1 :
71  (_f > 0 ? Math::asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) /
72  sqrt(abs(_e2))))/2) // authalic radius squared
73  // The sig12 threshold for "really short". Using the auxiliary sphere
74  // solution with dnm computed at (bet1 + bet2) / 2, the relative error in
75  // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
76  // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a
77  // given f and sig12, the max error occurs for lines near the pole. If
78  // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
79  // increases by a factor of 2.) Setting this equal to epsilon gives
80  // sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100)
81  // and max(0.001, abs(f)) stops etol2 getting too large in the nearly
82  // spherical case.
83  , _etol2(0.1 * tol2_ /
84  sqrt( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
85  {
86  if (!(Math::isfinite(_a) && _a > 0))
87  throw GeographicErr("Major radius is not positive");
88  if (!(Math::isfinite(_b) && _b > 0))
89  throw GeographicErr("Minor radius is not positive");
90  C4coeff();
91  }
92 
94  static const GeodesicExact wgs84(Constants::WGS84_a(),
96  return wgs84;
97  }
98 
99  Math::real GeodesicExact::CosSeries(real sinx, real cosx,
100  const real c[], int n) {
101  // Evaluate
102  // y = sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
103  // using Clenshaw summation.
104  // Approx operation count = (n + 5) mult and (2 * n + 2) add
105  c += n ; // Point to one beyond last element
106  real
107  ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
108  y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum
109  // Now n is even
110  n /= 2;
111  while (n--) {
112  // Unroll loop x 2, so accumulators return to their original role
113  y1 = ar * y0 - y1 + *--c;
114  y0 = ar * y1 - y0 + *--c;
115  }
116  return cosx * (y0 - y1); // cos(x) * (y0 - y1)
117  }
118 
119  GeodesicLineExact GeodesicExact::Line(real lat1, real lon1, real azi1,
120  unsigned caps) const {
121  return GeodesicLineExact(*this, lat1, lon1, azi1, caps);
122  }
123 
124  Math::real GeodesicExact::GenDirect(real lat1, real lon1, real azi1,
125  bool arcmode, real s12_a12,
126  unsigned outmask,
127  real& lat2, real& lon2, real& azi2,
128  real& s12, real& m12,
129  real& M12, real& M21,
130  real& S12) const {
131  // Automatically supply DISTANCE_IN if necessary
132  if (!arcmode) outmask |= DISTANCE_IN;
133  return GeodesicLineExact(*this, lat1, lon1, azi1, outmask)
134  . // Note the dot!
135  GenPosition(arcmode, s12_a12, outmask,
136  lat2, lon2, azi2, s12, m12, M12, M21, S12);
137  }
138 
140  real azi1,
141  bool arcmode, real s12_a12,
142  unsigned caps) const {
143  azi1 = Math::AngNormalize(azi1);
144  real salp1, calp1;
145  // Guard against underflow in salp0. Also -0 is converted to +0.
146  Math::sincosd(Math::AngRound(azi1), salp1, calp1);
147  // Automatically supply DISTANCE_IN if necessary
148  if (!arcmode) caps |= DISTANCE_IN;
149  return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1,
150  caps, arcmode, s12_a12);
151  }
152 
154  real azi1, real s12,
155  unsigned caps) const {
156  return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
157  }
158 
160  real azi1, real a12,
161  unsigned caps) const {
162  return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
163  }
164 
165  Math::real GeodesicExact::GenInverse(real lat1, real lon1,
166  real lat2, real lon2,
167  unsigned outmask, real& s12,
168  real& salp1, real& calp1,
169  real& salp2, real& calp2,
170  real& m12, real& M12, real& M21,
171  real& S12) const {
172  // Compute longitude difference (AngDiff does this carefully). Result is
173  // in [-180, 180] but -180 is only for west-going geodesics. 180 is for
174  // east-going and meridional geodesics.
175  real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
176  // Make longitude difference positive.
177  int lonsign = lon12 >= 0 ? 1 : -1;
178  // If very close to being on the same half-meridian, then make it so.
179  lon12 = lonsign * Math::AngRound(lon12);
180  lon12s = Math::AngRound((180 - lon12) - lonsign * lon12s);
181  real
182  lam12 = lon12 * Math::degree(),
183  slam12, clam12;
184  if (lon12 > 90) {
185  Math::sincosd(lon12s, slam12, clam12);
186  clam12 = -clam12;
187  } else
188  Math::sincosd(lon12, slam12, clam12);
189 
190  // If really close to the equator, treat as on equator.
191  lat1 = Math::AngRound(Math::LatFix(lat1));
192  lat2 = Math::AngRound(Math::LatFix(lat2));
193  // Swap points so that point with higher (abs) latitude is point 1
194  // If one latitude is a nan, then it becomes lat1.
195  int swapp = abs(lat1) < abs(lat2) ? -1 : 1;
196  if (swapp < 0) {
197  lonsign *= -1;
198  swap(lat1, lat2);
199  }
200  // Make lat1 <= 0
201  int latsign = lat1 < 0 ? 1 : -1;
202  lat1 *= latsign;
203  lat2 *= latsign;
204  // Now we have
205  //
206  // 0 <= lon12 <= 180
207  // -90 <= lat1 <= 0
208  // lat1 <= lat2 <= -lat1
209  //
210  // longsign, swapp, latsign register the transformation to bring the
211  // coordinates to this canonical form. In all cases, 1 means no change was
212  // made. We make these transformations so that there are few cases to
213  // check, e.g., on verifying quadrants in atan2. In addition, this
214  // enforces some symmetries in the results returned.
215 
216  real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
217  // Initialize for the meridian. No longitude calculation is done in this
218  // case to let the parameter default to 0.
219  EllipticFunction E(-_ep2);
220 
221  Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
222  // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
223  // will be <= 2*tiny for two points at the same pole.
224  Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
225 
226  Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
227  // Ensure cbet2 = +epsilon at poles
228  Math::norm(sbet2, cbet2); cbet2 = max(tiny_, cbet2);
229 
230  // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
231  // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
232  // a better measure. This logic is used in assigning calp2 in Lambda12.
233  // Sometimes these quantities vanish and in that case we force bet2 = +/-
234  // bet1 exactly. An example where is is necessary is the inverse problem
235  // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
236  // which failed with Visual Studio 10 (Release and Debug)
237 
238  if (cbet1 < -sbet1) {
239  if (cbet2 == cbet1)
240  sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
241  } else {
242  if (abs(sbet2) == -sbet1)
243  cbet2 = cbet1;
244  }
245 
246  real
247  dn1 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet1)) :
248  sqrt(1 - _e2 * Math::sq(cbet1)) / _f1),
249  dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) :
250  sqrt(1 - _e2 * Math::sq(cbet2)) / _f1);
251 
252  real a12, sig12;
253 
254  bool meridian = lat1 == -90 || slam12 == 0;
255 
256  if (meridian) {
257 
258  // Endpoints are on a single full meridian, so the geodesic might lie on
259  // a meridian.
260 
261  calp1 = clam12; salp1 = slam12; // Head to the target longitude
262  calp2 = 1; salp2 = 0; // At the target we're heading north
263 
264  real
265  // tan(bet) = tan(sig) * cos(alp)
266  ssig1 = sbet1, csig1 = calp1 * cbet1,
267  ssig2 = sbet2, csig2 = calp2 * cbet2;
268 
269  // sig12 = sig2 - sig1
270  sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
271  csig1 * csig2 + ssig1 * ssig2);
272  {
273  real dummy;
274  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
275  cbet1, cbet2, outmask | REDUCEDLENGTH,
276  s12x, m12x, dummy, M12, M21);
277  }
278  // Add the check for sig12 since zero length geodesics might yield m12 <
279  // 0. Test case was
280  //
281  // echo 20.001 0 20.001 0 | GeodSolve -i
282  //
283  // In fact, we will have sig12 > pi/2 for meridional geodesic which is
284  // not a shortest path.
285  if (sig12 < 1 || m12x >= 0) {
286  // Need at least 2, to handle 90 0 90 180
287  if (sig12 < 3 * tiny_)
288  sig12 = m12x = s12x = 0;
289  m12x *= _b;
290  s12x *= _b;
291  a12 = sig12 / Math::degree();
292  } else
293  // m12 < 0, i.e., prolate and too close to anti-podal
294  meridian = false;
295  }
296 
297  // somg12 > 1 marks that it needs to be calculated
298  real omg12 = 0, somg12 = 2, comg12 = 0;
299  if (!meridian &&
300  sbet1 == 0 && // and sbet2 == 0
301  (_f <= 0 || lon12s >= _f * 180)) {
302 
303  // Geodesic runs along equator
304  calp1 = calp2 = 0; salp1 = salp2 = 1;
305  s12x = _a * lam12;
306  sig12 = omg12 = lam12 / _f1;
307  m12x = _b * sin(sig12);
308  if (outmask & GEODESICSCALE)
309  M12 = M21 = cos(sig12);
310  a12 = lon12 / _f1;
311 
312  } else if (!meridian) {
313 
314  // Now point1 and point2 belong within a hemisphere bounded by a
315  // meridian and geodesic is neither meridional or equatorial.
316 
317  // Figure a starting point for Newton's method
318  real dnm;
319  sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
320  lam12, slam12, clam12,
321  salp1, calp1, salp2, calp2, dnm);
322 
323  if (sig12 >= 0) {
324  // Short lines (InverseStart sets salp2, calp2, dnm)
325  s12x = sig12 * _b * dnm;
326  m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
327  if (outmask & GEODESICSCALE)
328  M12 = M21 = cos(sig12 / dnm);
329  a12 = sig12 / Math::degree();
330  omg12 = lam12 / (_f1 * dnm);
331  } else {
332 
333  // Newton's method. This is a straightforward solution of f(alp1) =
334  // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
335  // root in the interval (0, pi) and its derivative is positive at the
336  // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
337  // alp1. During the course of the iteration, a range (alp1a, alp1b) is
338  // maintained which brackets the root and with each evaluation of
339  // f(alp) the range is shrunk, if possible. Newton's method is
340  // restarted whenever the derivative of f is negative (because the new
341  // value of alp1 is then further from the solution) or if the new
342  // estimate of alp1 lies outside (0,pi); in this case, the new starting
343  // guess is taken to be (alp1a + alp1b) / 2.
344  //
345  // initial values to suppress warnings (if loop is executed 0 times)
346  real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0;
347  unsigned numit = 0;
348  // Bracketing range
349  real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
350  for (bool tripn = false, tripb = false;
351  numit < maxit2_ || GEOGRAPHICLIB_PANIC;
352  ++numit) {
353  // 1/4 meridan = 10e6 m and random input. max err is estimated max
354  // error in nm (checking solution of inverse problem by direct
355  // solution). iter is mean and sd of number of iterations
356  //
357  // max iter
358  // log2(b/a) err mean sd
359  // -7 387 5.33 3.68
360  // -6 345 5.19 3.43
361  // -5 269 5.00 3.05
362  // -4 210 4.76 2.44
363  // -3 115 4.55 1.87
364  // -2 69 4.35 1.38
365  // -1 36 4.05 1.03
366  // 0 15 0.01 0.13
367  // 1 25 5.10 1.53
368  // 2 96 5.61 2.09
369  // 3 318 6.02 2.74
370  // 4 985 6.24 3.22
371  // 5 2352 6.32 3.44
372  // 6 6008 6.30 3.45
373  // 7 19024 6.19 3.30
374  real dv;
375  real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
376  slam12, clam12,
377  salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
378  E, somg12, comg12, numit < maxit1_, dv);
379  // Reversed test to allow escape with NaNs
380  if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_)) break;
381  // Update bracketing values
382  if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
383  { salp1b = salp1; calp1b = calp1; }
384  else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
385  { salp1a = salp1; calp1a = calp1; }
386  if (numit < maxit1_ && dv > 0) {
387  real
388  dalp1 = -v/dv;
389  real
390  sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
391  nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
392  if (nsalp1 > 0 && abs(dalp1) < Math::pi()) {
393  calp1 = calp1 * cdalp1 - salp1 * sdalp1;
394  salp1 = nsalp1;
395  Math::norm(salp1, calp1);
396  // In some regimes we don't get quadratic convergence because
397  // slope -> 0. So use convergence conditions based on epsilon
398  // instead of sqrt(epsilon).
399  tripn = abs(v) <= 16 * tol0_;
400  continue;
401  }
402  }
403  // Either dv was not postive or updated value was outside legal
404  // range. Use the midpoint of the bracket as the next estimate.
405  // This mechanism is not needed for the WGS84 ellipsoid, but it does
406  // catch problems with more eccentric ellipsoids. Its efficacy is
407  // such for the WGS84 test set with the starting guess set to alp1 =
408  // 90deg:
409  // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
410  // WGS84 and random input: mean = 4.74, sd = 0.99
411  salp1 = (salp1a + salp1b)/2;
412  calp1 = (calp1a + calp1b)/2;
413  Math::norm(salp1, calp1);
414  tripn = false;
415  tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
416  abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
417  }
418  {
419  real dummy;
420  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
421  cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
422  }
423  m12x *= _b;
424  s12x *= _b;
425  a12 = sig12 / Math::degree();
426  }
427  }
428 
429  if (outmask & DISTANCE)
430  s12 = 0 + s12x; // Convert -0 to 0
431 
432  if (outmask & REDUCEDLENGTH)
433  m12 = 0 + m12x; // Convert -0 to 0
434 
435  if (outmask & AREA) {
436  real
437  // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
438  salp0 = salp1 * cbet1,
439  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
440  real alp12;
441  if (calp0 != 0 && salp0 != 0) {
442  real
443  // From Lambda12: tan(bet) = tan(sig) * cos(alp)
444  ssig1 = sbet1, csig1 = calp1 * cbet1,
445  ssig2 = sbet2, csig2 = calp2 * cbet2,
446  k2 = Math::sq(calp0) * _ep2,
447  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
448  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
449  A4 = Math::sq(_a) * calp0 * salp0 * _e2;
450  Math::norm(ssig1, csig1);
451  Math::norm(ssig2, csig2);
452  real C4a[nC4_];
453  C4f(eps, C4a);
454  real
455  B41 = CosSeries(ssig1, csig1, C4a, nC4_),
456  B42 = CosSeries(ssig2, csig2, C4a, nC4_);
457  S12 = A4 * (B42 - B41);
458  } else
459  // Avoid problems with indeterminate sig1, sig2 on equator
460  S12 = 0;
461 
462  if (!meridian) {
463  if (somg12 > 1) {
464  somg12 = sin(omg12); comg12 = cos(omg12);
465  } else
466  Math::norm(somg12, comg12);
467  }
468 
469  if (!meridian &&
470  // omg12 < 3/4 * pi
471  comg12 > -real(0.7071) && // Long difference not too big
472  sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
473  // Use tan(Gamma/2) = tan(omg12/2)
474  // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
475  // with tan(x/2) = sin(x)/(1+cos(x))
476  real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
477  alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
478  domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
479  } else {
480  // alp12 = alp2 - alp1, used in atan2 so no need to normalize
481  real
482  salp12 = salp2 * calp1 - calp2 * salp1,
483  calp12 = calp2 * calp1 + salp2 * salp1;
484  // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
485  // salp12 = -0 and alp12 = -180. However this depends on the sign
486  // being attached to 0 correctly. The following ensures the correct
487  // behavior.
488  if (salp12 == 0 && calp12 < 0) {
489  salp12 = tiny_ * calp1;
490  calp12 = -1;
491  }
492  alp12 = atan2(salp12, calp12);
493  }
494  S12 += _c2 * alp12;
495  S12 *= swapp * lonsign * latsign;
496  // Convert -0 to 0
497  S12 += 0;
498  }
499 
500  // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
501  if (swapp < 0) {
502  swap(salp1, salp2);
503  swap(calp1, calp2);
504  if (outmask & GEODESICSCALE)
505  swap(M12, M21);
506  }
507 
508  salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
509  salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
510 
511  // Returned value in [0, 180]
512  return a12;
513  }
514 
515  Math::real GeodesicExact::GenInverse(real lat1, real lon1,
516  real lat2, real lon2,
517  unsigned outmask,
518  real& s12, real& azi1, real& azi2,
519  real& m12, real& M12, real& M21,
520  real& S12)
521  const {
522  outmask &= OUT_MASK;
523  real salp1, calp1, salp2, calp2,
524  a12 = GenInverse(lat1, lon1, lat2, lon2,
525  outmask, s12, salp1, calp1, salp2, calp2,
526  m12, M12, M21, S12);
527  if (outmask & AZIMUTH) {
528  azi1 = Math::atan2d(salp1, calp1);
529  azi2 = Math::atan2d(salp2, calp2);
530  }
531  return a12;
532  }
533 
535  real lat2, real lon2,
536  unsigned caps) const {
537  real t, salp1, calp1, salp2, calp2,
538  a12 = GenInverse(lat1, lon1, lat2, lon2,
539  // No need to specify AZIMUTH here
540  0u, t, salp1, calp1, salp2, calp2,
541  t, t, t, t),
542  azi1 = Math::atan2d(salp1, calp1);
543  // Ensure that a12 can be converted to a distance
544  if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
545  return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1, caps,
546  true, a12);
547  }
548 
549  void GeodesicExact::Lengths(const EllipticFunction& E,
550  real sig12,
551  real ssig1, real csig1, real dn1,
552  real ssig2, real csig2, real dn2,
553  real cbet1, real cbet2, unsigned outmask,
554  real& s12b, real& m12b, real& m0,
555  real& M12, real& M21) const {
556  // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
557  // and m0 = coefficient of secular term in expression for reduced length.
558 
559  outmask &= OUT_ALL;
560  // outmask & DISTANCE: set s12b
561  // outmask & REDUCEDLENGTH: set m12b & m0
562  // outmask & GEODESICSCALE: set M12 & M21
563 
564  // It's OK to have repeated dummy arguments,
565  // e.g., s12b = m0 = M12 = M21 = dummy
566 
567  if (outmask & DISTANCE)
568  // Missing a factor of _b
569  s12b = E.E() / (Math::pi() / 2) *
570  (sig12 + (E.deltaE(ssig2, csig2, dn2) - E.deltaE(ssig1, csig1, dn1)));
571  if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
572  real
573  m0x = - E.k2() * E.D() / (Math::pi() / 2),
574  J12 = m0x *
575  (sig12 + (E.deltaD(ssig2, csig2, dn2) - E.deltaD(ssig1, csig1, dn1)));
576  if (outmask & REDUCEDLENGTH) {
577  m0 = m0x;
578  // Missing a factor of _b. Add parens around (csig1 * ssig2) and
579  // (ssig1 * csig2) to ensure accurate cancellation in the case of
580  // coincident points.
581  m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
582  csig1 * csig2 * J12;
583  }
584  if (outmask & GEODESICSCALE) {
585  real csig12 = csig1 * csig2 + ssig1 * ssig2;
586  real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
587  M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
588  M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
589  }
590  }
591  }
592 
593  Math::real GeodesicExact::Astroid(real x, real y) {
594  // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
595  // This solution is adapted from Geocentric::Reverse.
596  real k;
597  real
598  p = Math::sq(x),
599  q = Math::sq(y),
600  r = (p + q - 1) / 6;
601  if ( !(q == 0 && r <= 0) ) {
602  real
603  // Avoid possible division by zero when r = 0 by multiplying equations
604  // for s and t by r^3 and r, resp.
605  S = p * q / 4, // S = r^3 * s
606  r2 = Math::sq(r),
607  r3 = r * r2,
608  // The discriminant of the quadratic equation for T3. This is zero on
609  // the evolute curve p^(1/3)+q^(1/3) = 1
610  disc = S * (S + 2 * r3);
611  real u = r;
612  if (disc >= 0) {
613  real T3 = S + r3;
614  // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
615  // of precision due to cancellation. The result is unchanged because
616  // of the way the T is used in definition of u.
617  T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
618  // N.B. cbrt always returns the real root. cbrt(-8) = -2.
619  real T = Math::cbrt(T3); // T = r * t
620  // T can be zero; but then r2 / T -> 0.
621  u += T + (T ? r2 / T : 0);
622  } else {
623  // T is complex, but the way u is defined the result is real.
624  real ang = atan2(sqrt(-disc), -(S + r3));
625  // There are three possible cube roots. We choose the root which
626  // avoids cancellation. Note that disc < 0 implies that r < 0.
627  u += 2 * r * cos(ang / 3);
628  }
629  real
630  v = sqrt(Math::sq(u) + q), // guaranteed positive
631  // Avoid loss of accuracy when u < 0.
632  uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
633  w = (uv - q) / (2 * v); // positive?
634  // Rearrange expression for k to avoid loss of accuracy due to
635  // subtraction. Division by 0 not possible because uv > 0, w >= 0.
636  k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
637  } else { // q == 0 && r <= 0
638  // y = 0 with |x| <= 1. Handle this case directly.
639  // for y small, positive root is k = abs(y)/sqrt(1-x^2)
640  k = 0;
641  }
642  return k;
643  }
644 
645  Math::real GeodesicExact::InverseStart(EllipticFunction& E,
646  real sbet1, real cbet1, real dn1,
647  real sbet2, real cbet2, real dn2,
648  real lam12, real slam12, real clam12,
649  real& salp1, real& calp1,
650  // Only updated if return val >= 0
651  real& salp2, real& calp2,
652  // Only updated for short lines
653  real& dnm)
654  const {
655  // Return a starting point for Newton's method in salp1 and calp1 (function
656  // value is -1). If Newton's method doesn't need to be used, return also
657  // salp2 and calp2 and function value is sig12.
658  real
659  sig12 = -1, // Return value
660  // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
661  sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
662  cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
663 #if defined(__GNUC__) && __GNUC__ == 4 && \
664  (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
665  // Volatile declaration needed to fix inverse cases
666  // 88.202499451857 0 -88.202499451857 179.981022032992859592
667  // 89.262080389218 0 -89.262080389218 179.992207982775375662
668  // 89.333123580033 0 -89.333123580032997687 179.99295812360148422
669  // which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
670  // and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw).
671  real sbet12a;
672  {
673  GEOGRAPHICLIB_VOLATILE real xx1 = sbet2 * cbet1;
674  GEOGRAPHICLIB_VOLATILE real xx2 = cbet2 * sbet1;
675  sbet12a = xx1 + xx2;
676  }
677 #else
678  real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
679 #endif
680  bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
681  cbet2 * lam12 < real(0.5);
682  real somg12, comg12;
683  if (shortline) {
684  real sbetm2 = Math::sq(sbet1 + sbet2);
685  // sin((bet1+bet2)/2)^2
686  // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
687  sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
688  dnm = sqrt(1 + _ep2 * sbetm2);
689  real omg12 = lam12 / (_f1 * dnm);
690  somg12 = sin(omg12); comg12 = cos(omg12);
691  } else {
692  somg12 = slam12; comg12 = clam12;
693  }
694 
695  salp1 = cbet2 * somg12;
696  calp1 = comg12 >= 0 ?
697  sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
698  sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
699 
700  real
701  ssig12 = Math::hypot(salp1, calp1),
702  csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
703 
704  if (shortline && ssig12 < _etol2) {
705  // really short lines
706  salp2 = cbet1 * somg12;
707  calp2 = sbet12 - cbet1 * sbet2 *
708  (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
709  Math::norm(salp2, calp2);
710  // Set return value
711  sig12 = atan2(ssig12, csig12);
712  } else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
713  csig12 >= 0 ||
714  ssig12 >= 6 * abs(_n) * Math::pi() * Math::sq(cbet1)) {
715  // Nothing to do, zeroth order spherical approximation is OK
716  } else {
717  // Scale lam12 and bet2 to x, y coordinate system where antipodal point
718  // is at origin and singular point is at y = 0, x = -1.
719  real y, lamscale, betscale;
720  // Volatile declaration needed to fix inverse case
721  // 56.320923501171 0 -56.320923501171 179.664747671772880215
722  // which otherwise fails with g++ 4.4.4 x86 -O3
723  GEOGRAPHICLIB_VOLATILE real x;
724  real lam12x = atan2(-slam12, -clam12); // lam12 - pi
725  if (_f >= 0) { // In fact f == 0 does not get here
726  // x = dlong, y = dlat
727  {
728  real k2 = Math::sq(sbet1) * _ep2;
729  E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
730  lamscale = _e2/_f1 * cbet1 * 2 * E.H();
731  }
732  betscale = lamscale * cbet1;
733 
734  x = lam12x / lamscale;
735  y = sbet12a / betscale;
736  } else { // _f < 0
737  // x = dlat, y = dlong
738  real
739  cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
740  bet12a = atan2(sbet12a, cbet12a);
741  real m12b, m0, dummy;
742  // In the case of lon12 = 180, this repeats a calculation made in
743  // Inverse.
744  Lengths(E, Math::pi() + bet12a,
745  sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
746  cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy);
747  x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
748  betscale = x < -real(0.01) ? sbet12a / x :
749  -_f * Math::sq(cbet1) * Math::pi();
750  lamscale = betscale / cbet1;
751  y = lam12x / lamscale;
752  }
753 
754  if (y > -tol1_ && x > -1 - xthresh_) {
755  // strip near cut
756  // Need real(x) here to cast away the volatility of x for min/max
757  if (_f >= 0) {
758  salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 - Math::sq(salp1));
759  } else {
760  calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
761  salp1 = sqrt(1 - Math::sq(calp1));
762  }
763  } else {
764  // Estimate alp1, by solving the astroid problem.
765  //
766  // Could estimate alpha1 = theta + pi/2, directly, i.e.,
767  // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
768  // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
769  //
770  // However, it's better to estimate omg12 from astroid and use
771  // spherical formula to compute alp1. This reduces the mean number of
772  // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
773  // (min 0 max 5). The changes in the number of iterations are as
774  // follows:
775  //
776  // change percent
777  // 1 5
778  // 0 78
779  // -1 16
780  // -2 0.6
781  // -3 0.04
782  // -4 0.002
783  //
784  // The histogram of iterations is (m = number of iterations estimating
785  // alp1 directly, n = number of iterations estimating via omg12, total
786  // number of trials = 148605):
787  //
788  // iter m n
789  // 0 148 186
790  // 1 13046 13845
791  // 2 93315 102225
792  // 3 36189 32341
793  // 4 5396 7
794  // 5 455 1
795  // 6 56 0
796  //
797  // Because omg12 is near pi, estimate work with omg12a = pi - omg12
798  real k = Astroid(x, y);
799  real
800  omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
801  somg12 = sin(omg12a); comg12 = -cos(omg12a);
802  // Update spherical estimate of alp1 using omg12 instead of lam12
803  salp1 = cbet2 * somg12;
804  calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
805  }
806  }
807  // Sanity check on starting guess. Backwards check allows NaN through.
808  if (!(salp1 <= 0))
809  Math::norm(salp1, calp1);
810  else {
811  salp1 = 1; calp1 = 0;
812  }
813  return sig12;
814  }
815 
816  Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1,
817  real sbet2, real cbet2, real dn2,
818  real salp1, real calp1,
819  real slam120, real clam120,
820  real& salp2, real& calp2,
821  real& sig12,
822  real& ssig1, real& csig1,
823  real& ssig2, real& csig2,
824  EllipticFunction& E,
825  real& somg12, real& comg12,
826  bool diffp, real& dlam12) const
827  {
828 
829  if (sbet1 == 0 && calp1 == 0)
830  // Break degeneracy of equatorial line. This case has already been
831  // handled.
832  calp1 = -tiny_;
833 
834  real
835  // sin(alp1) * cos(bet1) = sin(alp0)
836  salp0 = salp1 * cbet1,
837  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
838 
839  real somg1, comg1, somg2, comg2, cchi1, cchi2, lam12;
840  // tan(bet1) = tan(sig1) * cos(alp1)
841  // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
842  ssig1 = sbet1; somg1 = salp0 * sbet1;
843  csig1 = comg1 = calp1 * cbet1;
844  // Without normalization we have schi1 = somg1.
845  cchi1 = _f1 * dn1 * comg1;
846  Math::norm(ssig1, csig1);
847  // Math::norm(somg1, comg1); -- don't need to normalize!
848  // Math::norm(schi1, cchi1); -- don't need to normalize!
849 
850  // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
851  // about this case, since this can yield singularities in the Newton
852  // iteration.
853  // sin(alp2) * cos(bet2) = sin(alp0)
854  salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
855  // calp2 = sqrt(1 - sq(salp2))
856  // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
857  // and subst for calp0 and rearrange to give (choose positive sqrt
858  // to give alp2 in [0, pi/2]).
859  calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
860  sqrt(Math::sq(calp1 * cbet1) +
861  (cbet1 < -sbet1 ?
862  (cbet2 - cbet1) * (cbet1 + cbet2) :
863  (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
864  abs(calp1);
865  // tan(bet2) = tan(sig2) * cos(alp2)
866  // tan(omg2) = sin(alp0) * tan(sig2).
867  ssig2 = sbet2; somg2 = salp0 * sbet2;
868  csig2 = comg2 = calp2 * cbet2;
869  // Without normalization we have schi2 = somg2.
870  cchi2 = _f1 * dn2 * comg2;
871  Math::norm(ssig2, csig2);
872  // Math::norm(somg2, comg2); -- don't need to normalize!
873  // Math::norm(schi2, cchi2); -- don't need to normalize!
874 
875  // sig12 = sig2 - sig1, limit to [0, pi]
876  sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
877  csig1 * csig2 + ssig1 * ssig2);
878 
879  // omg12 = omg2 - omg1, limit to [0, pi]
880  somg12 = max(real(0), comg1 * somg2 - somg1 * comg2);
881  comg12 = comg1 * comg2 + somg1 * somg2;
882  real k2 = Math::sq(calp0) * _ep2;
883  E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
884  // chi12 = chi2 - chi1, limit to [0, pi]
885  real
886  schi12 = max(real(0), cchi1 * somg2 - somg1 * cchi2),
887  cchi12 = cchi1 * cchi2 + somg1 * somg2;
888  // eta = chi12 - lam120
889  real eta = atan2(schi12 * clam120 - cchi12 * slam120,
890  cchi12 * clam120 + schi12 * slam120);
891 
892  lam12 = eta -
893  _e2/_f1 * salp0 * E.H() / (Math::pi() / 2) *
894  (sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1)));
895 
896  if (diffp) {
897  if (calp2 == 0)
898  dlam12 = - 2 * _f1 * dn1 / sbet1;
899  else {
900  real dummy;
901  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
902  cbet1, cbet2, REDUCEDLENGTH,
903  dummy, dlam12, dummy, dummy, dummy);
904  dlam12 *= _f1 / (calp2 * cbet2);
905  }
906  }
907 
908  return lam12;
909  }
910 
911  void GeodesicExact::C4f(real eps, real c[]) const {
912  // Evaluate C4 coeffs
913  // Elements c[0] thru c[nC4_ - 1] are set
914  real mult = 1;
915  int o = 0;
916  for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
917  int m = nC4_ - l - 1; // order of polynomial in eps
918  c[l] = mult * Math::polyval(m, _C4x + o, eps);
919  o += m + 1;
920  mult *= eps;
921  }
922  // Post condition: o == nC4x_
923  if (!(o == nC4x_))
924  throw GeographicErr("C4 misalignment");
925  }
926 
927 } // namespace GeographicLib
static T AngNormalize(T x)
Definition: Math.hpp:437
GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
static T pi()
Definition: Math.hpp:202
void Reset(real k2=0, real alpha2=0)
static T cbrt(T x)
Definition: Math.hpp:345
static bool isfinite(T x)
Definition: Math.hpp:781
static T LatFix(T x)
Definition: Math.hpp:456
GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:102
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.hpp:545
static T AngDiff(T x, T y, T &e)
Definition: Math.hpp:475
Elliptic integrals and functions.
static void norm(T &x, T &y)
Definition: Math.hpp:384
#define GEOGRAPHICLIB_VOLATILE
Definition: Math.hpp:84
static T hypot(T x, T y)
Definition: Math.hpp:243
static T sq(T x)
Definition: Math.hpp:232
GeodesicLineExact Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Header for GeographicLib::GeodesicLineExact class.
static T atan2d(T y, T x)
Definition: Math.hpp:671
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:425
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
static T degree()
Definition: Math.hpp:216
GeodesicLineExact DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
Exact geodesic calculations.
Math::real deltaE(real sn, real cn, real dn) const
Math::real deltaH(real sn, real cn, real dn) const
Header for GeographicLib::GeodesicExact class.
GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Exception handling for GeographicLib.
Definition: Constants.hpp:373
static T AngRound(T x)
Definition: Math.hpp:524
Math::real deltaD(real sn, real cn, real dn) const
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:87
static const GeodesicExact & WGS84()