GeographicLib  1.30
Geodesic.cpp
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1 /**
2  * \file Geodesic.cpp
3  * \brief Implementation for GeographicLib::Geodesic class
4  *
5  * Copyright (c) Charles Karney (2009-2013) <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 
29 #include <GeographicLib/Geodesic.hpp>
30 #include <GeographicLib/GeodesicLine.hpp>
31 
32 #if defined(_MSC_VER)
33 // Squelch warnings about potentially uninitialized local variables
34 # pragma warning (disable: 4701)
35 #endif
36 
37 namespace GeographicLib {
38 
39  using namespace std;
40 
41  // Underflow guard. We require
42  // tiny_ * epsilon() > 0
43  // tiny_ + epsilon() == epsilon()
44  const Math::real Geodesic::tiny_ = sqrt(numeric_limits<real>::min());
45  const Math::real Geodesic::tol0_ = numeric_limits<real>::epsilon();
46  // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse case
47  // 52.784459512564 0 -52.784459512563990912 179.634407464943777557
48  // which otherwise failed for Visual Studio 10 (Release and Debug)
49  const Math::real Geodesic::tol1_ = 200 * tol0_;
50  const Math::real Geodesic::tol2_ = sqrt(tol0_);
51  // Check on bisection interval
52  const Math::real Geodesic::tolb_ = tol0_ * tol2_;
53  const Math::real Geodesic::xthresh_ = 1000 * tol2_;
54 
55  Geodesic::Geodesic(real a, real f)
56  : _a(a)
57  , _f(f <= 1 ? f : 1/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  , _c2((Math::sq(_a) + Math::sq(_b) *
64  (_e2 == 0 ? 1 :
65  (_e2 > 0 ? Math::atanh(sqrt(_e2)) : atan(sqrt(-_e2))) /
66  sqrt(abs(_e2))))/2) // authalic radius squared
67  // The sig12 threshold for "really short"
68  , _etol2(0.01 * tol2_ / max(real(0.1), sqrt(abs(_e2))))
69  {
70  if (!(Math::isfinite(_a) && _a > 0))
71  throw GeographicErr("Major radius is not positive");
72  if (!(Math::isfinite(_b) && _b > 0))
73  throw GeographicErr("Minor radius is not positive");
74  A3coeff();
75  C3coeff();
76  C4coeff();
77  }
78 
79  const Geodesic Geodesic::WGS84(Constants::WGS84_a<real>(),
80  Constants::WGS84_f<real>());
81 
82  Math::real Geodesic::SinCosSeries(bool sinp,
83  real sinx, real cosx,
84  const real c[], int n) throw() {
85  // Evaluate
86  // y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
87  // sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
88  // using Clenshaw summation. N.B. c[0] is unused for sin series
89  // Approx operation count = (n + 5) mult and (2 * n + 2) add
90  c += (n + sinp); // Point to one beyond last element
91  real
92  ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
93  y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum
94  // Now n is even
95  n /= 2;
96  while (n--) {
97  // Unroll loop x 2, so accumulators return to their original role
98  y1 = ar * y0 - y1 + *--c;
99  y0 = ar * y1 - y0 + *--c;
100  }
101  return sinp
102  ? 2 * sinx * cosx * y0 // sin(2 * x) * y0
103  : cosx * (y0 - y1); // cos(x) * (y0 - y1)
104  }
105 
106  GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1, unsigned caps)
107  const throw() {
108  return GeodesicLine(*this, lat1, lon1, azi1, caps);
109  }
110 
111  Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1,
112  bool arcmode, real s12_a12, unsigned outmask,
113  real& lat2, real& lon2, real& azi2,
114  real& s12, real& m12, real& M12, real& M21,
115  real& S12) const throw() {
116  return GeodesicLine(*this, lat1, lon1, azi1,
117  // Automatically supply DISTANCE_IN if necessary
118  outmask | (arcmode ? NONE : DISTANCE_IN))
119  . // Note the dot!
120  GenPosition(arcmode, s12_a12, outmask,
121  lat2, lon2, azi2, s12, m12, M12, M21, S12);
122  }
123 
124  Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
125  unsigned outmask,
126  real& s12, real& azi1, real& azi2,
127  real& m12, real& M12, real& M21, real& S12)
128  const throw() {
129  outmask &= OUT_ALL;
130  // Compute longitude difference (AngDiff does this carefully). Result is
131  // in [-180, 180] but -180 is only for west-going geodesics. 180 is for
132  // east-going and meridional geodesics.
133  real lon12 = Math::AngDiff(Math::AngNormalize(lon1),
134  Math::AngNormalize(lon2));
135  // If very close to being on the same half-meridian, then make it so.
136  lon12 = AngRound(lon12);
137  // Make longitude difference positive.
138  int lonsign = lon12 >= 0 ? 1 : -1;
139  lon12 *= lonsign;
140  // If really close to the equator, treat as on equator.
141  lat1 = AngRound(lat1);
142  lat2 = AngRound(lat2);
143  // Swap points so that point with higher (abs) latitude is point 1
144  int swapp = abs(lat1) >= abs(lat2) ? 1 : -1;
145  if (swapp < 0) {
146  lonsign *= -1;
147  swap(lat1, lat2);
148  }
149  // Make lat1 <= 0
150  int latsign = lat1 < 0 ? 1 : -1;
151  lat1 *= latsign;
152  lat2 *= latsign;
153  // Now we have
154  //
155  // 0 <= lon12 <= 180
156  // -90 <= lat1 <= 0
157  // lat1 <= lat2 <= -lat1
158  //
159  // longsign, swapp, latsign register the transformation to bring the
160  // coordinates to this canonical form. In all cases, 1 means no change was
161  // made. We make these transformations so that there are few cases to
162  // check, e.g., on verifying quadrants in atan2. In addition, this
163  // enforces some symmetries in the results returned.
164 
165  real phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x;
166 
167  phi = lat1 * Math::degree<real>();
168  // Ensure cbet1 = +epsilon at poles
169  sbet1 = _f1 * sin(phi);
170  cbet1 = lat1 == -90 ? tiny_ : cos(phi);
171  SinCosNorm(sbet1, cbet1);
172 
173  phi = lat2 * Math::degree<real>();
174  // Ensure cbet2 = +epsilon at poles
175  sbet2 = _f1 * sin(phi);
176  cbet2 = abs(lat2) == 90 ? tiny_ : cos(phi);
177  SinCosNorm(sbet2, cbet2);
178 
179  // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
180  // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
181  // a better measure. This logic is used in assigning calp2 in Lambda12.
182  // Sometimes these quantities vanish and in that case we force bet2 = +/-
183  // bet1 exactly. An example where is is necessary is the inverse problem
184  // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
185  // which failed with Visual Studio 10 (Release and Debug)
186 
187  if (cbet1 < -sbet1) {
188  if (cbet2 == cbet1)
189  sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
190  } else {
191  if (abs(sbet2) == -sbet1)
192  cbet2 = cbet1;
193  }
194 
195  real
196  dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
197  dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
198 
199  real
200  lam12 = lon12 * Math::degree<real>(),
201  slam12 = abs(lon12) == 180 ? 0 : sin(lam12),
202  clam12 = cos(lam12); // lon12 == 90 isn't interesting
203 
204  real a12, sig12, calp1, salp1, calp2, salp2;
205  // index zero elements of these arrays are unused
206  real C1a[nC1_ + 1], C2a[nC2_ + 1], C3a[nC3_];
207 
208  bool meridian = lat1 == -90 || slam12 == 0;
209 
210  if (meridian) {
211 
212  // Endpoints are on a single full meridian, so the geodesic might lie on
213  // a meridian.
214 
215  calp1 = clam12; salp1 = slam12; // Head to the target longitude
216  calp2 = 1; salp2 = 0; // At the target we're heading north
217 
218  real
219  // tan(bet) = tan(sig) * cos(alp)
220  ssig1 = sbet1, csig1 = calp1 * cbet1,
221  ssig2 = sbet2, csig2 = calp2 * cbet2;
222 
223  // sig12 = sig2 - sig1
224  sig12 = atan2(max(csig1 * ssig2 - ssig1 * csig2, real(0)),
225  csig1 * csig2 + ssig1 * ssig2);
226  {
227  real dummy;
228  Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
229  cbet1, cbet2, s12x, m12x, dummy,
230  (outmask & GEODESICSCALE) != 0U, M12, M21, C1a, C2a);
231  }
232  // Add the check for sig12 since zero length geodesics might yield m12 <
233  // 0. Test case was
234  //
235  // echo 20.001 0 20.001 0 | Geod -i
236  //
237  // In fact, we will have sig12 > pi/2 for meridional geodesic which is
238  // not a shortest path.
239  if (sig12 < 1 || m12x >= 0) {
240  m12x *= _b;
241  s12x *= _b;
242  a12 = sig12 / Math::degree<real>();
243  } else
244  // m12 < 0, i.e., prolate and too close to anti-podal
245  meridian = false;
246  }
247 
248  real omg12;
249  if (!meridian &&
250  sbet1 == 0 && // and sbet2 == 0
251  // Mimic the way Lambda12 works with calp1 = 0
252  (_f <= 0 || lam12 <= Math::pi<real>() - _f * Math::pi<real>())) {
253 
254  // Geodesic runs along equator
255  calp1 = calp2 = 0; salp1 = salp2 = 1;
256  s12x = _a * lam12;
257  sig12 = omg12 = lam12 / _f1;
258  m12x = _b * sin(sig12);
259  if (outmask & GEODESICSCALE)
260  M12 = M21 = cos(sig12);
261  a12 = lon12 / _f1;
262 
263  } else if (!meridian) {
264 
265  // Now point1 and point2 belong within a hemisphere bounded by a
266  // meridian and geodesic is neither meridional or equatorial.
267 
268  // Figure a starting point for Newton's method
269  sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
270  lam12,
271  salp1, calp1, salp2, calp2,
272  C1a, C2a);
273 
274  if (sig12 >= 0) {
275  // Short lines (InverseStart sets salp2, calp2)
276  real dnm = (dn1 + dn2) / 2;
277  s12x = sig12 * _b * dnm;
278  m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
279  if (outmask & GEODESICSCALE)
280  M12 = M21 = cos(sig12 / dnm);
281  a12 = sig12 / Math::degree<real>();
282  omg12 = lam12 / (_f1 * dnm);
283  } else {
284 
285  // Newton's method. This is a straightforward solution of f(alp1) =
286  // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
287  // root in the interval (0, pi) and its derivative is positive at the
288  // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
289  // alp1. During the course of the iteration, a range (alp1a, alp1b) is
290  // maintained which brackets the root and with each evaluation of
291  // f(alp) the range is shrunk, if possible. Newton's method is
292  // restarted whenever the derivative of f is negative (because the new
293  // value of alp1 is then further from the solution) or if the new
294  // estimate of alp1 lies outside (0,pi); in this case, the new starting
295  // guess is taken to be (alp1a + alp1b) / 2.
296  real ssig1, csig1, ssig2, csig2, eps;
297  unsigned numit = 0;
298  // Bracketing range
299  real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
300  for (bool tripn = false, tripb = false; numit < maxit2_; ++numit) {
301  // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
302  // WGS84 and random input: mean = 2.85, sd = 0.60
303  real dv;
304  real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
305  salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
306  eps, omg12, numit < maxit1_, dv, C1a, C2a, C3a)
307  - lam12;
308  // 2 * tol0 is approximately 1 ulp for a number in [0, pi].
309  // Reversed test to allow escape with NaNs
310  if (tripb || !(abs(v) >= (tripn ? 8 : 2) * tol0_)) break;
311  // Update bracketing values
312  if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
313  { salp1b = salp1; calp1b = calp1; }
314  else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
315  { salp1a = salp1; calp1a = calp1; }
316  if (numit < maxit1_ && dv > 0) {
317  real
318  dalp1 = -v/dv;
319  real
320  sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
321  nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
322  if (nsalp1 > 0 && abs(dalp1) < Math::pi<real>()) {
323  calp1 = calp1 * cdalp1 - salp1 * sdalp1;
324  salp1 = nsalp1;
325  SinCosNorm(salp1, calp1);
326  // In some regimes we don't get quadratic convergence because
327  // slope -> 0. So use convergence conditions based on epsilon
328  // instead of sqrt(epsilon).
329  tripn = abs(v) <= 16 * tol0_;
330  continue;
331  }
332  }
333  // Either dv was not postive or updated value was outside legal
334  // range. Use the midpoint of the bracket as the next estimate.
335  // This mechanism is not needed for the WGS84 ellipsoid, but it does
336  // catch problems with more eccentric ellipsoids. Its efficacy is
337  // such for the WGS84 test set with the starting guess set to alp1 =
338  // 90deg:
339  // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
340  // WGS84 and random input: mean = 4.74, sd = 0.99
341  salp1 = (salp1a + salp1b)/2;
342  calp1 = (calp1a + calp1b)/2;
343  SinCosNorm(salp1, calp1);
344  tripn = false;
345  tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
346  abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
347  }
348  {
349  real dummy;
350  Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
351  cbet1, cbet2, s12x, m12x, dummy,
352  (outmask & GEODESICSCALE) != 0U, M12, M21, C1a, C2a);
353  }
354  m12x *= _b;
355  s12x *= _b;
356  a12 = sig12 / Math::degree<real>();
357  omg12 = lam12 - omg12;
358  }
359  }
360 
361  if (outmask & DISTANCE)
362  s12 = 0 + s12x; // Convert -0 to 0
363 
364  if (outmask & REDUCEDLENGTH)
365  m12 = 0 + m12x; // Convert -0 to 0
366 
367  if (outmask & AREA) {
368  real
369  // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
370  salp0 = salp1 * cbet1,
371  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
372  real alp12;
373  if (calp0 != 0 && salp0 != 0) {
374  real
375  // From Lambda12: tan(bet) = tan(sig) * cos(alp)
376  ssig1 = sbet1, csig1 = calp1 * cbet1,
377  ssig2 = sbet2, csig2 = calp2 * cbet2,
378  k2 = Math::sq(calp0) * _ep2,
379  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
380  // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
381  A4 = Math::sq(_a) * calp0 * salp0 * _e2;
382  SinCosNorm(ssig1, csig1);
383  SinCosNorm(ssig2, csig2);
384  real C4a[nC4_];
385  C4f(eps, C4a);
386  real
387  B41 = SinCosSeries(false, ssig1, csig1, C4a, nC4_),
388  B42 = SinCosSeries(false, ssig2, csig2, C4a, nC4_);
389  S12 = A4 * (B42 - B41);
390  } else
391  // Avoid problems with indeterminate sig1, sig2 on equator
392  S12 = 0;
393 
394  if (!meridian &&
395  omg12 < real(0.75) * Math::pi<real>() && // Long difference too big
396  sbet2 - sbet1 < real(1.75)) { // Lat difference too big
397  // Use tan(Gamma/2) = tan(omg12/2)
398  // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
399  // with tan(x/2) = sin(x)/(1+cos(x))
400  real
401  somg12 = sin(omg12), domg12 = 1 + cos(omg12),
402  dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
403  alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
404  domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
405  } else {
406  // alp12 = alp2 - alp1, used in atan2 so no need to normalize
407  real
408  salp12 = salp2 * calp1 - calp2 * salp1,
409  calp12 = calp2 * calp1 + salp2 * salp1;
410  // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
411  // salp12 = -0 and alp12 = -180. However this depends on the sign
412  // being attached to 0 correctly. The following ensures the correct
413  // behavior.
414  if (salp12 == 0 && calp12 < 0) {
415  salp12 = tiny_ * calp1;
416  calp12 = -1;
417  }
418  alp12 = atan2(salp12, calp12);
419  }
420  S12 += _c2 * alp12;
421  S12 *= swapp * lonsign * latsign;
422  // Convert -0 to 0
423  S12 += 0;
424  }
425 
426  // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
427  if (swapp < 0) {
428  swap(salp1, salp2);
429  swap(calp1, calp2);
430  if (outmask & GEODESICSCALE)
431  swap(M12, M21);
432  }
433 
434  salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
435  salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
436 
437  if (outmask & AZIMUTH) {
438  // minus signs give range [-180, 180). 0- converts -0 to +0.
439  azi1 = 0 - atan2(-salp1, calp1) / Math::degree<real>();
440  azi2 = 0 - atan2(-salp2, calp2) / Math::degree<real>();
441  }
442 
443  // Returned value in [0, 180]
444  return a12;
445  }
446 
447  void Geodesic::Lengths(real eps, real sig12,
448  real ssig1, real csig1, real dn1,
449  real ssig2, real csig2, real dn2,
450  real cbet1, real cbet2,
451  real& s12b, real& m12b, real& m0,
452  bool scalep, real& M12, real& M21,
453  // Scratch areas of the right size
454  real C1a[], real C2a[]) const throw() {
455  // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
456  // and m0 = coefficient of secular term in expression for reduced length.
457  C1f(eps, C1a);
458  C2f(eps, C2a);
459  real
460  A1m1 = A1m1f(eps),
461  AB1 = (1 + A1m1) * (SinCosSeries(true, ssig2, csig2, C1a, nC1_) -
462  SinCosSeries(true, ssig1, csig1, C1a, nC1_)),
463  A2m1 = A2m1f(eps),
464  AB2 = (1 + A2m1) * (SinCosSeries(true, ssig2, csig2, C2a, nC2_) -
465  SinCosSeries(true, ssig1, csig1, C2a, nC2_));
466  m0 = A1m1 - A2m1;
467  real J12 = m0 * sig12 + (AB1 - AB2);
468  // Missing a factor of _b.
469  // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure accurate
470  // cancellation in the case of coincident points.
471  m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
472  // Missing a factor of _b
473  s12b = (1 + A1m1) * sig12 + AB1;
474  if (scalep) {
475  real csig12 = csig1 * csig2 + ssig1 * ssig2;
476  real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
477  M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
478  M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
479  }
480  }
481 
482  Math::real Geodesic::Astroid(real x, real y) throw() {
483  // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
484  // This solution is adapted from Geocentric::Reverse.
485  real k;
486  real
487  p = Math::sq(x),
488  q = Math::sq(y),
489  r = (p + q - 1) / 6;
490  if ( !(q == 0 && r <= 0) ) {
491  real
492  // Avoid possible division by zero when r = 0 by multiplying equations
493  // for s and t by r^3 and r, resp.
494  S = p * q / 4, // S = r^3 * s
495  r2 = Math::sq(r),
496  r3 = r * r2,
497  // The discrimant of the quadratic equation for T3. This is zero on
498  // the evolute curve p^(1/3)+q^(1/3) = 1
499  disc = S * (S + 2 * r3);
500  real u = r;
501  if (disc >= 0) {
502  real T3 = S + r3;
503  // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
504  // of precision due to cancellation. The result is unchanged because
505  // of the way the T is used in definition of u.
506  T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
507  // N.B. cbrt always returns the real root. cbrt(-8) = -2.
508  real T = Math::cbrt(T3); // T = r * t
509  // T can be zero; but then r2 / T -> 0.
510  u += T + (T != 0 ? r2 / T : 0);
511  } else {
512  // T is complex, but the way u is defined the result is real.
513  real ang = atan2(sqrt(-disc), -(S + r3));
514  // There are three possible cube roots. We choose the root which
515  // avoids cancellation. Note that disc < 0 implies that r < 0.
516  u += 2 * r * cos(ang / 3);
517  }
518  real
519  v = sqrt(Math::sq(u) + q), // guaranteed positive
520  // Avoid loss of accuracy when u < 0.
521  uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
522  w = (uv - q) / (2 * v); // positive?
523  // Rearrange expression for k to avoid loss of accuracy due to
524  // subtraction. Division by 0 not possible because uv > 0, w >= 0.
525  k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
526  } else { // q == 0 && r <= 0
527  // y = 0 with |x| <= 1. Handle this case directly.
528  // for y small, positive root is k = abs(y)/sqrt(1-x^2)
529  k = 0;
530  }
531  return k;
532  }
533 
534  Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
535  real sbet2, real cbet2, real dn2,
536  real lam12,
537  real& salp1, real& calp1,
538  // Only updated if return val >= 0
539  real& salp2, real& calp2,
540  // Scratch areas of the right size
541  real C1a[], real C2a[]) const throw() {
542  // Return a starting point for Newton's method in salp1 and calp1 (function
543  // value is -1). If Newton's method doesn't need to be used, return also
544  // salp2 and calp2 and function value is sig12.
545  real
546  sig12 = -1, // Return value
547  // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
548  sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
549  cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
550 #if defined(__GNUC__) && __GNUC__ == 4 && \
551  (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
552  // Volatile declaration needed to fix inverse cases
553  // 88.202499451857 0 -88.202499451857 179.981022032992859592
554  // 89.262080389218 0 -89.262080389218 179.992207982775375662
555  // 89.333123580033 0 -89.333123580032997687 179.99295812360148422
556  // which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
557  // and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw).
558  real sbet12a;
559  {
560  volatile real xx1 = sbet2 * cbet1;
561  volatile real xx2 = cbet2 * sbet1;
562  sbet12a = xx1 + xx2;
563  }
564 #else
565  real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
566 #endif
567  bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
568  lam12 <= Math::pi<real>() / 6;
569  real
570  omg12 = !shortline ? lam12 : lam12 / (_f1 * (dn1 + dn2) / 2),
571  somg12 = sin(omg12), comg12 = cos(omg12);
572 
573  salp1 = cbet2 * somg12;
574  calp1 = comg12 >= 0 ?
575  sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
576  sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
577 
578  real
579  ssig12 = Math::hypot(salp1, calp1),
580  csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
581 
582  if (shortline && ssig12 < _etol2) {
583  // really short lines
584  salp2 = cbet1 * somg12;
585  calp2 = sbet12 - cbet1 * sbet2 * Math::sq(somg12) / (1 + comg12);
586  SinCosNorm(salp2, calp2);
587  // Set return value
588  sig12 = atan2(ssig12, csig12);
589  } else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
590  csig12 >= 0 ||
591  ssig12 >= 6 * abs(_n) * Math::pi<real>() * Math::sq(cbet1)) {
592  // Nothing to do, zeroth order spherical approximation is OK
593  } else {
594  // Scale lam12 and bet2 to x, y coordinate system where antipodal point
595  // is at origin and singular point is at y = 0, x = -1.
596  real y, lamscale, betscale;
597  // Volatile declaration needed to fix inverse case
598  // 56.320923501171 0 -56.320923501171 179.664747671772880215
599  // which otherwise fails with g++ 4.4.4 x86 -O3
600  volatile real x;
601  if (_f >= 0) { // In fact f == 0 does not get here
602  // x = dlong, y = dlat
603  {
604  real
605  k2 = Math::sq(sbet1) * _ep2,
606  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
607  lamscale = _f * cbet1 * A3f(eps) * Math::pi<real>();
608  }
609  betscale = lamscale * cbet1;
610 
611  x = (lam12 - Math::pi<real>()) / lamscale;
612  y = sbet12a / betscale;
613  } else { // _f < 0
614  // x = dlat, y = dlong
615  real
616  cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
617  bet12a = atan2(sbet12a, cbet12a);
618  real m12b, m0, dummy;
619  // In the case of lon12 = 180, this repeats a calculation made in
620  // Inverse.
621  Lengths(_n, Math::pi<real>() + bet12a,
622  sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
623  cbet1, cbet2, dummy, m12b, m0, false,
624  dummy, dummy, C1a, C2a);
625  x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi<real>());
626  betscale = x < -real(0.01) ? sbet12a / x :
627  -_f * Math::sq(cbet1) * Math::pi<real>();
628  lamscale = betscale / cbet1;
629  y = (lam12 - Math::pi<real>()) / lamscale;
630  }
631 
632  if (y > -tol1_ && x > -1 - xthresh_) {
633  // strip near cut
634  // Need real(x) here to cast away the volatility of x for min/max
635  if (_f >= 0) {
636  salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 - Math::sq(salp1));
637  } else {
638  calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
639  salp1 = sqrt(1 - Math::sq(calp1));
640  }
641  } else {
642  // Estimate alp1, by solving the astroid problem.
643  //
644  // Could estimate alpha1 = theta + pi/2, directly, i.e.,
645  // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
646  // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
647  //
648  // However, it's better to estimate omg12 from astroid and use
649  // spherical formula to compute alp1. This reduces the mean number of
650  // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
651  // (min 0 max 5). The changes in the number of iterations are as
652  // follows:
653  //
654  // change percent
655  // 1 5
656  // 0 78
657  // -1 16
658  // -2 0.6
659  // -3 0.04
660  // -4 0.002
661  //
662  // The histogram of iterations is (m = number of iterations estimating
663  // alp1 directly, n = number of iterations estimating via omg12, total
664  // number of trials = 148605):
665  //
666  // iter m n
667  // 0 148 186
668  // 1 13046 13845
669  // 2 93315 102225
670  // 3 36189 32341
671  // 4 5396 7
672  // 5 455 1
673  // 6 56 0
674  //
675  // Because omg12 is near pi, estimate work with omg12a = pi - omg12
676  real k = Astroid(x, y);
677  real
678  omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
679  somg12 = sin(omg12a); comg12 = -cos(omg12a);
680  // Update spherical estimate of alp1 using omg12 instead of lam12
681  salp1 = cbet2 * somg12;
682  calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
683  }
684  }
685  if (salp1 > 0) // Sanity check on starting guess
686  SinCosNorm(salp1, calp1);
687  else {
688  salp1 = 1; calp1 = 0;
689  }
690  return sig12;
691  }
692 
693  Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
694  real sbet2, real cbet2, real dn2,
695  real salp1, real calp1,
696  real& salp2, real& calp2,
697  real& sig12,
698  real& ssig1, real& csig1,
699  real& ssig2, real& csig2,
700  real& eps, real& domg12,
701  bool diffp, real& dlam12,
702  // Scratch areas of the right size
703  real C1a[], real C2a[], real C3a[]) const
704  throw() {
705 
706  if (sbet1 == 0 && calp1 == 0)
707  // Break degeneracy of equatorial line. This case has already been
708  // handled.
709  calp1 = -tiny_;
710 
711  real
712  // sin(alp1) * cos(bet1) = sin(alp0)
713  salp0 = salp1 * cbet1,
714  calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
715 
716  real somg1, comg1, somg2, comg2, omg12, lam12;
717  // tan(bet1) = tan(sig1) * cos(alp1)
718  // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
719  ssig1 = sbet1; somg1 = salp0 * sbet1;
720  csig1 = comg1 = calp1 * cbet1;
721  SinCosNorm(ssig1, csig1);
722  // SinCosNorm(somg1, comg1); -- don't need to normalize!
723 
724  // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
725  // about this case, since this can yield singularities in the Newton
726  // iteration.
727  // sin(alp2) * cos(bet2) = sin(alp0)
728  salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
729  // calp2 = sqrt(1 - sq(salp2))
730  // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
731  // and subst for calp0 and rearrange to give (choose positive sqrt
732  // to give alp2 in [0, pi/2]).
733  calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
734  sqrt(Math::sq(calp1 * cbet1) +
735  (cbet1 < -sbet1 ?
736  (cbet2 - cbet1) * (cbet1 + cbet2) :
737  (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
738  abs(calp1);
739  // tan(bet2) = tan(sig2) * cos(alp2)
740  // tan(omg2) = sin(alp0) * tan(sig2).
741  ssig2 = sbet2; somg2 = salp0 * sbet2;
742  csig2 = comg2 = calp2 * cbet2;
743  SinCosNorm(ssig2, csig2);
744  // SinCosNorm(somg2, comg2); -- don't need to normalize!
745 
746  // sig12 = sig2 - sig1, limit to [0, pi]
747  sig12 = atan2(max(csig1 * ssig2 - ssig1 * csig2, real(0)),
748  csig1 * csig2 + ssig1 * ssig2);
749 
750  // omg12 = omg2 - omg1, limit to [0, pi]
751  omg12 = atan2(max(comg1 * somg2 - somg1 * comg2, real(0)),
752  comg1 * comg2 + somg1 * somg2);
753  real B312, h0;
754  real k2 = Math::sq(calp0) * _ep2;
755  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
756  C3f(eps, C3a);
757  B312 = (SinCosSeries(true, ssig2, csig2, C3a, nC3_-1) -
758  SinCosSeries(true, ssig1, csig1, C3a, nC3_-1));
759  h0 = -_f * A3f(eps);
760  domg12 = salp0 * h0 * (sig12 + B312);
761  lam12 = omg12 + domg12;
762 
763  if (diffp) {
764  if (calp2 == 0)
765  dlam12 = - 2 * _f1 * dn1 / sbet1;
766  else {
767  real dummy;
768  Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
769  cbet1, cbet2, dummy, dlam12, dummy,
770  false, dummy, dummy, C1a, C2a);
771  dlam12 *= _f1 / (calp2 * cbet2);
772  }
773  }
774 
775  return lam12;
776  }
777 
778  Math::real Geodesic::A3f(real eps) const throw() {
779  // Evaluate sum(_A3x[k] * eps^k, k, 0, nA3x_-1) by Horner's method
780  real v = 0;
781  for (int i = nA3x_; i; )
782  v = eps * v + _A3x[--i];
783  return v;
784  }
785 
786  void Geodesic::C3f(real eps, real c[]) const throw() {
787  // Evaluate C3 coeffs by Horner's method
788  // Elements c[1] thru c[nC3_ - 1] are set
789  for (int j = nC3x_, k = nC3_ - 1; k; ) {
790  real t = 0;
791  for (int i = nC3_ - k; i; --i)
792  t = eps * t + _C3x[--j];
793  c[k--] = t;
794  }
795 
796  real mult = 1;
797  for (int k = 1; k < nC3_; ) {
798  mult *= eps;
799  c[k++] *= mult;
800  }
801  }
802 
803  void Geodesic::C4f(real eps, real c[]) const throw() {
804  // Evaluate C4 coeffs by Horner's method
805  // Elements c[0] thru c[nC4_ - 1] are set
806  for (int j = nC4x_, k = nC4_; k; ) {
807  real t = 0;
808  for (int i = nC4_ - k + 1; i; --i)
809  t = eps * t + _C4x[--j];
810  c[--k] = t;
811  }
812 
813  real mult = 1;
814  for (int k = 1; k < nC4_; ) {
815  mult *= eps;
816  c[k++] *= mult;
817  }
818  }
819 
820  // Generated by Maxima on 2010-09-04 10:26:17-04:00
821 
822  // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
823  Math::real Geodesic::A1m1f(real eps) throw() {
824  real
825  eps2 = Math::sq(eps),
826  t;
827  switch (nA1_/2) {
828  case 0:
829  t = 0;
830  break;
831  case 1:
832  t = eps2/4;
833  break;
834  case 2:
835  t = eps2*(eps2+16)/64;
836  break;
837  case 3:
838  t = eps2*(eps2*(eps2+4)+64)/256;
839  break;
840  case 4:
841  t = eps2*(eps2*(eps2*(25*eps2+64)+256)+4096)/16384;
842  break;
843  default:
844  STATIC_ASSERT(nA1_ >= 0 && nA1_ <= 8, "Bad value of nA1_");
845  t = 0;
846  }
847  return (t + eps) / (1 - eps);
848  }
849 
850  // The coefficients C1[l] in the Fourier expansion of B1
851  void Geodesic::C1f(real eps, real c[]) throw() {
852  real
853  eps2 = Math::sq(eps),
854  d = eps;
855  switch (nC1_) {
856  case 0:
857  break;
858  case 1:
859  c[1] = -d/2;
860  break;
861  case 2:
862  c[1] = -d/2;
863  d *= eps;
864  c[2] = -d/16;
865  break;
866  case 3:
867  c[1] = d*(3*eps2-8)/16;
868  d *= eps;
869  c[2] = -d/16;
870  d *= eps;
871  c[3] = -d/48;
872  break;
873  case 4:
874  c[1] = d*(3*eps2-8)/16;
875  d *= eps;
876  c[2] = d*(eps2-2)/32;
877  d *= eps;
878  c[3] = -d/48;
879  d *= eps;
880  c[4] = -5*d/512;
881  break;
882  case 5:
883  c[1] = d*((6-eps2)*eps2-16)/32;
884  d *= eps;
885  c[2] = d*(eps2-2)/32;
886  d *= eps;
887  c[3] = d*(9*eps2-16)/768;
888  d *= eps;
889  c[4] = -5*d/512;
890  d *= eps;
891  c[5] = -7*d/1280;
892  break;
893  case 6:
894  c[1] = d*((6-eps2)*eps2-16)/32;
895  d *= eps;
896  c[2] = d*((64-9*eps2)*eps2-128)/2048;
897  d *= eps;
898  c[3] = d*(9*eps2-16)/768;
899  d *= eps;
900  c[4] = d*(3*eps2-5)/512;
901  d *= eps;
902  c[5] = -7*d/1280;
903  d *= eps;
904  c[6] = -7*d/2048;
905  break;
906  case 7:
907  c[1] = d*(eps2*(eps2*(19*eps2-64)+384)-1024)/2048;
908  d *= eps;
909  c[2] = d*((64-9*eps2)*eps2-128)/2048;
910  d *= eps;
911  c[3] = d*((72-9*eps2)*eps2-128)/6144;
912  d *= eps;
913  c[4] = d*(3*eps2-5)/512;
914  d *= eps;
915  c[5] = d*(35*eps2-56)/10240;
916  d *= eps;
917  c[6] = -7*d/2048;
918  d *= eps;
919  c[7] = -33*d/14336;
920  break;
921  case 8:
922  c[1] = d*(eps2*(eps2*(19*eps2-64)+384)-1024)/2048;
923  d *= eps;
924  c[2] = d*(eps2*(eps2*(7*eps2-18)+128)-256)/4096;
925  d *= eps;
926  c[3] = d*((72-9*eps2)*eps2-128)/6144;
927  d *= eps;
928  c[4] = d*((96-11*eps2)*eps2-160)/16384;
929  d *= eps;
930  c[5] = d*(35*eps2-56)/10240;
931  d *= eps;
932  c[6] = d*(9*eps2-14)/4096;
933  d *= eps;
934  c[7] = -33*d/14336;
935  d *= eps;
936  c[8] = -429*d/262144;
937  break;
938  default:
939  STATIC_ASSERT(nC1_ >= 0 && nC1_ <= 8, "Bad value of nC1_");
940  }
941  }
942 
943  // The coefficients C1p[l] in the Fourier expansion of B1p
944  void Geodesic::C1pf(real eps, real c[]) throw() {
945  real
946  eps2 = Math::sq(eps),
947  d = eps;
948  switch (nC1p_) {
949  case 0:
950  break;
951  case 1:
952  c[1] = d/2;
953  break;
954  case 2:
955  c[1] = d/2;
956  d *= eps;
957  c[2] = 5*d/16;
958  break;
959  case 3:
960  c[1] = d*(16-9*eps2)/32;
961  d *= eps;
962  c[2] = 5*d/16;
963  d *= eps;
964  c[3] = 29*d/96;
965  break;
966  case 4:
967  c[1] = d*(16-9*eps2)/32;
968  d *= eps;
969  c[2] = d*(30-37*eps2)/96;
970  d *= eps;
971  c[3] = 29*d/96;
972  d *= eps;
973  c[4] = 539*d/1536;
974  break;
975  case 5:
976  c[1] = d*(eps2*(205*eps2-432)+768)/1536;
977  d *= eps;
978  c[2] = d*(30-37*eps2)/96;
979  d *= eps;
980  c[3] = d*(116-225*eps2)/384;
981  d *= eps;
982  c[4] = 539*d/1536;
983  d *= eps;
984  c[5] = 3467*d/7680;
985  break;
986  case 6:
987  c[1] = d*(eps2*(205*eps2-432)+768)/1536;
988  d *= eps;
989  c[2] = d*(eps2*(4005*eps2-4736)+3840)/12288;
990  d *= eps;
991  c[3] = d*(116-225*eps2)/384;
992  d *= eps;
993  c[4] = d*(2695-7173*eps2)/7680;
994  d *= eps;
995  c[5] = 3467*d/7680;
996  d *= eps;
997  c[6] = 38081*d/61440;
998  break;
999  case 7:
1000  c[1] = d*(eps2*((9840-4879*eps2)*eps2-20736)+36864)/73728;
1001  d *= eps;
1002  c[2] = d*(eps2*(4005*eps2-4736)+3840)/12288;
1003  d *= eps;
1004  c[3] = d*(eps2*(8703*eps2-7200)+3712)/12288;
1005  d *= eps;
1006  c[4] = d*(2695-7173*eps2)/7680;
1007  d *= eps;
1008  c[5] = d*(41604-141115*eps2)/92160;
1009  d *= eps;
1010  c[6] = 38081*d/61440;
1011  d *= eps;
1012  c[7] = 459485*d/516096;
1013  break;
1014  case 8:
1015  c[1] = d*(eps2*((9840-4879*eps2)*eps2-20736)+36864)/73728;
1016  d *= eps;
1017  c[2] = d*(eps2*((120150-86171*eps2)*eps2-142080)+115200)/368640;
1018  d *= eps;
1019  c[3] = d*(eps2*(8703*eps2-7200)+3712)/12288;
1020  d *= eps;
1021  c[4] = d*(eps2*(1082857*eps2-688608)+258720)/737280;
1022  d *= eps;
1023  c[5] = d*(41604-141115*eps2)/92160;
1024  d *= eps;
1025  c[6] = d*(533134-2200311*eps2)/860160;
1026  d *= eps;
1027  c[7] = 459485*d/516096;
1028  d *= eps;
1029  c[8] = 109167851*d/82575360;
1030  break;
1031  default:
1032  STATIC_ASSERT(nC1p_ >= 0 && nC1p_ <= 8, "Bad value of nC1p_");
1033  }
1034  }
1035 
1036  // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
1037  Math::real Geodesic::A2m1f(real eps) throw() {
1038  real
1039  eps2 = Math::sq(eps),
1040  t;
1041  switch (nA2_/2) {
1042  case 0:
1043  t = 0;
1044  break;
1045  case 1:
1046  t = eps2/4;
1047  break;
1048  case 2:
1049  t = eps2*(9*eps2+16)/64;
1050  break;
1051  case 3:
1052  t = eps2*(eps2*(25*eps2+36)+64)/256;
1053  break;
1054  case 4:
1055  t = eps2*(eps2*(eps2*(1225*eps2+1600)+2304)+4096)/16384;
1056  break;
1057  default:
1058  STATIC_ASSERT(nA2_ >= 0 && nA2_ <= 8, "Bad value of nA2_");
1059  t = 0;
1060  }
1061  return t * (1 - eps) - eps;
1062  }
1063 
1064  // The coefficients C2[l] in the Fourier expansion of B2
1065  void Geodesic::C2f(real eps, real c[]) throw() {
1066  real
1067  eps2 = Math::sq(eps),
1068  d = eps;
1069  switch (nC2_) {
1070  case 0:
1071  break;
1072  case 1:
1073  c[1] = d/2;
1074  break;
1075  case 2:
1076  c[1] = d/2;
1077  d *= eps;
1078  c[2] = 3*d/16;
1079  break;
1080  case 3:
1081  c[1] = d*(eps2+8)/16;
1082  d *= eps;
1083  c[2] = 3*d/16;
1084  d *= eps;
1085  c[3] = 5*d/48;
1086  break;
1087  case 4:
1088  c[1] = d*(eps2+8)/16;
1089  d *= eps;
1090  c[2] = d*(eps2+6)/32;
1091  d *= eps;
1092  c[3] = 5*d/48;
1093  d *= eps;
1094  c[4] = 35*d/512;
1095  break;
1096  case 5:
1097  c[1] = d*(eps2*(eps2+2)+16)/32;
1098  d *= eps;
1099  c[2] = d*(eps2+6)/32;
1100  d *= eps;
1101  c[3] = d*(15*eps2+80)/768;
1102  d *= eps;
1103  c[4] = 35*d/512;
1104  d *= eps;
1105  c[5] = 63*d/1280;
1106  break;
1107  case 6:
1108  c[1] = d*(eps2*(eps2+2)+16)/32;
1109  d *= eps;
1110  c[2] = d*(eps2*(35*eps2+64)+384)/2048;
1111  d *= eps;
1112  c[3] = d*(15*eps2+80)/768;
1113  d *= eps;
1114  c[4] = d*(7*eps2+35)/512;
1115  d *= eps;
1116  c[5] = 63*d/1280;
1117  d *= eps;
1118  c[6] = 77*d/2048;
1119  break;
1120  case 7:
1121  c[1] = d*(eps2*(eps2*(41*eps2+64)+128)+1024)/2048;
1122  d *= eps;
1123  c[2] = d*(eps2*(35*eps2+64)+384)/2048;
1124  d *= eps;
1125  c[3] = d*(eps2*(69*eps2+120)+640)/6144;
1126  d *= eps;
1127  c[4] = d*(7*eps2+35)/512;
1128  d *= eps;
1129  c[5] = d*(105*eps2+504)/10240;
1130  d *= eps;
1131  c[6] = 77*d/2048;
1132  d *= eps;
1133  c[7] = 429*d/14336;
1134  break;
1135  case 8:
1136  c[1] = d*(eps2*(eps2*(41*eps2+64)+128)+1024)/2048;
1137  d *= eps;
1138  c[2] = d*(eps2*(eps2*(47*eps2+70)+128)+768)/4096;
1139  d *= eps;
1140  c[3] = d*(eps2*(69*eps2+120)+640)/6144;
1141  d *= eps;
1142  c[4] = d*(eps2*(133*eps2+224)+1120)/16384;
1143  d *= eps;
1144  c[5] = d*(105*eps2+504)/10240;
1145  d *= eps;
1146  c[6] = d*(33*eps2+154)/4096;
1147  d *= eps;
1148  c[7] = 429*d/14336;
1149  d *= eps;
1150  c[8] = 6435*d/262144;
1151  break;
1152  default:
1153  STATIC_ASSERT(nC2_ >= 0 && nC2_ <= 8, "Bad value of nC2_");
1154  }
1155  }
1156 
1157  // The scale factor A3 = mean value of (d/dsigma)I3
1158  void Geodesic::A3coeff() throw() {
1159  switch (nA3_) {
1160  case 0:
1161  break;
1162  case 1:
1163  _A3x[0] = 1;
1164  break;
1165  case 2:
1166  _A3x[0] = 1;
1167  _A3x[1] = -1/real(2);
1168  break;
1169  case 3:
1170  _A3x[0] = 1;
1171  _A3x[1] = (_n-1)/2;
1172  _A3x[2] = -1/real(4);
1173  break;
1174  case 4:
1175  _A3x[0] = 1;
1176  _A3x[1] = (_n-1)/2;
1177  _A3x[2] = (-_n-2)/8;
1178  _A3x[3] = -1/real(16);
1179  break;
1180  case 5:
1181  _A3x[0] = 1;
1182  _A3x[1] = (_n-1)/2;
1183  _A3x[2] = (_n*(3*_n-1)-2)/8;
1184  _A3x[3] = (-3*_n-1)/16;
1185  _A3x[4] = -3/real(64);
1186  break;
1187  case 6:
1188  _A3x[0] = 1;
1189  _A3x[1] = (_n-1)/2;
1190  _A3x[2] = (_n*(3*_n-1)-2)/8;
1191  _A3x[3] = ((-_n-3)*_n-1)/16;
1192  _A3x[4] = (-2*_n-3)/64;
1193  _A3x[5] = -3/real(128);
1194  break;
1195  case 7:
1196  _A3x[0] = 1;
1197  _A3x[1] = (_n-1)/2;
1198  _A3x[2] = (_n*(3*_n-1)-2)/8;
1199  _A3x[3] = (_n*(_n*(5*_n-1)-3)-1)/16;
1200  _A3x[4] = ((-10*_n-2)*_n-3)/64;
1201  _A3x[5] = (-5*_n-3)/128;
1202  _A3x[6] = -5/real(256);
1203  break;
1204  case 8:
1205  _A3x[0] = 1;
1206  _A3x[1] = (_n-1)/2;
1207  _A3x[2] = (_n*(3*_n-1)-2)/8;
1208  _A3x[3] = (_n*(_n*(5*_n-1)-3)-1)/16;
1209  _A3x[4] = (_n*((-5*_n-20)*_n-4)-6)/128;
1210  _A3x[5] = ((-5*_n-10)*_n-6)/256;
1211  _A3x[6] = (-15*_n-20)/1024;
1212  _A3x[7] = -25/real(2048);
1213  break;
1214  default:
1215  STATIC_ASSERT(nA3_ >= 0 && nA3_ <= 8, "Bad value of nA3_");
1216  }
1217  }
1218 
1219  // The coefficients C3[l] in the Fourier expansion of B3
1220  void Geodesic::C3coeff() throw() {
1221  switch (nC3_) {
1222  case 0:
1223  break;
1224  case 1:
1225  break;
1226  case 2:
1227  _C3x[0] = 1/real(4);
1228  break;
1229  case 3:
1230  _C3x[0] = (1-_n)/4;
1231  _C3x[1] = 1/real(8);
1232  _C3x[2] = 1/real(16);
1233  break;
1234  case 4:
1235  _C3x[0] = (1-_n)/4;
1236  _C3x[1] = 1/real(8);
1237  _C3x[2] = 3/real(64);
1238  _C3x[3] = (2-3*_n)/32;
1239  _C3x[4] = 3/real(64);
1240  _C3x[5] = 5/real(192);
1241  break;
1242  case 5:
1243  _C3x[0] = (1-_n)/4;
1244  _C3x[1] = (1-_n*_n)/8;
1245  _C3x[2] = (3*_n+3)/64;
1246  _C3x[3] = 5/real(128);
1247  _C3x[4] = ((_n-3)*_n+2)/32;
1248  _C3x[5] = (3-2*_n)/64;
1249  _C3x[6] = 3/real(128);
1250  _C3x[7] = (5-9*_n)/192;
1251  _C3x[8] = 3/real(128);
1252  _C3x[9] = 7/real(512);
1253  break;
1254  case 6:
1255  _C3x[0] = (1-_n)/4;
1256  _C3x[1] = (1-_n*_n)/8;
1257  _C3x[2] = ((3-_n)*_n+3)/64;
1258  _C3x[3] = (2*_n+5)/128;
1259  _C3x[4] = 3/real(128);
1260  _C3x[5] = ((_n-3)*_n+2)/32;
1261  _C3x[6] = ((-3*_n-2)*_n+3)/64;
1262  _C3x[7] = (_n+3)/128;
1263  _C3x[8] = 5/real(256);
1264  _C3x[9] = (_n*(5*_n-9)+5)/192;
1265  _C3x[10] = (9-10*_n)/384;
1266  _C3x[11] = 7/real(512);
1267  _C3x[12] = (7-14*_n)/512;
1268  _C3x[13] = 7/real(512);
1269  _C3x[14] = 21/real(2560);
1270  break;
1271  case 7:
1272  _C3x[0] = (1-_n)/4;
1273  _C3x[1] = (1-_n*_n)/8;
1274  _C3x[2] = (_n*((-5*_n-1)*_n+3)+3)/64;
1275  _C3x[3] = (_n*(2*_n+2)+5)/128;
1276  _C3x[4] = (11*_n+12)/512;
1277  _C3x[5] = 21/real(1024);
1278  _C3x[6] = ((_n-3)*_n+2)/32;
1279  _C3x[7] = (_n*(_n*(2*_n-3)-2)+3)/64;
1280  _C3x[8] = ((2-9*_n)*_n+6)/256;
1281  _C3x[9] = (_n+5)/256;
1282  _C3x[10] = 27/real(2048);
1283  _C3x[11] = (_n*((5-_n)*_n-9)+5)/192;
1284  _C3x[12] = ((-6*_n-10)*_n+9)/384;
1285  _C3x[13] = (21-4*_n)/1536;
1286  _C3x[14] = 3/real(256);
1287  _C3x[15] = (_n*(10*_n-14)+7)/512;
1288  _C3x[16] = (7-10*_n)/512;
1289  _C3x[17] = 9/real(1024);
1290  _C3x[18] = (21-45*_n)/2560;
1291  _C3x[19] = 9/real(1024);
1292  _C3x[20] = 11/real(2048);
1293  break;
1294  case 8:
1295  _C3x[0] = (1-_n)/4;
1296  _C3x[1] = (1-_n*_n)/8;
1297  _C3x[2] = (_n*((-5*_n-1)*_n+3)+3)/64;
1298  _C3x[3] = (_n*((2-2*_n)*_n+2)+5)/128;
1299  _C3x[4] = (_n*(3*_n+11)+12)/512;
1300  _C3x[5] = (10*_n+21)/1024;
1301  _C3x[6] = 243/real(16384);
1302  _C3x[7] = ((_n-3)*_n+2)/32;
1303  _C3x[8] = (_n*(_n*(2*_n-3)-2)+3)/64;
1304  _C3x[9] = (_n*((-6*_n-9)*_n+2)+6)/256;
1305  _C3x[10] = ((1-2*_n)*_n+5)/256;
1306  _C3x[11] = (69*_n+108)/8192;
1307  _C3x[12] = 187/real(16384);
1308  _C3x[13] = (_n*((5-_n)*_n-9)+5)/192;
1309  _C3x[14] = (_n*(_n*(10*_n-6)-10)+9)/384;
1310  _C3x[15] = ((-77*_n-8)*_n+42)/3072;
1311  _C3x[16] = (12-_n)/1024;
1312  _C3x[17] = 139/real(16384);
1313  _C3x[18] = (_n*((20-7*_n)*_n-28)+14)/1024;
1314  _C3x[19] = ((-7*_n-40)*_n+28)/2048;
1315  _C3x[20] = (72-43*_n)/8192;
1316  _C3x[21] = 127/real(16384);
1317  _C3x[22] = (_n*(75*_n-90)+42)/5120;
1318  _C3x[23] = (9-15*_n)/1024;
1319  _C3x[24] = 99/real(16384);
1320  _C3x[25] = (44-99*_n)/8192;
1321  _C3x[26] = 99/real(16384);
1322  _C3x[27] = 429/real(114688);
1323  break;
1324  default:
1325  STATIC_ASSERT(nC3_ >= 0 && nC3_ <= 8, "Bad value of nC3_");
1326  }
1327  }
1328 
1329  // Generated by Maxima on 2012-10-19 08:02:34-04:00
1330 
1331  // The coefficients C4[l] in the Fourier expansion of I4
1332  void Geodesic::C4coeff() throw() {
1333  switch (nC4_) {
1334  case 0:
1335  break;
1336  case 1:
1337  _C4x[0] = 2/real(3);
1338  break;
1339  case 2:
1340  _C4x[0] = (10-4*_n)/15;
1341  _C4x[1] = -1/real(5);
1342  _C4x[2] = 1/real(45);
1343  break;
1344  case 3:
1345  _C4x[0] = (_n*(8*_n-28)+70)/105;
1346  _C4x[1] = (16*_n-7)/35;
1347  _C4x[2] = -2/real(105);
1348  _C4x[3] = (7-16*_n)/315;
1349  _C4x[4] = -2/real(105);
1350  _C4x[5] = 4/real(525);
1351  break;
1352  case 4:
1353  _C4x[0] = (_n*(_n*(4*_n+24)-84)+210)/315;
1354  _C4x[1] = ((48-32*_n)*_n-21)/105;
1355  _C4x[2] = (-32*_n-6)/315;
1356  _C4x[3] = 11/real(315);
1357  _C4x[4] = (_n*(32*_n-48)+21)/945;
1358  _C4x[5] = (64*_n-18)/945;
1359  _C4x[6] = -1/real(105);
1360  _C4x[7] = (12-32*_n)/1575;
1361  _C4x[8] = -8/real(1575);
1362  _C4x[9] = 8/real(2205);
1363  break;
1364  case 5:
1365  _C4x[0] = (_n*(_n*(_n*(16*_n+44)+264)-924)+2310)/3465;
1366  _C4x[1] = (_n*(_n*(48*_n-352)+528)-231)/1155;
1367  _C4x[2] = (_n*(1088*_n-352)-66)/3465;
1368  _C4x[3] = (121-368*_n)/3465;
1369  _C4x[4] = 4/real(1155);
1370  _C4x[5] = (_n*((352-48*_n)*_n-528)+231)/10395;
1371  _C4x[6] = ((704-896*_n)*_n-198)/10395;
1372  _C4x[7] = (80*_n-99)/10395;
1373  _C4x[8] = 4/real(1155);
1374  _C4x[9] = (_n*(320*_n-352)+132)/17325;
1375  _C4x[10] = (384*_n-88)/17325;
1376  _C4x[11] = -8/real(1925);
1377  _C4x[12] = (88-256*_n)/24255;
1378  _C4x[13] = -16/real(8085);
1379  _C4x[14] = 64/real(31185);
1380  break;
1381  case 6:
1382  _C4x[0] = (_n*(_n*(_n*(_n*(100*_n+208)+572)+3432)-12012)+30030)/45045;
1383  _C4x[1] = (_n*(_n*(_n*(64*_n+624)-4576)+6864)-3003)/15015;
1384  _C4x[2] = (_n*((14144-10656*_n)*_n-4576)-858)/45045;
1385  _C4x[3] = ((-224*_n-4784)*_n+1573)/45045;
1386  _C4x[4] = (1088*_n+156)/45045;
1387  _C4x[5] = 97/real(15015);
1388  _C4x[6] = (_n*(_n*((-64*_n-624)*_n+4576)-6864)+3003)/135135;
1389  _C4x[7] = (_n*(_n*(5952*_n-11648)+9152)-2574)/135135;
1390  _C4x[8] = (_n*(5792*_n+1040)-1287)/135135;
1391  _C4x[9] = (468-2944*_n)/135135;
1392  _C4x[10] = 1/real(9009);
1393  _C4x[11] = (_n*((4160-1440*_n)*_n-4576)+1716)/225225;
1394  _C4x[12] = ((4992-8448*_n)*_n-1144)/225225;
1395  _C4x[13] = (1856*_n-936)/225225;
1396  _C4x[14] = 8/real(10725);
1397  _C4x[15] = (_n*(3584*_n-3328)+1144)/315315;
1398  _C4x[16] = (1024*_n-208)/105105;
1399  _C4x[17] = -136/real(63063);
1400  _C4x[18] = (832-2560*_n)/405405;
1401  _C4x[19] = -128/real(135135);
1402  _C4x[20] = 128/real(99099);
1403  break;
1404  case 7:
1405  _C4x[0] = (_n*(_n*(_n*(_n*(_n*(56*_n+100)+208)+572)+3432)-12012)+30030)/
1406  45045;
1407  _C4x[1] = (_n*(_n*(_n*(_n*(16*_n+64)+624)-4576)+6864)-3003)/15015;
1408  _C4x[2] = (_n*(_n*(_n*(1664*_n-10656)+14144)-4576)-858)/45045;
1409  _C4x[3] = (_n*(_n*(10736*_n-224)-4784)+1573)/45045;
1410  _C4x[4] = ((1088-4480*_n)*_n+156)/45045;
1411  _C4x[5] = (291-464*_n)/45045;
1412  _C4x[6] = 10/real(9009);
1413  _C4x[7] = (_n*(_n*(_n*((-16*_n-64)*_n-624)+4576)-6864)+3003)/135135;
1414  _C4x[8] = (_n*(_n*((5952-768*_n)*_n-11648)+9152)-2574)/135135;
1415  _C4x[9] = (_n*((5792-10704*_n)*_n+1040)-1287)/135135;
1416  _C4x[10] = (_n*(3840*_n-2944)+468)/135135;
1417  _C4x[11] = (112*_n+15)/135135;
1418  _C4x[12] = 10/real(9009);
1419  _C4x[13] = (_n*(_n*(_n*(128*_n-1440)+4160)-4576)+1716)/225225;
1420  _C4x[14] = (_n*(_n*(6784*_n-8448)+4992)-1144)/225225;
1421  _C4x[15] = (_n*(1664*_n+1856)-936)/225225;
1422  _C4x[16] = (168-1664*_n)/225225;
1423  _C4x[17] = -4/real(25025);
1424  _C4x[18] = (_n*((3584-1792*_n)*_n-3328)+1144)/315315;
1425  _C4x[19] = ((1024-2048*_n)*_n-208)/105105;
1426  _C4x[20] = (1792*_n-680)/315315;
1427  _C4x[21] = 64/real(315315);
1428  _C4x[22] = (_n*(3072*_n-2560)+832)/405405;
1429  _C4x[23] = (2048*_n-384)/405405;
1430  _C4x[24] = -512/real(405405);
1431  _C4x[25] = (640-2048*_n)/495495;
1432  _C4x[26] = -256/real(495495);
1433  _C4x[27] = 512/real(585585);
1434  break;
1435  case 8:
1436  _C4x[0] = (_n*(_n*(_n*(_n*(_n*(_n*(588*_n+952)+1700)+3536)+9724)+58344)-
1437  204204)+510510)/765765;
1438  _C4x[1] = (_n*(_n*(_n*(_n*(_n*(96*_n+272)+1088)+10608)-77792)+116688)-
1439  51051)/255255;
1440  _C4x[2] = (_n*(_n*(_n*(_n*(3232*_n+28288)-181152)+240448)-77792)-14586)/
1441  765765;
1442  _C4x[3] = (_n*(_n*((182512-154048*_n)*_n-3808)-81328)+26741)/765765;
1443  _C4x[4] = (_n*(_n*(12480*_n-76160)+18496)+2652)/765765;
1444  _C4x[5] = (_n*(20960*_n-7888)+4947)/765765;
1445  _C4x[6] = (4192*_n+850)/765765;
1446  _C4x[7] = 193/real(85085);
1447  _C4x[8] = (_n*(_n*(_n*(_n*((-96*_n-272)*_n-1088)-10608)+77792)-116688)+
1448  51051)/2297295;
1449  _C4x[9] = (_n*(_n*(_n*((-1344*_n-13056)*_n+101184)-198016)+155584)-43758)/
1450  2297295;
1451  _C4x[10] = (_n*(_n*(_n*(103744*_n-181968)+98464)+17680)-21879)/2297295;
1452  _C4x[11] = (_n*(_n*(52608*_n+65280)-50048)+7956)/2297295;
1453  _C4x[12] = ((1904-39840*_n)*_n+255)/2297295;
1454  _C4x[13] = (510-1472*_n)/459459;
1455  _C4x[14] = 349/real(2297295);
1456  _C4x[15] = (_n*(_n*(_n*(_n*(160*_n+2176)-24480)+70720)-77792)+29172)/
1457  3828825;
1458  _C4x[16] = (_n*(_n*((115328-41472*_n)*_n-143616)+84864)-19448)/3828825;
1459  _C4x[17] = (_n*((28288-126528*_n)*_n+31552)-15912)/3828825;
1460  _C4x[18] = (_n*(64256*_n-28288)+2856)/3828825;
1461  _C4x[19] = (-928*_n-612)/3828825;
1462  _C4x[20] = 464/real(1276275);
1463  _C4x[21] = (_n*(_n*(_n*(7168*_n-30464)+60928)-56576)+19448)/5360355;
1464  _C4x[22] = (_n*(_n*(35840*_n-34816)+17408)-3536)/1786785;
1465  _C4x[23] = ((30464-2560*_n)*_n-11560)/5360355;
1466  _C4x[24] = (1088-16384*_n)/5360355;
1467  _C4x[25] = -16/real(97461);
1468  _C4x[26] = (_n*((52224-32256*_n)*_n-43520)+14144)/6891885;
1469  _C4x[27] = ((34816-77824*_n)*_n-6528)/6891885;
1470  _C4x[28] = (26624*_n-8704)/6891885;
1471  _C4x[29] = 128/real(2297295);
1472  _C4x[30] = (_n*(45056*_n-34816)+10880)/8423415;
1473  _C4x[31] = (24576*_n-4352)/8423415;
1474  _C4x[32] = -6784/real(8423415);
1475  _C4x[33] = (8704-28672*_n)/9954945;
1476  _C4x[34] = -1024/real(3318315);
1477  _C4x[35] = 1024/real(1640925);
1478  break;
1479  default:
1480  STATIC_ASSERT(nC4_ >= 0 && nC4_ <= 8, "Bad value of nC4_");
1481  }
1482  }
1483 
1484 } // namespace GeographicLib
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