GeographicLib  1.45
GeodesicExact.cpp
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1 /**
2  * \file GeodesicExact.cpp
3  * \brief Implementation for GeographicLib::GeodesicExact class
4  *
5  * Copyright (c) Charles Karney (2012-2015) <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 
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 <= 1 ? f : 1/f) // f > 1 behavior is DEPRECATED
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  return GeodesicLineExact(*this, lat1, lon1, azi1,
132  // Automatically supply DISTANCE_IN if necessary
133  outmask | (arcmode ? NONE : DISTANCE_IN))
134  . // Note the dot!
135  GenPosition(arcmode, s12_a12, outmask,
136  lat2, lon2, azi2, s12, m12, M12, M21, S12);
137  }
138 
139  Math::real GeodesicExact::GenInverse(real lat1, real lon1,
140  real lat2, real lon2,
141  unsigned outmask,
142  real& s12, real& azi1, real& azi2,
143  real& m12, real& M12, real& M21,
144  real& S12) const {
145  outmask &= OUT_ALL;
146  // Compute longitude difference (AngDiff does this carefully). Result is
147  // in [-180, 180] but -180 is only for west-going geodesics. 180 is for
148  // east-going and meridional geodesics.
149  // If very close to being on the same half-meridian, then make it so.
150  real lon12 = Math::AngRound(Math::AngDiff(lon1, lon2));
151  // Make longitude difference positive.
152  int lonsign = lon12 >= 0 ? 1 : -1;
153  lon12 *= lonsign;
154  // If really close to the equator, treat as on equator.
155  lat1 = Math::AngRound(Math::LatFix(lat1));
156  lat2 = Math::AngRound(Math::LatFix(lat2));
157  // Swap points so that point with higher (abs) latitude is point 1
158  // If one latitude is a nan, then it becomes lat1.
159  int swapp = abs(lat1) < abs(lat2) ? -1 : 1;
160  if (swapp < 0) {
161  lonsign *= -1;
162  swap(lat1, lat2);
163  }
164  // Make lat1 <= 0
165  int latsign = lat1 < 0 ? 1 : -1;
166  lat1 *= latsign;
167  lat2 *= latsign;
168  // Now we have
169  //
170  // 0 <= lon12 <= 180
171  // -90 <= lat1 <= 0
172  // lat1 <= lat2 <= -lat1
173  //
174  // longsign, swapp, latsign register the transformation to bring the
175  // coordinates to this canonical form. In all cases, 1 means no change was
176  // made. We make these transformations so that there are few cases to
177  // check, e.g., on verifying quadrants in atan2. In addition, this
178  // enforces some symmetries in the results returned.
179 
180  real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
181  // Initialize for the meridian. No longitude calculation is done in this
182  // case to let the parameter default to 0.
183  EllipticFunction E(-_ep2);
184 
185  Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
186  // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
187  // will be <= 2*tiny for two points at the same pole.
188  Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
189 
190  Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
191  // Ensure cbet2 = +epsilon at poles
192  Math::norm(sbet2, cbet2); cbet2 = max(tiny_, cbet2);
193 
194  // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
195  // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
196  // a better measure. This logic is used in assigning calp2 in Lambda12.
197  // Sometimes these quantities vanish and in that case we force bet2 = +/-
198  // bet1 exactly. An example where is is necessary is the inverse problem
199  // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
200  // which failed with Visual Studio 10 (Release and Debug)
201 
202  if (cbet1 < -sbet1) {
203  if (cbet2 == cbet1)
204  sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
205  } else {
206  if (abs(sbet2) == -sbet1)
207  cbet2 = cbet1;
208  }
209 
210  real
211  dn1 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet1)) :
212  sqrt(1 - _e2 * Math::sq(cbet1)) / _f1),
213  dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) :
214  sqrt(1 - _e2 * Math::sq(cbet2)) / _f1);
215 
216  real
217  lam12 = lon12 * Math::degree(), slam12, clam12;
218  Math::sincosd(lon12, slam12, clam12);
219 
220  // initial values to suppress warning
221  real a12, sig12, calp1, salp1, calp2 = 0, salp2 = 0;
222 
223  bool meridian = lat1 == -90 || slam12 == 0;
224 
225  if (meridian) {
226 
227  // Endpoints are on a single full meridian, so the geodesic might lie on
228  // a meridian.
229 
230  calp1 = clam12; salp1 = slam12; // Head to the target longitude
231  calp2 = 1; salp2 = 0; // At the target we're heading north
232 
233  real
234  // tan(bet) = tan(sig) * cos(alp)
235  ssig1 = sbet1, csig1 = calp1 * cbet1,
236  ssig2 = sbet2, csig2 = calp2 * cbet2;
237 
238  // sig12 = sig2 - sig1
239  sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
240  csig1 * csig2 + ssig1 * ssig2);
241  {
242  real dummy;
243  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
244  cbet1, cbet2, outmask | REDUCEDLENGTH,
245  s12x, m12x, dummy, M12, M21);
246  }
247  // Add the check for sig12 since zero length geodesics might yield m12 <
248  // 0. Test case was
249  //
250  // echo 20.001 0 20.001 0 | GeodSolve -i
251  //
252  // In fact, we will have sig12 > pi/2 for meridional geodesic which is
253  // not a shortest path.
254  if (sig12 < 1 || m12x >= 0) {
255  // Need at least 2, to handle 90 0 90 180
256  if (sig12 < 3 * tiny_)
257  sig12 = m12x = s12x = 0;
258  m12x *= _b;
259  s12x *= _b;
260  a12 = sig12 / Math::degree();
261  } else
262  // m12 < 0, i.e., prolate and too close to anti-podal
263  meridian = false;
264  }
265 
266  real omg12 = 0; // initial value to suppress warning
267  if (!meridian &&
268  sbet1 == 0 && // and sbet2 == 0
269  // Mimic the way Lambda12 works with calp1 = 0
270  (_f <= 0 || lam12 <= Math::pi() - _f * Math::pi())) {
271 
272  // Geodesic runs along equator
273  calp1 = calp2 = 0; salp1 = salp2 = 1;
274  s12x = _a * lam12;
275  sig12 = omg12 = lam12 / _f1;
276  m12x = _b * sin(sig12);
277  if (outmask & GEODESICSCALE)
278  M12 = M21 = cos(sig12);
279  a12 = lon12 / _f1;
280 
281  } else if (!meridian) {
282 
283  // Now point1 and point2 belong within a hemisphere bounded by a
284  // meridian and geodesic is neither meridional or equatorial.
285 
286  // Figure a starting point for Newton's method
287  real dnm;
288  sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
289  lam12,
290  salp1, calp1, salp2, calp2, dnm);
291 
292  if (sig12 >= 0) {
293  // Short lines (InverseStart sets salp2, calp2, dnm)
294  s12x = sig12 * _b * dnm;
295  m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
296  if (outmask & GEODESICSCALE)
297  M12 = M21 = cos(sig12 / dnm);
298  a12 = sig12 / Math::degree();
299  omg12 = lam12 / (_f1 * dnm);
300  } else {
301 
302  // Newton's method. This is a straightforward solution of f(alp1) =
303  // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
304  // root in the interval (0, pi) and its derivative is positive at the
305  // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
306  // alp1. During the course of the iteration, a range (alp1a, alp1b) is
307  // maintained which brackets the root and with each evaluation of
308  // f(alp) the range is shrunk, if possible. Newton's method is
309  // restarted whenever the derivative of f is negative (because the new
310  // value of alp1 is then further from the solution) or if the new
311  // estimate of alp1 lies outside (0,pi); in this case, the new starting
312  // guess is taken to be (alp1a + alp1b) / 2.
313  //
314  // initial values to suppress warnings (if loop is executed 0 times)
315  real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0;
316  unsigned numit = 0;
317  // Bracketing range
318  real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
319  for (bool tripn = false, tripb = false;
320  numit < maxit2_ || GEOGRAPHICLIB_PANIC;
321  ++numit) {
322  // 1/4 meridan = 10e6 m and random input. max err is estimated max
323  // error in nm (checking solution of inverse problem by direct
324  // solution). iter is mean and sd of number of iterations
325  //
326  // max iter
327  // log2(b/a) err mean sd
328  // -7 387 5.33 3.68
329  // -6 345 5.19 3.43
330  // -5 269 5.00 3.05
331  // -4 210 4.76 2.44
332  // -3 115 4.55 1.87
333  // -2 69 4.35 1.38
334  // -1 36 4.05 1.03
335  // 0 15 0.01 0.13
336  // 1 25 5.10 1.53
337  // 2 96 5.61 2.09
338  // 3 318 6.02 2.74
339  // 4 985 6.24 3.22
340  // 5 2352 6.32 3.44
341  // 6 6008 6.30 3.45
342  // 7 19024 6.19 3.30
343  real dv;
344  real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
345  salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
346  E, omg12, numit < maxit1_, dv) - lam12;
347  // 2 * tol0 is approximately 1 ulp for a number in [0, pi].
348  // Reversed test to allow escape with NaNs
349  if (tripb || !(abs(v) >= (tripn ? 8 : 2) * tol0_)) break;
350  // Update bracketing values
351  if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
352  { salp1b = salp1; calp1b = calp1; }
353  else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
354  { salp1a = salp1; calp1a = calp1; }
355  if (numit < maxit1_ && dv > 0) {
356  real
357  dalp1 = -v/dv;
358  real
359  sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
360  nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
361  if (nsalp1 > 0 && abs(dalp1) < Math::pi()) {
362  calp1 = calp1 * cdalp1 - salp1 * sdalp1;
363  salp1 = nsalp1;
364  Math::norm(salp1, calp1);
365  // In some regimes we don't get quadratic convergence because
366  // slope -> 0. So use convergence conditions based on epsilon
367  // instead of sqrt(epsilon).
368  tripn = abs(v) <= 16 * tol0_;
369  continue;
370  }
371  }
372  // Either dv was not postive or updated value was outside legal
373  // range. Use the midpoint of the bracket as the next estimate.
374  // This mechanism is not needed for the WGS84 ellipsoid, but it does
375  // catch problems with more eccentric ellipsoids. Its efficacy is
376  // such for the WGS84 test set with the starting guess set to alp1 =
377  // 90deg:
378  // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
379  // WGS84 and random input: mean = 4.74, sd = 0.99
380  salp1 = (salp1a + salp1b)/2;
381  calp1 = (calp1a + calp1b)/2;
382  Math::norm(salp1, calp1);
383  tripn = false;
384  tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
385  abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
386  }
387  {
388  real dummy;
389  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
390  cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
391  }
392  m12x *= _b;
393  s12x *= _b;
394  a12 = sig12 / Math::degree();
395  }
396  }
397 
398  if (outmask & DISTANCE)
399  s12 = 0 + s12x; // Convert -0 to 0
400 
401  if (outmask & REDUCEDLENGTH)
402  m12 = 0 + m12x; // Convert -0 to 0
403 
404  if (outmask & AREA) {
405  real
406  // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
407  salp0 = salp1 * cbet1,
408  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
409  real alp12;
410  if (calp0 != 0 && salp0 != 0) {
411  real
412  // From Lambda12: tan(bet) = tan(sig) * cos(alp)
413  ssig1 = sbet1, csig1 = calp1 * cbet1,
414  ssig2 = sbet2, csig2 = calp2 * cbet2,
415  k2 = Math::sq(calp0) * _ep2,
416  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
417  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
418  A4 = Math::sq(_a) * calp0 * salp0 * _e2;
419  Math::norm(ssig1, csig1);
420  Math::norm(ssig2, csig2);
421  real C4a[nC4_];
422  C4f(eps, C4a);
423  real
424  B41 = CosSeries(ssig1, csig1, C4a, nC4_),
425  B42 = CosSeries(ssig2, csig2, C4a, nC4_);
426  S12 = A4 * (B42 - B41);
427  } else
428  // Avoid problems with indeterminate sig1, sig2 on equator
429  S12 = 0;
430 
431  if (!meridian &&
432  omg12 < real(0.75) * Math::pi() && // Long difference too big
433  sbet2 - sbet1 < real(1.75)) { // Lat difference too big
434  // Use tan(Gamma/2) = tan(omg12/2)
435  // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
436  // with tan(x/2) = sin(x)/(1+cos(x))
437  real
438  somg12 = sin(omg12), domg12 = 1 + cos(omg12),
439  dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
440  alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
441  domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
442  } else {
443  // alp12 = alp2 - alp1, used in atan2 so no need to normalize
444  real
445  salp12 = salp2 * calp1 - calp2 * salp1,
446  calp12 = calp2 * calp1 + salp2 * salp1;
447  // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
448  // salp12 = -0 and alp12 = -180. However this depends on the sign
449  // being attached to 0 correctly. The following ensures the correct
450  // behavior.
451  if (salp12 == 0 && calp12 < 0) {
452  salp12 = tiny_ * calp1;
453  calp12 = -1;
454  }
455  alp12 = atan2(salp12, calp12);
456  }
457  S12 += _c2 * alp12;
458  S12 *= swapp * lonsign * latsign;
459  // Convert -0 to 0
460  S12 += 0;
461  }
462 
463  // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
464  if (swapp < 0) {
465  swap(salp1, salp2);
466  swap(calp1, calp2);
467  if (outmask & GEODESICSCALE)
468  swap(M12, M21);
469  }
470 
471  salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
472  salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
473 
474  if (outmask & AZIMUTH) {
475  azi1 = Math::atan2d(salp1, calp1);
476  azi2 = Math::atan2d(salp2, calp2);
477  }
478 
479  // Returned value in [0, 180]
480  return a12;
481  }
482 
483  void GeodesicExact::Lengths(const EllipticFunction& E,
484  real sig12,
485  real ssig1, real csig1, real dn1,
486  real ssig2, real csig2, real dn2,
487  real cbet1, real cbet2, unsigned outmask,
488  real& s12b, real& m12b, real& m0,
489  real& M12, real& M21) const {
490  // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
491  // and m0 = coefficient of secular term in expression for reduced length.
492 
493  outmask &= OUT_ALL;
494  // outmask & DISTANCE: set s12b
495  // outmask & REDUCEDLENGTH: set m12b & m0
496  // outmask & GEODESICSCALE: set M12 & M21
497 
498  // It's OK to have repeated dummy arguments,
499  // e.g., s12b = m0 = M12 = M21 = dummy
500 
501  if (outmask & DISTANCE)
502  // Missing a factor of _b
503  s12b = E.E() / (Math::pi() / 2) *
504  (sig12 + (E.deltaE(ssig2, csig2, dn2) - E.deltaE(ssig1, csig1, dn1)));
505  if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
506  real
507  m0x = - E.k2() * E.D() / (Math::pi() / 2),
508  J12 = m0x *
509  (sig12 + (E.deltaD(ssig2, csig2, dn2) - E.deltaD(ssig1, csig1, dn1)));
510  if (outmask & REDUCEDLENGTH) {
511  m0 = m0x;
512  // Missing a factor of _b. Add parens around (csig1 * ssig2) and
513  // (ssig1 * csig2) to ensure accurate cancellation in the case of
514  // coincident points.
515  m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
516  csig1 * csig2 * J12;
517  }
518  if (outmask & GEODESICSCALE) {
519  real csig12 = csig1 * csig2 + ssig1 * ssig2;
520  real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
521  M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
522  M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
523  }
524  }
525  }
526 
527  Math::real GeodesicExact::Astroid(real x, real y) {
528  // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
529  // This solution is adapted from Geocentric::Reverse.
530  real k;
531  real
532  p = Math::sq(x),
533  q = Math::sq(y),
534  r = (p + q - 1) / 6;
535  if ( !(q == 0 && r <= 0) ) {
536  real
537  // Avoid possible division by zero when r = 0 by multiplying equations
538  // for s and t by r^3 and r, resp.
539  S = p * q / 4, // S = r^3 * s
540  r2 = Math::sq(r),
541  r3 = r * r2,
542  // The discriminant of the quadratic equation for T3. This is zero on
543  // the evolute curve p^(1/3)+q^(1/3) = 1
544  disc = S * (S + 2 * r3);
545  real u = r;
546  if (disc >= 0) {
547  real T3 = S + r3;
548  // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
549  // of precision due to cancellation. The result is unchanged because
550  // of the way the T is used in definition of u.
551  T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
552  // N.B. cbrt always returns the real root. cbrt(-8) = -2.
553  real T = Math::cbrt(T3); // T = r * t
554  // T can be zero; but then r2 / T -> 0.
555  u += T + (T ? r2 / T : 0);
556  } else {
557  // T is complex, but the way u is defined the result is real.
558  real ang = atan2(sqrt(-disc), -(S + r3));
559  // There are three possible cube roots. We choose the root which
560  // avoids cancellation. Note that disc < 0 implies that r < 0.
561  u += 2 * r * cos(ang / 3);
562  }
563  real
564  v = sqrt(Math::sq(u) + q), // guaranteed positive
565  // Avoid loss of accuracy when u < 0.
566  uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
567  w = (uv - q) / (2 * v); // positive?
568  // Rearrange expression for k to avoid loss of accuracy due to
569  // subtraction. Division by 0 not possible because uv > 0, w >= 0.
570  k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
571  } else { // q == 0 && r <= 0
572  // y = 0 with |x| <= 1. Handle this case directly.
573  // for y small, positive root is k = abs(y)/sqrt(1-x^2)
574  k = 0;
575  }
576  return k;
577  }
578 
579  Math::real GeodesicExact::InverseStart(EllipticFunction& E,
580  real sbet1, real cbet1, real dn1,
581  real sbet2, real cbet2, real dn2,
582  real lam12,
583  real& salp1, real& calp1,
584  // Only updated if return val >= 0
585  real& salp2, real& calp2,
586  // Only updated for short lines
587  real& dnm)
588  const {
589  // Return a starting point for Newton's method in salp1 and calp1 (function
590  // value is -1). If Newton's method doesn't need to be used, return also
591  // salp2 and calp2 and function value is sig12.
592  real
593  sig12 = -1, // Return value
594  // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
595  sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
596  cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
597 #if defined(__GNUC__) && __GNUC__ == 4 && \
598  (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
599  // Volatile declaration needed to fix inverse cases
600  // 88.202499451857 0 -88.202499451857 179.981022032992859592
601  // 89.262080389218 0 -89.262080389218 179.992207982775375662
602  // 89.333123580033 0 -89.333123580032997687 179.99295812360148422
603  // which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
604  // and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw).
605  real sbet12a;
606  {
607  GEOGRAPHICLIB_VOLATILE real xx1 = sbet2 * cbet1;
608  GEOGRAPHICLIB_VOLATILE real xx2 = cbet2 * sbet1;
609  sbet12a = xx1 + xx2;
610  }
611 #else
612  real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
613 #endif
614  bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
615  cbet2 * lam12 < real(0.5);
616  real omg12 = lam12;
617  if (shortline) {
618  real sbetm2 = Math::sq(sbet1 + sbet2);
619  // sin((bet1+bet2)/2)^2
620  // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
621  sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
622  dnm = sqrt(1 + _ep2 * sbetm2);
623  omg12 /= _f1 * dnm;
624  }
625  real somg12 = sin(omg12), comg12 = cos(omg12);
626 
627  salp1 = cbet2 * somg12;
628  calp1 = comg12 >= 0 ?
629  sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
630  sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
631 
632  real
633  ssig12 = Math::hypot(salp1, calp1),
634  csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
635 
636  if (shortline && ssig12 < _etol2) {
637  // really short lines
638  salp2 = cbet1 * somg12;
639  calp2 = sbet12 - cbet1 * sbet2 *
640  (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
641  Math::norm(salp2, calp2);
642  // Set return value
643  sig12 = atan2(ssig12, csig12);
644  } else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
645  csig12 >= 0 ||
646  ssig12 >= 6 * abs(_n) * Math::pi() * Math::sq(cbet1)) {
647  // Nothing to do, zeroth order spherical approximation is OK
648  } else {
649  // Scale lam12 and bet2 to x, y coordinate system where antipodal point
650  // is at origin and singular point is at y = 0, x = -1.
651  real y, lamscale, betscale;
652  // Volatile declaration needed to fix inverse case
653  // 56.320923501171 0 -56.320923501171 179.664747671772880215
654  // which otherwise fails with g++ 4.4.4 x86 -O3
656  if (_f >= 0) { // In fact f == 0 does not get here
657  // x = dlong, y = dlat
658  {
659  real k2 = Math::sq(sbet1) * _ep2;
660  E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
661  lamscale = _e2/_f1 * cbet1 * 2 * E.H();
662  }
663  betscale = lamscale * cbet1;
664 
665  x = (lam12 - Math::pi()) / lamscale;
666  y = sbet12a / betscale;
667  } else { // _f < 0
668  // x = dlat, y = dlong
669  real
670  cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
671  bet12a = atan2(sbet12a, cbet12a);
672  real m12b, m0, dummy;
673  // In the case of lon12 = 180, this repeats a calculation made in
674  // Inverse.
675  Lengths(E, Math::pi() + bet12a,
676  sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
677  cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy);
678  x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
679  betscale = x < -real(0.01) ? sbet12a / x :
680  -_f * Math::sq(cbet1) * Math::pi();
681  lamscale = betscale / cbet1;
682  y = (lam12 - Math::pi()) / lamscale;
683  }
684 
685  if (y > -tol1_ && x > -1 - xthresh_) {
686  // strip near cut
687  // Need real(x) here to cast away the volatility of x for min/max
688  if (_f >= 0) {
689  salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 - Math::sq(salp1));
690  } else {
691  calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
692  salp1 = sqrt(1 - Math::sq(calp1));
693  }
694  } else {
695  // Estimate alp1, by solving the astroid problem.
696  //
697  // Could estimate alpha1 = theta + pi/2, directly, i.e.,
698  // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
699  // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
700  //
701  // However, it's better to estimate omg12 from astroid and use
702  // spherical formula to compute alp1. This reduces the mean number of
703  // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
704  // (min 0 max 5). The changes in the number of iterations are as
705  // follows:
706  //
707  // change percent
708  // 1 5
709  // 0 78
710  // -1 16
711  // -2 0.6
712  // -3 0.04
713  // -4 0.002
714  //
715  // The histogram of iterations is (m = number of iterations estimating
716  // alp1 directly, n = number of iterations estimating via omg12, total
717  // number of trials = 148605):
718  //
719  // iter m n
720  // 0 148 186
721  // 1 13046 13845
722  // 2 93315 102225
723  // 3 36189 32341
724  // 4 5396 7
725  // 5 455 1
726  // 6 56 0
727  //
728  // Because omg12 is near pi, estimate work with omg12a = pi - omg12
729  real k = Astroid(x, y);
730  real
731  omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
732  somg12 = sin(omg12a); comg12 = -cos(omg12a);
733  // Update spherical estimate of alp1 using omg12 instead of lam12
734  salp1 = cbet2 * somg12;
735  calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
736  }
737  }
738  // Sanity check on starting guess. Backwards check allows NaN through.
739  if (!(salp1 <= 0))
740  Math::norm(salp1, calp1);
741  else {
742  salp1 = 1; calp1 = 0;
743  }
744  return sig12;
745  }
746 
747  Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1,
748  real sbet2, real cbet2, real dn2,
749  real salp1, real calp1,
750  real& salp2, real& calp2,
751  real& sig12,
752  real& ssig1, real& csig1,
753  real& ssig2, real& csig2,
754  EllipticFunction& E,
755  real& omg12,
756  bool diffp, real& dlam12) const
757  {
758 
759  if (sbet1 == 0 && calp1 == 0)
760  // Break degeneracy of equatorial line. This case has already been
761  // handled.
762  calp1 = -tiny_;
763 
764  real
765  // sin(alp1) * cos(bet1) = sin(alp0)
766  salp0 = salp1 * cbet1,
767  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
768 
769  real somg1, comg1, somg2, comg2, cchi1, cchi2, lam12;
770  // tan(bet1) = tan(sig1) * cos(alp1)
771  // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
772  ssig1 = sbet1; somg1 = salp0 * sbet1;
773  csig1 = comg1 = calp1 * cbet1;
774  // Without normalization we have schi1 = somg1.
775  cchi1 = _f1 * dn1 * comg1;
776  Math::norm(ssig1, csig1);
777  // Math::norm(somg1, comg1); -- don't need to normalize!
778  // Math::norm(schi1, cchi1); -- don't need to normalize!
779 
780  // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
781  // about this case, since this can yield singularities in the Newton
782  // iteration.
783  // sin(alp2) * cos(bet2) = sin(alp0)
784  salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
785  // calp2 = sqrt(1 - sq(salp2))
786  // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
787  // and subst for calp0 and rearrange to give (choose positive sqrt
788  // to give alp2 in [0, pi/2]).
789  calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
790  sqrt(Math::sq(calp1 * cbet1) +
791  (cbet1 < -sbet1 ?
792  (cbet2 - cbet1) * (cbet1 + cbet2) :
793  (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
794  abs(calp1);
795  // tan(bet2) = tan(sig2) * cos(alp2)
796  // tan(omg2) = sin(alp0) * tan(sig2).
797  ssig2 = sbet2; somg2 = salp0 * sbet2;
798  csig2 = comg2 = calp2 * cbet2;
799  // Without normalization we have schi2 = somg2.
800  cchi2 = _f1 * dn2 * comg2;
801  Math::norm(ssig2, csig2);
802  // Math::norm(somg2, comg2); -- don't need to normalize!
803  // Math::norm(schi2, cchi2); -- don't need to normalize!
804 
805  // sig12 = sig2 - sig1, limit to [0, pi]
806  sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
807  csig1 * csig2 + ssig1 * ssig2);
808 
809  // omg12 = omg2 - omg1, limit to [0, pi]
810  omg12 = atan2(max(real(0), comg1 * somg2 - somg1 * comg2),
811  comg1 * comg2 + somg1 * somg2);
812  real k2 = Math::sq(calp0) * _ep2;
813  E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
814  real chi12 = atan2(max(real(0), cchi1 * somg2 - somg1 * cchi2),
815  cchi1 * cchi2 + somg1 * somg2);
816  lam12 = chi12 -
817  _e2/_f1 * salp0 * E.H() / (Math::pi() / 2) *
818  (sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1)));
819 
820  if (diffp) {
821  if (calp2 == 0)
822  dlam12 = - 2 * _f1 * dn1 / sbet1;
823  else {
824  real dummy;
825  Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
826  cbet1, cbet2, REDUCEDLENGTH,
827  dummy, dlam12, dummy, dummy, dummy);
828  dlam12 *= _f1 / (calp2 * cbet2);
829  }
830  }
831 
832  return lam12;
833  }
834 
835  void GeodesicExact::C4f(real eps, real c[]) const {
836  // Evaluate C4 coeffs
837  // Elements c[0] thru c[nC4_ - 1] are set
838  real mult = 1;
839  int o = 0;
840  for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
841  int m = nC4_ - l - 1; // order of polynomial in eps
842  c[l] = mult * Math::polyval(m, _C4x + o, eps);
843  o += m + 1;
844  mult *= eps;
845  }
846  // Post condition: o == nC4x_
847  if (!(o == nC4x_))
848  throw GeographicErr("C4 misalignment");
849  }
850 
851 } // namespace GeographicLib
static T pi()
Definition: Math.hpp:216
GeographicLib::Math::real real
Definition: GeodSolve.cpp:32
static T cbrt(T x)
Definition: Math.hpp:359
static bool isfinite(T x)
Definition: Math.hpp:768
static T LatFix(T x)
Definition: Math.hpp:482
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:102
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.hpp:559
Elliptic integrals and functions.
static void norm(T &x, T &y)
Definition: Math.hpp:398
#define GEOGRAPHICLIB_VOLATILE
Definition: Math.hpp:84
Math::real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21, real &S12) const
static T hypot(T x, T y)
Definition: Math.hpp:257
static T sq(T x)
Definition: Math.hpp:246
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:676
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:439
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
static T degree()
Definition: Math.hpp:230
static T AngDiff(T x, T y)
Definition: Math.hpp:499
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
Header for GeographicLib::GeodesicExact class.
Exception handling for GeographicLib.
Definition: Constants.hpp:386
static T AngRound(T x)
Definition: Math.hpp:530
Math::real deltaD(real sn, real cn, real dn) const
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:87
static const GeodesicExact & WGS84()