GeographicLib 2.7

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Geodesic.cpp
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1/**
2 * \file Geodesic.cpp
3 * \brief Implementation for GeographicLib::Geodesic class
4 *
5 * Copyright (c) Charles Karney (2009-2025) <karney@alum.mit.edu> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 *
9 * This is a reformulation of the geodesic problem. The notation is as
10 * follows:
11 * - at a general point (no suffix or 1 or 2 as suffix)
12 * - phi = latitude
13 * - beta = latitude on auxiliary sphere
14 * - omega = longitude on auxiliary sphere
15 * - lambda = longitude
16 * - alpha = azimuth of great circle
17 * - sigma = arc length along great circle
18 * - s = distance
19 * - tau = scaled distance (= sigma at multiples of pi/2)
20 * - at northwards equator crossing
21 * - beta = phi = 0
22 * - omega = lambda = 0
23 * - alpha = alpha0
24 * - sigma = s = 0
25 * - a 12 suffix means a difference, e.g., s12 = s2 - s1.
26 * - s and c prefixes mean sin and cos
27 **********************************************************************/
28
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 Geodesic::Geodesic(real a, real f, bool exact)
42 : maxit2_(maxit1_ + Math::digits() + 10)
43 // Underflow guard. We require
44 // tiny_ * epsilon() > 0
45 // tiny_ + epsilon() == epsilon()
46 , tiny_(sqrt(numeric_limits<real>::min()))
47 , tol0_(numeric_limits<real>::epsilon())
48 // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
49 // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
50 // which otherwise failed for Visual Studio 10 (Release and Debug)
51 , tol1_(200 * tol0_)
52 , tol2_(sqrt(tol0_))
53 , tolb_(tol0_) // Check on bisection interval
54 , xthresh_(1000 * tol2_)
55 , _a(a)
56 , _f(f)
57 , _exact(exact)
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 Math::eatanhe(real(1), (_f < 0 ? -1 : 1) * sqrt(fabs(_e2))) / _e2))
66 / 2) // authalic radius squared
67 // The sig12 threshold for "really short". Using the auxiliary sphere
68 // solution with dnm computed at (bet1 + bet2) / 2, the relative error in
69 // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
70 // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a
71 // given f and sig12, the max error occurs for lines near the pole. If
72 // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
73 // increases by a factor of 2.) Setting this equal to epsilon gives
74 // sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100)
75 // and max(0.001, abs(f)) stops etol2 getting too large in the nearly
76 // spherical case.
77 , _etol2(real(0.1) * tol2_ /
78 sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 ))
79 , _geodexact(_exact ? GeodesicExact(a, f) : GeodesicExact())
80 {
81 if (_exact)
82 _c2 = _geodexact._c2;
83 else {
84 if (!(isfinite(_a) && _a > 0))
85 throw GeographicErr("Equatorial radius is not positive");
86 if (!(isfinite(_b) && _b > 0))
87 throw GeographicErr("Polar semiaxis is not positive");
88 A3coeff();
89 C3coeff();
90 C4coeff();
91 }
92 }
93
94 const Geodesic& Geodesic::WGS84() {
95 static const Geodesic wgs84(Constants::WGS84_a(), Constants::WGS84_f());
96 return wgs84;
97 }
98
99 Math::real Geodesic::SinCosSeries(bool sinp,
100 real sinx, real cosx,
101 const real c[], int n) {
102 // Evaluate
103 // y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
104 // sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
105 // using Clenshaw summation. N.B. c[0] is unused for sin series
106 // Approx operation count = (n + 5) mult and (2 * n + 2) add
107 c += (n + sinp); // Point to one beyond last element
108 real
109 ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
110 y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum
111 // Now n is even
112 n /= 2;
113 while (n--) {
114 // Unroll loop x 2, so accumulators return to their original role
115 y1 = ar * y0 - y1 + *--c;
116 y0 = ar * y1 - y0 + *--c;
117 }
118 return sinp
119 ? 2 * sinx * cosx * y0 // sin(2 * x) * y0
120 : cosx * (y0 - y1); // cos(x) * (y0 - y1)
121 }
122
123 GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1,
124 unsigned caps) const {
125 return GeodesicLine(*this, lat1, lon1, azi1, caps);
126 }
127
128 Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1,
129 bool arcmode, real s12_a12, unsigned outmask,
130 real& lat2, real& lon2, real& azi2,
131 real& s12, real& m12, real& M12, real& M21,
132 real& S12) const {
133 if (_exact)
134 return _geodexact.GenDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask,
135 lat2, lon2, azi2,
136 s12, m12, M12, M21, S12);
137 // Automatically supply DISTANCE_IN if necessary
138 if (!arcmode) outmask |= DISTANCE_IN;
139 return GeodesicLine(*this, lat1, lon1, azi1, outmask)
140 . // Note the dot!
141 GenPosition(arcmode, s12_a12, outmask,
142 lat2, lon2, azi2, s12, m12, M12, M21, S12);
143 }
144
145 GeodesicLine Geodesic::GenDirectLine(real lat1, real lon1, real azi1,
146 bool arcmode, real s12_a12,
147 unsigned caps) const {
148 azi1 = Math::AngNormalize(azi1);
149 real salp1, calp1;
150 // Guard against underflow in salp0. Also -0 is converted to +0.
151 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
152 // Automatically supply DISTANCE_IN if necessary
153 if (!arcmode) caps |= DISTANCE_IN;
154 return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1,
155 caps, arcmode, s12_a12);
156 }
157
158 GeodesicLine Geodesic::DirectLine(real lat1, real lon1, real azi1, real s12,
159 unsigned caps) const {
160 return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
161 }
162
163 GeodesicLine Geodesic::ArcDirectLine(real lat1, real lon1, real azi1,
164 real a12, unsigned caps) const {
165 return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
166 }
167
168 Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
169 unsigned outmask, real& s12,
170 real& salp1, real& calp1,
171 real& salp2, real& calp2,
172 real& m12, real& M12, real& M21,
173 real& S12) const {
174 if (_exact)
175 return _geodexact.GenInverse(lat1, lon1, lat2, lon2,
176 outmask, s12,
177 salp1, calp1, salp2, calp2,
178 m12, M12, M21, S12);
179 // Compute longitude difference (AngDiff does this carefully).
180 real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
181 // Make longitude difference positive.
182 int lonsign = signbit(lon12) ? -1 : 1;
183 lon12 *= lonsign; lon12s *= lonsign;
184 real
185 lam12 = lon12 * Math::degree(),
186 slam12, clam12;
187 // Calculate sincos of lon12 + error (this applies AngRound internally).
188 Math::sincosde(lon12, lon12s, slam12, clam12);
189 // the supplementary longitude difference
190 lon12s = (Math::hd - lon12) - lon12s;
191
192 // If really close to the equator, treat as on equator.
193 lat1 = Math::AngRound(Math::LatFix(lat1));
194 lat2 = Math::AngRound(Math::LatFix(lat2));
195 // Swap points so that point with higher (abs) latitude is point 1.
196 // If one latitude is a nan, then it becomes lat1.
197 int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1;
198 if (swapp < 0) {
199 lonsign *= -1;
200 swap(lat1, lat2);
201 }
202 // Make lat1 <= -0
203 int latsign = signbit(lat1) ? 1 : -1;
204 lat1 *= latsign;
205 lat2 *= latsign;
206 // Now we have
207 //
208 // 0 <= lon12 <= 180
209 // -90 <= lat1 <= -0
210 // lat1 <= lat2 <= -lat1
211 //
212 // longsign, swapp, latsign register the transformation to bring the
213 // coordinates to this canonical form. In all cases, 1 means no change was
214 // made. We make these transformations so that there are few cases to
215 // check, e.g., on verifying quadrants in atan2. In addition, this
216 // enforces some symmetries in the results returned.
217
218 real sbet1, cbet1, sbet2, cbet2, s12x, m12x = Math::NaN();
219
220 Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
221 // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
222 // will be <= 2*tiny for two points at the same pole.
223 Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
224
225 Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
226 // Ensure cbet2 = +epsilon at poles
227 Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2);
228
229 // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
230 // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
231 // a better measure. This logic is used in assigning calp2 in Lambda12.
232 // Sometimes these quantities vanish and in that case we force bet2 = +/-
233 // bet1 exactly. An example where is is necessary is the inverse problem
234 // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
235 // which failed with Visual Studio 10 (Release and Debug)
236
237 if (cbet1 < -sbet1) {
238 if (cbet2 == cbet1)
239 sbet2 = copysign(sbet1, sbet2);
240 } else {
241 if (fabs(sbet2) == -sbet1)
242 cbet2 = cbet1;
243 }
244
245 real
246 dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
247 dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
248
249 real a12, sig12;
250 // index zero element of this array is unused
251 real Ca[nC_];
252
253 bool meridian = lat1 == -Math::qd || slam12 == 0;
254
255 if (meridian) {
256
257 // Endpoints are on a single full meridian, so the geodesic might lie on
258 // a meridian.
259
260 calp1 = clam12; salp1 = slam12; // Head to the target longitude
261 calp2 = 1; salp2 = 0; // At the target we're heading north
262
263 real
264 // tan(bet) = tan(sig) * cos(alp)
265 ssig1 = sbet1, csig1 = calp1 * cbet1,
266 ssig2 = sbet2, csig2 = calp2 * cbet2;
267
268 // sig12 = sig2 - sig1
269 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
270 csig1 * csig2 + ssig1 * ssig2);
271 {
272 real dummy;
273 Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
274 outmask | DISTANCE | REDUCEDLENGTH,
275 s12x, m12x, dummy, M12, M21, Ca);
276 }
277 // Add the check for sig12 since zero length geodesics might yield m12 <
278 // 0. Test case was
279 //
280 // echo 20.001 0 20.001 0 | GeodSolve -i
281 if (sig12 < tol2_ || m12x >= 0) {
282 // Need at least 2, to handle 90 0 90 180
283 if (sig12 < 3 * tiny_ ||
284 // Prevent negative s12 or m12 for short lines
285 (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
286 sig12 = m12x = s12x = 0;
287 m12x *= _b;
288 s12x *= _b;
289 a12 = sig12 / Math::degree();
290 } else
291 // m12 < 0, i.e., prolate and too close to anti-podal
292 meridian = false;
293 }
294
295 // somg12 == 2 marks that it needs to be calculated
296 real omg12 = 0, somg12 = 2, comg12 = 0;
297 if (!meridian &&
298 sbet1 == 0 && // and sbet2 == 0
299 (_f <= 0 || lon12s >= _f * Math::hd)) {
300
301 // Geodesic runs along equator
302 calp1 = calp2 = 0; salp1 = salp2 = 1;
303 s12x = _a * lam12;
304 sig12 = omg12 = lam12 / _f1;
305 m12x = _b * sin(sig12);
306 if (outmask & GEODESICSCALE)
307 M12 = M21 = cos(sig12);
308 a12 = lon12 / _f1;
309
310 } else if (!meridian) {
311
312 // Now point1 and point2 belong within a hemisphere bounded by a
313 // meridian and geodesic is neither meridional or equatorial.
314
315 // Figure a starting point for Newton's method
316 real dnm;
317 sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
318 lam12, slam12, clam12,
319 salp1, calp1, salp2, calp2, dnm,
320 Ca);
321
322 if (sig12 >= 0) {
323 // Short lines (InverseStart sets salp2, calp2, dnm)
324 s12x = sig12 * _b * dnm;
325 m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
326 if (outmask & GEODESICSCALE)
327 M12 = M21 = cos(sig12 / dnm);
328 a12 = sig12 / Math::degree();
329 omg12 = lam12 / (_f1 * dnm);
330 } else {
331
332 // Newton's method. This is a straightforward solution of f(alp1) =
333 // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
334 // root in the interval (0, pi) and its derivative is positive at the
335 // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
336 // alp1. During the course of the iteration, a range (alp1a, alp1b) is
337 // maintained which brackets the root and with each evaluation of
338 // f(alp) the range is shrunk, if possible. Newton's method is
339 // restarted whenever the derivative of f is negative (because the new
340 // value of alp1 is then further from the solution) or if the new
341 // estimate of alp1 lies outside (0,pi); in this case, the new starting
342 // guess is taken to be (alp1a + alp1b) / 2.
343 //
344 // initial values to suppress warnings (if loop is executed 0 times)
345 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
346 unsigned numit = 0;
347 // Bracketing range
348 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
349 for (bool tripn = false, tripb = false;; ++numit) {
350 // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
351 // WGS84 and random input: mean = 2.85, sd = 0.60
352 real dv;
353 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
354 slam12, clam12,
355 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
356 eps, domg12, numit < maxit1_, dv, Ca);
357 if (tripb ||
358 // Reversed test to allow escape with NaNs
359 !(fabs(v) >= (tripn ? 8 : 1) * tol0_) ||
360 // Enough bisections to get accurate result
361 numit == maxit2_)
362 break;
363 // Update bracketing values
364 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
365 { salp1b = salp1; calp1b = calp1; }
366 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
367 { salp1a = salp1; calp1a = calp1; }
368 if (numit < maxit1_ && dv > 0) {
369 real
370 dalp1 = -v/dv;
371 // |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION
372 // = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages
373 // (because of the need to do accurate range reduction).
374 if (fabs(dalp1) < Math::pi()) {
375 real
376 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
377 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
378 if (nsalp1 > 0) {
379 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
380 salp1 = nsalp1;
381 Math::norm(salp1, calp1);
382 // In some regimes we don't get quadratic convergence because
383 // slope -> 0. So use convergence conditions based on epsilon
384 // instead of sqrt(epsilon).
385 tripn = fabs(v) <= 16 * tol0_;
386 continue;
387 }
388 }
389 }
390 // Either dv was not positive or updated value was outside legal
391 // range. Use the midpoint of the bracket as the next estimate.
392 // This mechanism is not needed for the WGS84 ellipsoid, but it does
393 // catch problems with more eccentric ellipsoids. Its efficacy is
394 // such for the WGS84 test set with the starting guess set to alp1 =
395 // 90deg:
396 // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
397 // WGS84 and random input: mean = 4.74, sd = 0.99
398 salp1 = (salp1a + salp1b)/2;
399 calp1 = (calp1a + calp1b)/2;
400 Math::norm(salp1, calp1);
401 tripn = false;
402 tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
403 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
404 }
405 {
406 real dummy;
407 // Ensure that the reduced length and geodesic scale are computed in
408 // a "canonical" way, with the I2 integral.
409 unsigned lengthmask = outmask |
410 (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE);
411 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
412 cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
413 }
414 m12x *= _b;
415 s12x *= _b;
416 a12 = sig12 / Math::degree();
417 if (outmask & AREA) {
418 // omg12 = lam12 - domg12
419 real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
420 somg12 = slam12 * cdomg12 - clam12 * sdomg12;
421 comg12 = clam12 * cdomg12 + slam12 * sdomg12;
422 }
423 }
424 }
425
426 if (outmask & DISTANCE)
427 s12 = real(0) + s12x; // Convert -0 to 0
428
429 if (outmask & REDUCEDLENGTH)
430 m12 = real(0) + m12x; // Convert -0 to 0
431
432 if (outmask & AREA) {
433 real
434 // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
435 salp0 = salp1 * cbet1,
436 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
437 real alp12;
438 if (calp0 != 0 && salp0 != 0) {
439 real
440 // From Lambda12: tan(bet) = tan(sig) * cos(alp)
441 ssig1 = sbet1, csig1 = calp1 * cbet1,
442 ssig2 = sbet2, csig2 = calp2 * cbet2,
443 k2 = Math::sq(calp0) * _ep2,
444 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
445 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
446 A4 = Math::sq(_a) * calp0 * salp0 * _e2;
447 Math::norm(ssig1, csig1);
448 Math::norm(ssig2, csig2);
449 C4f(eps, Ca);
450 real
451 B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
452 B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
453 S12 = A4 * (B42 - B41);
454 } else
455 // Avoid problems with indeterminate sig1, sig2 on equator
456 S12 = 0;
457 if (!meridian && somg12 == 2) {
458 somg12 = sin(omg12); comg12 = cos(omg12);
459 }
460
461 if (!meridian &&
462 // omg12 < 3/4 * pi
463 comg12 > -real(0.7071) && // Long difference not too big
464 sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
465 // Use tan(Gamma/2) = tan(omg12/2)
466 // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
467 // with tan(x/2) = sin(x)/(1+cos(x))
468 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
469 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
470 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
471 } else {
472 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
473 real
474 salp12 = salp2 * calp1 - calp2 * salp1,
475 calp12 = calp2 * calp1 + salp2 * salp1;
476 // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
477 // salp12 = -0 and alp12 = -180. However this depends on the sign
478 // being attached to 0 correctly. The following ensures the correct
479 // behavior.
480 if (salp12 == 0 && calp12 < 0) {
481 salp12 = tiny_ * calp1;
482 calp12 = -1;
483 }
484 alp12 = atan2(salp12, calp12);
485 }
486 S12 += _c2 * alp12;
487 S12 *= swapp * lonsign * latsign;
488 // Convert -0 to 0
489 S12 += 0;
490 }
491
492 // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
493 if (swapp < 0) {
494 swap(salp1, salp2);
495 swap(calp1, calp2);
496 if (outmask & GEODESICSCALE)
497 swap(M12, M21);
498 }
499
500 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
501 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
502 // Returned value in [0, 180]
503 return a12;
504 }
505
506 Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
507 unsigned outmask,
508 real& s12, real& azi1, real& azi2,
509 real& m12, real& M12, real& M21,
510 real& S12) const {
511 outmask &= OUT_MASK;
512 real salp1, calp1, salp2, calp2,
513 a12 = GenInverse(lat1, lon1, lat2, lon2,
514 outmask, s12, salp1, calp1, salp2, calp2,
515 m12, M12, M21, S12);
516 if (outmask & AZIMUTH) {
517 azi1 = Math::atan2d(salp1, calp1);
518 azi2 = Math::atan2d(salp2, calp2);
519 }
520 return a12;
521 }
522
523 GeodesicLine Geodesic::InverseLine(real lat1, real lon1,
524 real lat2, real lon2,
525 unsigned caps) const {
526 real t, salp1, calp1, salp2, calp2,
527 a12 = GenInverse(lat1, lon1, lat2, lon2,
528 // No need to specify AZIMUTH here
529 0u, t, salp1, calp1, salp2, calp2,
530 t, t, t, t),
531 azi1 = Math::atan2d(salp1, calp1);
532 // Ensure that a12 can be converted to a distance
533 if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
534 return
535 GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12);
536 }
537
538 void Geodesic::Lengths(real eps, real sig12,
539 real ssig1, real csig1, real dn1,
540 real ssig2, real csig2, real dn2,
541 real cbet1, real cbet2, unsigned outmask,
542 real& s12b, real& m12b, real& m0,
543 real& M12, real& M21,
544 // Scratch area of the right size
545 real Ca[]) const {
546 // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
547 // and m0 = coefficient of secular term in expression for reduced length.
548
549 outmask &= OUT_MASK;
550 // outmask & DISTANCE: set s12b
551 // outmask & REDUCEDLENGTH: set m12b & m0
552 // outmask & GEODESICSCALE: set M12 & M21
553
554 real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
555 real Cb[nC2_ + 1];
556 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
557 A1 = A1m1f(eps);
558 C1f(eps, Ca);
559 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
560 A2 = A2m1f(eps);
561 C2f(eps, Cb);
562 m0x = A1 - A2;
563 A2 = 1 + A2;
564 }
565 A1 = 1 + A1;
566 }
567 if (outmask & DISTANCE) {
568 real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
569 SinCosSeries(true, ssig1, csig1, Ca, nC1_);
570 // Missing a factor of _b
571 s12b = A1 * (sig12 + B1);
572 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
573 real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
574 SinCosSeries(true, ssig1, csig1, Cb, nC2_);
575 J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
576 }
577 } else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
578 // Assume here that nC1_ >= nC2_
579 for (int l = 1; l <= nC2_; ++l)
580 Cb[l] = A1 * Ca[l] - A2 * Cb[l];
581 J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
582 SinCosSeries(true, ssig1, csig1, Cb, nC2_));
583 }
584 if (outmask & REDUCEDLENGTH) {
585 m0 = m0x;
586 // Missing a factor of _b.
587 // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
588 // accurate cancellation in the case of coincident points.
589 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
590 csig1 * csig2 * J12;
591 }
592 if (outmask & GEODESICSCALE) {
593 real csig12 = csig1 * csig2 + ssig1 * ssig2;
594 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
595 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
596 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
597 }
598 }
599
600 Math::real Geodesic::Astroid(real x, real y) {
601 // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
602 // This solution is adapted from Geocentric::Reverse.
603 real k;
604 real
605 p = Math::sq(x),
606 q = Math::sq(y),
607 r = (p + q - 1) / 6;
608 if ( !(q == 0 && r <= 0) ) {
609 real
610 // Avoid possible division by zero when r = 0 by multiplying equations
611 // for s and t by r^3 and r, resp.
612 S = p * q / 4, // S = r^3 * s
613 r2 = Math::sq(r),
614 r3 = r * r2,
615 // The discriminant of the quadratic equation for T3. This is zero on
616 // the evolute curve p^(1/3)+q^(1/3) = 1
617 disc = S * (S + 2 * r3);
618 real u = r;
619 if (disc >= 0) {
620 real T3 = S + r3;
621 // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
622 // of precision due to cancellation. The result is unchanged because
623 // of the way the T is used in definition of u.
624 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
625 // N.B. cbrt always returns the real root. cbrt(-8) = -2.
626 real T = cbrt(T3); // T = r * t
627 // T can be zero; but then r2 / T -> 0.
628 u += T + (T != 0 ? r2 / T : 0);
629 } else {
630 // T is complex, but the way u is defined the result is real.
631 real ang = atan2(sqrt(-disc), -(S + r3));
632 // There are three possible cube roots. We choose the root which
633 // avoids cancellation. Note that disc < 0 implies that r < 0.
634 u += 2 * r * cos(ang / 3);
635 }
636 real
637 v = sqrt(Math::sq(u) + q), // guaranteed positive
638 // Avoid loss of accuracy when u < 0.
639 uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
640 w = (uv - q) / (2 * v); // positive?
641 // Rearrange expression for k to avoid loss of accuracy due to
642 // subtraction. Division by 0 not possible because uv > 0, w >= 0.
643 k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
644 } else { // q == 0 && r <= 0
645 // y = 0 with |x| <= 1. Handle this case directly.
646 // for y small, positive root is k = abs(y)/sqrt(1-x^2)
647 k = 0;
648 }
649 return k;
650 }
651
652 Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
653 real sbet2, real cbet2, real dn2,
654 real lam12, real slam12, real clam12,
655 real& salp1, real& calp1,
656 // Only updated if return val >= 0
657 real& salp2, real& calp2,
658 // Only updated for short lines
659 real& dnm,
660 // Scratch area of the right size
661 real Ca[]) const {
662 // Return a starting point for Newton's method in salp1 and calp1 (function
663 // value is -1). If Newton's method doesn't need to be used, return also
664 // salp2 and calp2 and function value is sig12.
665 real
666 sig12 = -1, // Return value
667 // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
668 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
669 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
670 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
671 bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
672 cbet2 * lam12 < real(0.5);
673 real somg12, comg12;
674 if (shortline) {
675 real sbetm2 = Math::sq(sbet1 + sbet2);
676 // sin((bet1+bet2)/2)^2
677 // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
678 sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
679 dnm = sqrt(1 + _ep2 * sbetm2);
680 real omg12 = lam12 / (_f1 * dnm);
681 somg12 = sin(omg12); comg12 = cos(omg12);
682 } else {
683 somg12 = slam12; comg12 = clam12;
684 }
685
686 salp1 = cbet2 * somg12;
687 calp1 = comg12 >= 0 ?
688 sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
689 sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
690
691 real
692 ssig12 = hypot(salp1, calp1),
693 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
694
695 if (shortline && ssig12 < _etol2) {
696 // really short lines
697 salp2 = cbet1 * somg12;
698 calp2 = sbet12 - cbet1 * sbet2 *
699 (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
700 Math::norm(salp2, calp2);
701 // Set return value
702 sig12 = atan2(ssig12, csig12);
703 } else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric
704 csig12 >= 0 ||
705 ssig12 >= 6 * fabs(_n) * Math::pi() * Math::sq(cbet1)) {
706 // Nothing to do, zeroth order spherical approximation is OK
707 } else {
708 // Scale lam12 and bet2 to x, y coordinate system where antipodal point
709 // is at origin and singular point is at y = 0, x = -1.
710 real x, y, lamscale, betscale;
711 real lam12x = atan2(-slam12, -clam12); // lam12 - pi
712 if (_f >= 0) { // In fact f == 0 does not get here
713 // x = dlong, y = dlat
714 {
715 real
716 k2 = Math::sq(sbet1) * _ep2,
717 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
718 lamscale = _f * cbet1 * A3f(eps) * Math::pi();
719 }
720 betscale = lamscale * cbet1;
721
722 x = lam12x / lamscale;
723 y = sbet12a / betscale;
724 } else { // _f < 0
725 // x = dlat, y = dlong
726 real
727 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
728 bet12a = atan2(sbet12a, cbet12a);
729 real m12b, m0, dummy;
730 // In the case of lon12 = 180, this repeats a calculation made in
731 // Inverse.
732 Lengths(_n, Math::pi() + bet12a,
733 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
734 cbet1, cbet2,
735 REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca);
736 x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
737 betscale = x < -real(0.01) ? sbet12a / x :
738 -_f * Math::sq(cbet1) * Math::pi();
739 lamscale = betscale / cbet1;
740 y = lam12x / lamscale;
741 }
742
743 if (y > -tol1_ && x > -1 - xthresh_) {
744 // strip near cut
745 // Need real(x) here to cast away the volatility of x for min/max
746 if (_f >= 0) {
747 salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1));
748 } else {
749 calp1 = fmax(real(x > -tol1_ ? 0 : -1), x);
750 salp1 = sqrt(1 - Math::sq(calp1));
751 }
752 } else {
753 // Estimate alp1, by solving the astroid problem.
754 //
755 // Could estimate alpha1 = theta + pi/2, directly, i.e.,
756 // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
757 // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
758 //
759 // However, it's better to estimate omg12 from astroid and use
760 // spherical formula to compute alp1. This reduces the mean number of
761 // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
762 // (min 0 max 5). The changes in the number of iterations are as
763 // follows:
764 //
765 // change percent
766 // 1 5
767 // 0 78
768 // -1 16
769 // -2 0.6
770 // -3 0.04
771 // -4 0.002
772 //
773 // The histogram of iterations is (m = number of iterations estimating
774 // alp1 directly, n = number of iterations estimating via omg12, total
775 // number of trials = 148605):
776 //
777 // iter m n
778 // 0 148 186
779 // 1 13046 13845
780 // 2 93315 102225
781 // 3 36189 32341
782 // 4 5396 7
783 // 5 455 1
784 // 6 56 0
785 //
786 // Because omg12 is near pi, estimate work with omg12a = pi - omg12
787 real k = Astroid(x, y);
788 real
789 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
790 somg12 = sin(omg12a); comg12 = -cos(omg12a);
791 // Update spherical estimate of alp1 using omg12 instead of lam12
792 salp1 = cbet2 * somg12;
793 calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
794 }
795 }
796 // Sanity check on starting guess. Backwards check allows NaN through.
797 if (!(salp1 <= 0))
798 Math::norm(salp1, calp1);
799 else {
800 salp1 = 1; calp1 = 0;
801 }
802 return sig12;
803 }
804
805 Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
806 real sbet2, real cbet2, real dn2,
807 real salp1, real calp1,
808 real slam120, real clam120,
809 real& salp2, real& calp2,
810 real& sig12,
811 real& ssig1, real& csig1,
812 real& ssig2, real& csig2,
813 real& eps, real& domg12,
814 bool diffp, real& dlam12,
815 // Scratch area of the right size
816 real Ca[]) const {
817
818 if (sbet1 == 0 && calp1 == 0)
819 // Break degeneracy of equatorial line. This case has already been
820 // handled.
821 calp1 = -tiny_;
822
823 real
824 // sin(alp1) * cos(bet1) = sin(alp0)
825 salp0 = salp1 * cbet1,
826 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
827
828 real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
829 // tan(bet1) = tan(sig1) * cos(alp1)
830 // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
831 ssig1 = sbet1; somg1 = salp0 * sbet1;
832 csig1 = comg1 = calp1 * cbet1;
833 Math::norm(ssig1, csig1);
834 // Math::norm(somg1, comg1); -- don't need to normalize!
835
836 // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
837 // about this case, since this can yield singularities in the Newton
838 // iteration.
839 // sin(alp2) * cos(bet2) = sin(alp0)
840 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
841 // calp2 = sqrt(1 - sq(salp2))
842 // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
843 // and subst for calp0 and rearrange to give (choose positive sqrt
844 // to give alp2 in [0, pi/2]).
845 calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
846 sqrt(Math::sq(calp1 * cbet1) +
847 (cbet1 < -sbet1 ?
848 (cbet2 - cbet1) * (cbet1 + cbet2) :
849 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
850 fabs(calp1);
851 // tan(bet2) = tan(sig2) * cos(alp2)
852 // tan(omg2) = sin(alp0) * tan(sig2).
853 ssig2 = sbet2; somg2 = salp0 * sbet2;
854 csig2 = comg2 = calp2 * cbet2;
855 Math::norm(ssig2, csig2);
856 // Math::norm(somg2, comg2); -- don't need to normalize!
857
858 // sig12 = sig2 - sig1, limit to [0, pi]
859 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
860 csig1 * csig2 + ssig1 * ssig2);
861
862 // omg12 = omg2 - omg1, limit to [0, pi]
863 somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2) + real(0);
864 comg12 = comg1 * comg2 + somg1 * somg2;
865 // eta = omg12 - lam120
866 real eta = atan2(somg12 * clam120 - comg12 * slam120,
867 comg12 * clam120 + somg12 * slam120);
868 real B312;
869 real k2 = Math::sq(calp0) * _ep2;
870 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
871 C3f(eps, Ca);
872 B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) -
873 SinCosSeries(true, ssig1, csig1, Ca, nC3_-1));
874 domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
875 lam12 = eta + domg12;
876
877 if (diffp) {
878 if (calp2 == 0)
879 dlam12 = - 2 * _f1 * dn1 / sbet1;
880 else {
881 real dummy;
882 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
883 cbet1, cbet2, REDUCEDLENGTH,
884 dummy, dlam12, dummy, dummy, dummy, Ca);
885 dlam12 *= _f1 / (calp2 * cbet2);
886 }
887 }
888
889 return lam12;
890 }
891
892 Math::real Geodesic::A3f(real eps) const {
893 // Evaluate A3
894 return Math::polyval(nA3_ - 1, _aA3x, eps);
895 }
896
897 void Geodesic::C3f(real eps, real c[]) const {
898 // Evaluate C3 coeffs
899 // Elements c[1] thru c[nC3_ - 1] are set
900 real mult = 1;
901 int o = 0;
902 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
903 int m = nC3_ - l - 1; // order of polynomial in eps
904 mult *= eps;
905 c[l] = mult * Math::polyval(m, _cC3x + o, eps);
906 o += m + 1;
907 }
908 // Post condition: o == nC3x_
909 }
910
911 void Geodesic::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, _cC4x + o, eps);
919 o += m + 1;
920 mult *= eps;
921 }
922 // Post condition: o == nC4x_
923 }
924
925 // The static const coefficient arrays in the following functions are
926 // generated by Maxima and give the coefficients of the Taylor expansions for
927 // the geodesics. The convention on the order of these coefficients is as
928 // follows:
929 //
930 // ascending order in the trigonometric expansion,
931 // then powers of eps in descending order,
932 // finally powers of n in descending order.
933 //
934 // (For some expansions, only a subset of levels occur.) For each polynomial
935 // of order n at the lowest level, the (n+1) coefficients of the polynomial
936 // are followed by a divisor which is applied to the whole polynomial. In
937 // this way, the coefficients are expressible with no round off error. The
938 // sizes of the coefficient arrays are:
939 //
940 // A1m1f, A2m1f = floor(N/2) + 2
941 // C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
942 // C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
943 // C4coeff = N * (N + 1) * (N + 5) / 6
944 //
945 // where N = GEOGRAPHICLIB_GEODESIC_ORDER
946 // = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
947
948 // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
949 Math::real Geodesic::A1m1f(real eps) {
950 // Generated by Maxima on 2015-05-05 18:08:12-04:00
951#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
952 static const real coeff[] = {
953 // (1-eps)*A1-1, polynomial in eps2 of order 1
954 1, 0, 4,
955 };
956#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
957 static const real coeff[] = {
958 // (1-eps)*A1-1, polynomial in eps2 of order 2
959 1, 16, 0, 64,
960 };
961#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
962 static const real coeff[] = {
963 // (1-eps)*A1-1, polynomial in eps2 of order 3
964 1, 4, 64, 0, 256,
965 };
966#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
967 static const real coeff[] = {
968 // (1-eps)*A1-1, polynomial in eps2 of order 4
969 25, 64, 256, 4096, 0, 16384,
970 };
971#else
972#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
973#endif
974 static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2,
975 "Coefficient array size mismatch in A1m1f");
976 int m = nA1_/2;
977 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
978 return (t + eps) / (1 - eps);
979 }
980
981 // The coefficients C1[l] in the Fourier expansion of B1
982 void Geodesic::C1f(real eps, real c[]) {
983 // Generated by Maxima on 2015-05-05 18:08:12-04:00
984#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
985 static const real coeff[] = {
986 // C1[1]/eps^1, polynomial in eps2 of order 1
987 3, -8, 16,
988 // C1[2]/eps^2, polynomial in eps2 of order 0
989 -1, 16,
990 // C1[3]/eps^3, polynomial in eps2 of order 0
991 -1, 48,
992 };
993#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
994 static const real coeff[] = {
995 // C1[1]/eps^1, polynomial in eps2 of order 1
996 3, -8, 16,
997 // C1[2]/eps^2, polynomial in eps2 of order 1
998 1, -2, 32,
999 // C1[3]/eps^3, polynomial in eps2 of order 0
1000 -1, 48,
1001 // C1[4]/eps^4, polynomial in eps2 of order 0
1002 -5, 512,
1003 };
1004#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1005 static const real coeff[] = {
1006 // C1[1]/eps^1, polynomial in eps2 of order 2
1007 -1, 6, -16, 32,
1008 // C1[2]/eps^2, polynomial in eps2 of order 1
1009 1, -2, 32,
1010 // C1[3]/eps^3, polynomial in eps2 of order 1
1011 9, -16, 768,
1012 // C1[4]/eps^4, polynomial in eps2 of order 0
1013 -5, 512,
1014 // C1[5]/eps^5, polynomial in eps2 of order 0
1015 -7, 1280,
1016 };
1017#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1018 static const real coeff[] = {
1019 // C1[1]/eps^1, polynomial in eps2 of order 2
1020 -1, 6, -16, 32,
1021 // C1[2]/eps^2, polynomial in eps2 of order 2
1022 -9, 64, -128, 2048,
1023 // C1[3]/eps^3, polynomial in eps2 of order 1
1024 9, -16, 768,
1025 // C1[4]/eps^4, polynomial in eps2 of order 1
1026 3, -5, 512,
1027 // C1[5]/eps^5, polynomial in eps2 of order 0
1028 -7, 1280,
1029 // C1[6]/eps^6, polynomial in eps2 of order 0
1030 -7, 2048,
1031 };
1032#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1033 static const real coeff[] = {
1034 // C1[1]/eps^1, polynomial in eps2 of order 3
1035 19, -64, 384, -1024, 2048,
1036 // C1[2]/eps^2, polynomial in eps2 of order 2
1037 -9, 64, -128, 2048,
1038 // C1[3]/eps^3, polynomial in eps2 of order 2
1039 -9, 72, -128, 6144,
1040 // C1[4]/eps^4, polynomial in eps2 of order 1
1041 3, -5, 512,
1042 // C1[5]/eps^5, polynomial in eps2 of order 1
1043 35, -56, 10240,
1044 // C1[6]/eps^6, polynomial in eps2 of order 0
1045 -7, 2048,
1046 // C1[7]/eps^7, polynomial in eps2 of order 0
1047 -33, 14336,
1048 };
1049#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1050 static const real coeff[] = {
1051 // C1[1]/eps^1, polynomial in eps2 of order 3
1052 19, -64, 384, -1024, 2048,
1053 // C1[2]/eps^2, polynomial in eps2 of order 3
1054 7, -18, 128, -256, 4096,
1055 // C1[3]/eps^3, polynomial in eps2 of order 2
1056 -9, 72, -128, 6144,
1057 // C1[4]/eps^4, polynomial in eps2 of order 2
1058 -11, 96, -160, 16384,
1059 // C1[5]/eps^5, polynomial in eps2 of order 1
1060 35, -56, 10240,
1061 // C1[6]/eps^6, polynomial in eps2 of order 1
1062 9, -14, 4096,
1063 // C1[7]/eps^7, polynomial in eps2 of order 0
1064 -33, 14336,
1065 // C1[8]/eps^8, polynomial in eps2 of order 0
1066 -429, 262144,
1067 };
1068#else
1069#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1070#endif
1071 static_assert(sizeof(coeff) / sizeof(real) ==
1072 (nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4,
1073 "Coefficient array size mismatch in C1f");
1074 real
1075 eps2 = Math::sq(eps),
1076 d = eps;
1077 int o = 0;
1078 for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
1079 int m = (nC1_ - l) / 2; // order of polynomial in eps^2
1080 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1081 o += m + 2;
1082 d *= eps;
1083 }
1084 // Post condition: o == sizeof(coeff) / sizeof(real)
1085 }
1086
1087 // The coefficients C1p[l] in the Fourier expansion of B1p
1088 void Geodesic::C1pf(real eps, real c[]) {
1089 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1090#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1091 static const real coeff[] = {
1092 // C1p[1]/eps^1, polynomial in eps2 of order 1
1093 -9, 16, 32,
1094 // C1p[2]/eps^2, polynomial in eps2 of order 0
1095 5, 16,
1096 // C1p[3]/eps^3, polynomial in eps2 of order 0
1097 29, 96,
1098 };
1099#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1100 static const real coeff[] = {
1101 // C1p[1]/eps^1, polynomial in eps2 of order 1
1102 -9, 16, 32,
1103 // C1p[2]/eps^2, polynomial in eps2 of order 1
1104 -37, 30, 96,
1105 // C1p[3]/eps^3, polynomial in eps2 of order 0
1106 29, 96,
1107 // C1p[4]/eps^4, polynomial in eps2 of order 0
1108 539, 1536,
1109 };
1110#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1111 static const real coeff[] = {
1112 // C1p[1]/eps^1, polynomial in eps2 of order 2
1113 205, -432, 768, 1536,
1114 // C1p[2]/eps^2, polynomial in eps2 of order 1
1115 -37, 30, 96,
1116 // C1p[3]/eps^3, polynomial in eps2 of order 1
1117 -225, 116, 384,
1118 // C1p[4]/eps^4, polynomial in eps2 of order 0
1119 539, 1536,
1120 // C1p[5]/eps^5, polynomial in eps2 of order 0
1121 3467, 7680,
1122 };
1123#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1124 static const real coeff[] = {
1125 // C1p[1]/eps^1, polynomial in eps2 of order 2
1126 205, -432, 768, 1536,
1127 // C1p[2]/eps^2, polynomial in eps2 of order 2
1128 4005, -4736, 3840, 12288,
1129 // C1p[3]/eps^3, polynomial in eps2 of order 1
1130 -225, 116, 384,
1131 // C1p[4]/eps^4, polynomial in eps2 of order 1
1132 -7173, 2695, 7680,
1133 // C1p[5]/eps^5, polynomial in eps2 of order 0
1134 3467, 7680,
1135 // C1p[6]/eps^6, polynomial in eps2 of order 0
1136 38081, 61440,
1137 };
1138#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1139 static const real coeff[] = {
1140 // C1p[1]/eps^1, polynomial in eps2 of order 3
1141 -4879, 9840, -20736, 36864, 73728,
1142 // C1p[2]/eps^2, polynomial in eps2 of order 2
1143 4005, -4736, 3840, 12288,
1144 // C1p[3]/eps^3, polynomial in eps2 of order 2
1145 8703, -7200, 3712, 12288,
1146 // C1p[4]/eps^4, polynomial in eps2 of order 1
1147 -7173, 2695, 7680,
1148 // C1p[5]/eps^5, polynomial in eps2 of order 1
1149 -141115, 41604, 92160,
1150 // C1p[6]/eps^6, polynomial in eps2 of order 0
1151 38081, 61440,
1152 // C1p[7]/eps^7, polynomial in eps2 of order 0
1153 459485, 516096,
1154 };
1155#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1156 static const real coeff[] = {
1157 // C1p[1]/eps^1, polynomial in eps2 of order 3
1158 -4879, 9840, -20736, 36864, 73728,
1159 // C1p[2]/eps^2, polynomial in eps2 of order 3
1160 -86171, 120150, -142080, 115200, 368640,
1161 // C1p[3]/eps^3, polynomial in eps2 of order 2
1162 8703, -7200, 3712, 12288,
1163 // C1p[4]/eps^4, polynomial in eps2 of order 2
1164 1082857, -688608, 258720, 737280,
1165 // C1p[5]/eps^5, polynomial in eps2 of order 1
1166 -141115, 41604, 92160,
1167 // C1p[6]/eps^6, polynomial in eps2 of order 1
1168 -2200311, 533134, 860160,
1169 // C1p[7]/eps^7, polynomial in eps2 of order 0
1170 459485, 516096,
1171 // C1p[8]/eps^8, polynomial in eps2 of order 0
1172 109167851, 82575360,
1173 };
1174#else
1175#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1176#endif
1177 static_assert(sizeof(coeff) / sizeof(real) ==
1178 (nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4,
1179 "Coefficient array size mismatch in C1pf");
1180 real
1181 eps2 = Math::sq(eps),
1182 d = eps;
1183 int o = 0;
1184 for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
1185 int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
1186 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1187 o += m + 2;
1188 d *= eps;
1189 }
1190 // Post condition: o == sizeof(coeff) / sizeof(real)
1191 }
1192
1193 // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
1194 Math::real Geodesic::A2m1f(real eps) {
1195 // Generated by Maxima on 2015-05-29 08:09:47-04:00
1196#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
1197 static const real coeff[] = {
1198 // (eps+1)*A2-1, polynomial in eps2 of order 1
1199 -3, 0, 4,
1200 }; // count = 3
1201#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
1202 static const real coeff[] = {
1203 // (eps+1)*A2-1, polynomial in eps2 of order 2
1204 -7, -48, 0, 64,
1205 }; // count = 4
1206#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
1207 static const real coeff[] = {
1208 // (eps+1)*A2-1, polynomial in eps2 of order 3
1209 -11, -28, -192, 0, 256,
1210 }; // count = 5
1211#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
1212 static const real coeff[] = {
1213 // (eps+1)*A2-1, polynomial in eps2 of order 4
1214 -375, -704, -1792, -12288, 0, 16384,
1215 }; // count = 6
1216#else
1217#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1218#endif
1219 static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2,
1220 "Coefficient array size mismatch in A2m1f");
1221 int m = nA2_/2;
1222 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
1223 return (t - eps) / (1 + eps);
1224 }
1225
1226 // The coefficients C2[l] in the Fourier expansion of B2
1227 void Geodesic::C2f(real eps, real c[]) {
1228 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1229#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1230 static const real coeff[] = {
1231 // C2[1]/eps^1, polynomial in eps2 of order 1
1232 1, 8, 16,
1233 // C2[2]/eps^2, polynomial in eps2 of order 0
1234 3, 16,
1235 // C2[3]/eps^3, polynomial in eps2 of order 0
1236 5, 48,
1237 };
1238#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1239 static const real coeff[] = {
1240 // C2[1]/eps^1, polynomial in eps2 of order 1
1241 1, 8, 16,
1242 // C2[2]/eps^2, polynomial in eps2 of order 1
1243 1, 6, 32,
1244 // C2[3]/eps^3, polynomial in eps2 of order 0
1245 5, 48,
1246 // C2[4]/eps^4, polynomial in eps2 of order 0
1247 35, 512,
1248 };
1249#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1250 static const real coeff[] = {
1251 // C2[1]/eps^1, polynomial in eps2 of order 2
1252 1, 2, 16, 32,
1253 // C2[2]/eps^2, polynomial in eps2 of order 1
1254 1, 6, 32,
1255 // C2[3]/eps^3, polynomial in eps2 of order 1
1256 15, 80, 768,
1257 // C2[4]/eps^4, polynomial in eps2 of order 0
1258 35, 512,
1259 // C2[5]/eps^5, polynomial in eps2 of order 0
1260 63, 1280,
1261 };
1262#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1263 static const real coeff[] = {
1264 // C2[1]/eps^1, polynomial in eps2 of order 2
1265 1, 2, 16, 32,
1266 // C2[2]/eps^2, polynomial in eps2 of order 2
1267 35, 64, 384, 2048,
1268 // C2[3]/eps^3, polynomial in eps2 of order 1
1269 15, 80, 768,
1270 // C2[4]/eps^4, polynomial in eps2 of order 1
1271 7, 35, 512,
1272 // C2[5]/eps^5, polynomial in eps2 of order 0
1273 63, 1280,
1274 // C2[6]/eps^6, polynomial in eps2 of order 0
1275 77, 2048,
1276 };
1277#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1278 static const real coeff[] = {
1279 // C2[1]/eps^1, polynomial in eps2 of order 3
1280 41, 64, 128, 1024, 2048,
1281 // C2[2]/eps^2, polynomial in eps2 of order 2
1282 35, 64, 384, 2048,
1283 // C2[3]/eps^3, polynomial in eps2 of order 2
1284 69, 120, 640, 6144,
1285 // C2[4]/eps^4, polynomial in eps2 of order 1
1286 7, 35, 512,
1287 // C2[5]/eps^5, polynomial in eps2 of order 1
1288 105, 504, 10240,
1289 // C2[6]/eps^6, polynomial in eps2 of order 0
1290 77, 2048,
1291 // C2[7]/eps^7, polynomial in eps2 of order 0
1292 429, 14336,
1293 };
1294#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1295 static const real coeff[] = {
1296 // C2[1]/eps^1, polynomial in eps2 of order 3
1297 41, 64, 128, 1024, 2048,
1298 // C2[2]/eps^2, polynomial in eps2 of order 3
1299 47, 70, 128, 768, 4096,
1300 // C2[3]/eps^3, polynomial in eps2 of order 2
1301 69, 120, 640, 6144,
1302 // C2[4]/eps^4, polynomial in eps2 of order 2
1303 133, 224, 1120, 16384,
1304 // C2[5]/eps^5, polynomial in eps2 of order 1
1305 105, 504, 10240,
1306 // C2[6]/eps^6, polynomial in eps2 of order 1
1307 33, 154, 4096,
1308 // C2[7]/eps^7, polynomial in eps2 of order 0
1309 429, 14336,
1310 // C2[8]/eps^8, polynomial in eps2 of order 0
1311 6435, 262144,
1312 };
1313#else
1314#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1315#endif
1316 static_assert(sizeof(coeff) / sizeof(real) ==
1317 (nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4,
1318 "Coefficient array size mismatch in C2f");
1319 real
1320 eps2 = Math::sq(eps),
1321 d = eps;
1322 int o = 0;
1323 for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
1324 int m = (nC2_ - l) / 2; // order of polynomial in eps^2
1325 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1326 o += m + 2;
1327 d *= eps;
1328 }
1329 // Post condition: o == sizeof(coeff) / sizeof(real)
1330 }
1331
1332 // The scale factor A3 = mean value of (d/dsigma)I3
1333 void Geodesic::A3coeff() {
1334 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1335#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1336 static const real coeff[] = {
1337 // A3, coeff of eps^2, polynomial in n of order 0
1338 -1, 4,
1339 // A3, coeff of eps^1, polynomial in n of order 1
1340 1, -1, 2,
1341 // A3, coeff of eps^0, polynomial in n of order 0
1342 1, 1,
1343 };
1344#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1345 static const real coeff[] = {
1346 // A3, coeff of eps^3, polynomial in n of order 0
1347 -1, 16,
1348 // A3, coeff of eps^2, polynomial in n of order 1
1349 -1, -2, 8,
1350 // A3, coeff of eps^1, polynomial in n of order 1
1351 1, -1, 2,
1352 // A3, coeff of eps^0, polynomial in n of order 0
1353 1, 1,
1354 };
1355#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1356 static const real coeff[] = {
1357 // A3, coeff of eps^4, polynomial in n of order 0
1358 -3, 64,
1359 // A3, coeff of eps^3, polynomial in n of order 1
1360 -3, -1, 16,
1361 // A3, coeff of eps^2, polynomial in n of order 2
1362 3, -1, -2, 8,
1363 // A3, coeff of eps^1, polynomial in n of order 1
1364 1, -1, 2,
1365 // A3, coeff of eps^0, polynomial in n of order 0
1366 1, 1,
1367 };
1368#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1369 static const real coeff[] = {
1370 // A3, coeff of eps^5, polynomial in n of order 0
1371 -3, 128,
1372 // A3, coeff of eps^4, polynomial in n of order 1
1373 -2, -3, 64,
1374 // A3, coeff of eps^3, polynomial in n of order 2
1375 -1, -3, -1, 16,
1376 // A3, coeff of eps^2, polynomial in n of order 2
1377 3, -1, -2, 8,
1378 // A3, coeff of eps^1, polynomial in n of order 1
1379 1, -1, 2,
1380 // A3, coeff of eps^0, polynomial in n of order 0
1381 1, 1,
1382 };
1383#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1384 static const real coeff[] = {
1385 // A3, coeff of eps^6, polynomial in n of order 0
1386 -5, 256,
1387 // A3, coeff of eps^5, polynomial in n of order 1
1388 -5, -3, 128,
1389 // A3, coeff of eps^4, polynomial in n of order 2
1390 -10, -2, -3, 64,
1391 // A3, coeff of eps^3, polynomial in n of order 3
1392 5, -1, -3, -1, 16,
1393 // A3, coeff of eps^2, polynomial in n of order 2
1394 3, -1, -2, 8,
1395 // A3, coeff of eps^1, polynomial in n of order 1
1396 1, -1, 2,
1397 // A3, coeff of eps^0, polynomial in n of order 0
1398 1, 1,
1399 };
1400#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1401 static const real coeff[] = {
1402 // A3, coeff of eps^7, polynomial in n of order 0
1403 -25, 2048,
1404 // A3, coeff of eps^6, polynomial in n of order 1
1405 -15, -20, 1024,
1406 // A3, coeff of eps^5, polynomial in n of order 2
1407 -5, -10, -6, 256,
1408 // A3, coeff of eps^4, polynomial in n of order 3
1409 -5, -20, -4, -6, 128,
1410 // A3, coeff of eps^3, polynomial in n of order 3
1411 5, -1, -3, -1, 16,
1412 // A3, coeff of eps^2, polynomial in n of order 2
1413 3, -1, -2, 8,
1414 // A3, coeff of eps^1, polynomial in n of order 1
1415 1, -1, 2,
1416 // A3, coeff of eps^0, polynomial in n of order 0
1417 1, 1,
1418 };
1419#else
1420#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1421#endif
1422 static_assert(sizeof(coeff) / sizeof(real) ==
1423 (nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4,
1424 "Coefficient array size mismatch in A3f");
1425 int o = 0, k = 0;
1426 for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
1427 int m = min(nA3_ - j - 1, j); // order of polynomial in n
1428 _aA3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1429 o += m + 2;
1430 }
1431 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
1432 }
1433
1434 // The coefficients C3[l] in the Fourier expansion of B3
1435 void Geodesic::C3coeff() {
1436 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1437#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1438 static const real coeff[] = {
1439 // C3[1], coeff of eps^2, polynomial in n of order 0
1440 1, 8,
1441 // C3[1], coeff of eps^1, polynomial in n of order 1
1442 -1, 1, 4,
1443 // C3[2], coeff of eps^2, polynomial in n of order 0
1444 1, 16,
1445 };
1446#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1447 static const real coeff[] = {
1448 // C3[1], coeff of eps^3, polynomial in n of order 0
1449 3, 64,
1450 // C3[1], coeff of eps^2, polynomial in n of order 1
1451 // This is a case where a leading 0 term has been inserted to maintain the
1452 // pattern in the orders of the polynomials.
1453 0, 1, 8,
1454 // C3[1], coeff of eps^1, polynomial in n of order 1
1455 -1, 1, 4,
1456 // C3[2], coeff of eps^3, polynomial in n of order 0
1457 3, 64,
1458 // C3[2], coeff of eps^2, polynomial in n of order 1
1459 -3, 2, 32,
1460 // C3[3], coeff of eps^3, polynomial in n of order 0
1461 5, 192,
1462 };
1463#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1464 static const real coeff[] = {
1465 // C3[1], coeff of eps^4, polynomial in n of order 0
1466 5, 128,
1467 // C3[1], coeff of eps^3, polynomial in n of order 1
1468 3, 3, 64,
1469 // C3[1], coeff of eps^2, polynomial in n of order 2
1470 -1, 0, 1, 8,
1471 // C3[1], coeff of eps^1, polynomial in n of order 1
1472 -1, 1, 4,
1473 // C3[2], coeff of eps^4, polynomial in n of order 0
1474 3, 128,
1475 // C3[2], coeff of eps^3, polynomial in n of order 1
1476 -2, 3, 64,
1477 // C3[2], coeff of eps^2, polynomial in n of order 2
1478 1, -3, 2, 32,
1479 // C3[3], coeff of eps^4, polynomial in n of order 0
1480 3, 128,
1481 // C3[3], coeff of eps^3, polynomial in n of order 1
1482 -9, 5, 192,
1483 // C3[4], coeff of eps^4, polynomial in n of order 0
1484 7, 512,
1485 };
1486#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1487 static const real coeff[] = {
1488 // C3[1], coeff of eps^5, polynomial in n of order 0
1489 3, 128,
1490 // C3[1], coeff of eps^4, polynomial in n of order 1
1491 2, 5, 128,
1492 // C3[1], coeff of eps^3, polynomial in n of order 2
1493 -1, 3, 3, 64,
1494 // C3[1], coeff of eps^2, polynomial in n of order 2
1495 -1, 0, 1, 8,
1496 // C3[1], coeff of eps^1, polynomial in n of order 1
1497 -1, 1, 4,
1498 // C3[2], coeff of eps^5, polynomial in n of order 0
1499 5, 256,
1500 // C3[2], coeff of eps^4, polynomial in n of order 1
1501 1, 3, 128,
1502 // C3[2], coeff of eps^3, polynomial in n of order 2
1503 -3, -2, 3, 64,
1504 // C3[2], coeff of eps^2, polynomial in n of order 2
1505 1, -3, 2, 32,
1506 // C3[3], coeff of eps^5, polynomial in n of order 0
1507 7, 512,
1508 // C3[3], coeff of eps^4, polynomial in n of order 1
1509 -10, 9, 384,
1510 // C3[3], coeff of eps^3, polynomial in n of order 2
1511 5, -9, 5, 192,
1512 // C3[4], coeff of eps^5, polynomial in n of order 0
1513 7, 512,
1514 // C3[4], coeff of eps^4, polynomial in n of order 1
1515 -14, 7, 512,
1516 // C3[5], coeff of eps^5, polynomial in n of order 0
1517 21, 2560,
1518 };
1519#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1520 static const real coeff[] = {
1521 // C3[1], coeff of eps^6, polynomial in n of order 0
1522 21, 1024,
1523 // C3[1], coeff of eps^5, polynomial in n of order 1
1524 11, 12, 512,
1525 // C3[1], coeff of eps^4, polynomial in n of order 2
1526 2, 2, 5, 128,
1527 // C3[1], coeff of eps^3, polynomial in n of order 3
1528 -5, -1, 3, 3, 64,
1529 // C3[1], coeff of eps^2, polynomial in n of order 2
1530 -1, 0, 1, 8,
1531 // C3[1], coeff of eps^1, polynomial in n of order 1
1532 -1, 1, 4,
1533 // C3[2], coeff of eps^6, polynomial in n of order 0
1534 27, 2048,
1535 // C3[2], coeff of eps^5, polynomial in n of order 1
1536 1, 5, 256,
1537 // C3[2], coeff of eps^4, polynomial in n of order 2
1538 -9, 2, 6, 256,
1539 // C3[2], coeff of eps^3, polynomial in n of order 3
1540 2, -3, -2, 3, 64,
1541 // C3[2], coeff of eps^2, polynomial in n of order 2
1542 1, -3, 2, 32,
1543 // C3[3], coeff of eps^6, polynomial in n of order 0
1544 3, 256,
1545 // C3[3], coeff of eps^5, polynomial in n of order 1
1546 -4, 21, 1536,
1547 // C3[3], coeff of eps^4, polynomial in n of order 2
1548 -6, -10, 9, 384,
1549 // C3[3], coeff of eps^3, polynomial in n of order 3
1550 -1, 5, -9, 5, 192,
1551 // C3[4], coeff of eps^6, polynomial in n of order 0
1552 9, 1024,
1553 // C3[4], coeff of eps^5, polynomial in n of order 1
1554 -10, 7, 512,
1555 // C3[4], coeff of eps^4, polynomial in n of order 2
1556 10, -14, 7, 512,
1557 // C3[5], coeff of eps^6, polynomial in n of order 0
1558 9, 1024,
1559 // C3[5], coeff of eps^5, polynomial in n of order 1
1560 -45, 21, 2560,
1561 // C3[6], coeff of eps^6, polynomial in n of order 0
1562 11, 2048,
1563 };
1564#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1565 static const real coeff[] = {
1566 // C3[1], coeff of eps^7, polynomial in n of order 0
1567 243, 16384,
1568 // C3[1], coeff of eps^6, polynomial in n of order 1
1569 10, 21, 1024,
1570 // C3[1], coeff of eps^5, polynomial in n of order 2
1571 3, 11, 12, 512,
1572 // C3[1], coeff of eps^4, polynomial in n of order 3
1573 -2, 2, 2, 5, 128,
1574 // C3[1], coeff of eps^3, polynomial in n of order 3
1575 -5, -1, 3, 3, 64,
1576 // C3[1], coeff of eps^2, polynomial in n of order 2
1577 -1, 0, 1, 8,
1578 // C3[1], coeff of eps^1, polynomial in n of order 1
1579 -1, 1, 4,
1580 // C3[2], coeff of eps^7, polynomial in n of order 0
1581 187, 16384,
1582 // C3[2], coeff of eps^6, polynomial in n of order 1
1583 69, 108, 8192,
1584 // C3[2], coeff of eps^5, polynomial in n of order 2
1585 -2, 1, 5, 256,
1586 // C3[2], coeff of eps^4, polynomial in n of order 3
1587 -6, -9, 2, 6, 256,
1588 // C3[2], coeff of eps^3, polynomial in n of order 3
1589 2, -3, -2, 3, 64,
1590 // C3[2], coeff of eps^2, polynomial in n of order 2
1591 1, -3, 2, 32,
1592 // C3[3], coeff of eps^7, polynomial in n of order 0
1593 139, 16384,
1594 // C3[3], coeff of eps^6, polynomial in n of order 1
1595 -1, 12, 1024,
1596 // C3[3], coeff of eps^5, polynomial in n of order 2
1597 -77, -8, 42, 3072,
1598 // C3[3], coeff of eps^4, polynomial in n of order 3
1599 10, -6, -10, 9, 384,
1600 // C3[3], coeff of eps^3, polynomial in n of order 3
1601 -1, 5, -9, 5, 192,
1602 // C3[4], coeff of eps^7, polynomial in n of order 0
1603 127, 16384,
1604 // C3[4], coeff of eps^6, polynomial in n of order 1
1605 -43, 72, 8192,
1606 // C3[4], coeff of eps^5, polynomial in n of order 2
1607 -7, -40, 28, 2048,
1608 // C3[4], coeff of eps^4, polynomial in n of order 3
1609 -7, 20, -28, 14, 1024,
1610 // C3[5], coeff of eps^7, polynomial in n of order 0
1611 99, 16384,
1612 // C3[5], coeff of eps^6, polynomial in n of order 1
1613 -15, 9, 1024,
1614 // C3[5], coeff of eps^5, polynomial in n of order 2
1615 75, -90, 42, 5120,
1616 // C3[6], coeff of eps^7, polynomial in n of order 0
1617 99, 16384,
1618 // C3[6], coeff of eps^6, polynomial in n of order 1
1619 -99, 44, 8192,
1620 // C3[7], coeff of eps^7, polynomial in n of order 0
1621 429, 114688,
1622 };
1623#else
1624#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1625#endif
1626 static_assert(sizeof(coeff) / sizeof(real) ==
1627 ((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8,
1628 "Coefficient array size mismatch in C3coeff");
1629 int o = 0, k = 0;
1630 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
1631 for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
1632 int m = min(nC3_ - j - 1, j); // order of polynomial in n
1633 _cC3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1634 o += m + 2;
1635 }
1636 }
1637 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
1638 }
1639
1640 void Geodesic::C4coeff() {
1641 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1642#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1643 static const real coeff[] = {
1644 // C4[0], coeff of eps^2, polynomial in n of order 0
1645 -2, 105,
1646 // C4[0], coeff of eps^1, polynomial in n of order 1
1647 16, -7, 35,
1648 // C4[0], coeff of eps^0, polynomial in n of order 2
1649 8, -28, 70, 105,
1650 // C4[1], coeff of eps^2, polynomial in n of order 0
1651 -2, 105,
1652 // C4[1], coeff of eps^1, polynomial in n of order 1
1653 -16, 7, 315,
1654 // C4[2], coeff of eps^2, polynomial in n of order 0
1655 4, 525,
1656 };
1657#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1658 static const real coeff[] = {
1659 // C4[0], coeff of eps^3, polynomial in n of order 0
1660 11, 315,
1661 // C4[0], coeff of eps^2, polynomial in n of order 1
1662 -32, -6, 315,
1663 // C4[0], coeff of eps^1, polynomial in n of order 2
1664 -32, 48, -21, 105,
1665 // C4[0], coeff of eps^0, polynomial in n of order 3
1666 4, 24, -84, 210, 315,
1667 // C4[1], coeff of eps^3, polynomial in n of order 0
1668 -1, 105,
1669 // C4[1], coeff of eps^2, polynomial in n of order 1
1670 64, -18, 945,
1671 // C4[1], coeff of eps^1, polynomial in n of order 2
1672 32, -48, 21, 945,
1673 // C4[2], coeff of eps^3, polynomial in n of order 0
1674 -8, 1575,
1675 // C4[2], coeff of eps^2, polynomial in n of order 1
1676 -32, 12, 1575,
1677 // C4[3], coeff of eps^3, polynomial in n of order 0
1678 8, 2205,
1679 };
1680#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1681 static const real coeff[] = {
1682 // C4[0], coeff of eps^4, polynomial in n of order 0
1683 4, 1155,
1684 // C4[0], coeff of eps^3, polynomial in n of order 1
1685 -368, 121, 3465,
1686 // C4[0], coeff of eps^2, polynomial in n of order 2
1687 1088, -352, -66, 3465,
1688 // C4[0], coeff of eps^1, polynomial in n of order 3
1689 48, -352, 528, -231, 1155,
1690 // C4[0], coeff of eps^0, polynomial in n of order 4
1691 16, 44, 264, -924, 2310, 3465,
1692 // C4[1], coeff of eps^4, polynomial in n of order 0
1693 4, 1155,
1694 // C4[1], coeff of eps^3, polynomial in n of order 1
1695 80, -99, 10395,
1696 // C4[1], coeff of eps^2, polynomial in n of order 2
1697 -896, 704, -198, 10395,
1698 // C4[1], coeff of eps^1, polynomial in n of order 3
1699 -48, 352, -528, 231, 10395,
1700 // C4[2], coeff of eps^4, polynomial in n of order 0
1701 -8, 1925,
1702 // C4[2], coeff of eps^3, polynomial in n of order 1
1703 384, -88, 17325,
1704 // C4[2], coeff of eps^2, polynomial in n of order 2
1705 320, -352, 132, 17325,
1706 // C4[3], coeff of eps^4, polynomial in n of order 0
1707 -16, 8085,
1708 // C4[3], coeff of eps^3, polynomial in n of order 1
1709 -256, 88, 24255,
1710 // C4[4], coeff of eps^4, polynomial in n of order 0
1711 64, 31185,
1712 };
1713#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1714 static const real coeff[] = {
1715 // C4[0], coeff of eps^5, polynomial in n of order 0
1716 97, 15015,
1717 // C4[0], coeff of eps^4, polynomial in n of order 1
1718 1088, 156, 45045,
1719 // C4[0], coeff of eps^3, polynomial in n of order 2
1720 -224, -4784, 1573, 45045,
1721 // C4[0], coeff of eps^2, polynomial in n of order 3
1722 -10656, 14144, -4576, -858, 45045,
1723 // C4[0], coeff of eps^1, polynomial in n of order 4
1724 64, 624, -4576, 6864, -3003, 15015,
1725 // C4[0], coeff of eps^0, polynomial in n of order 5
1726 100, 208, 572, 3432, -12012, 30030, 45045,
1727 // C4[1], coeff of eps^5, polynomial in n of order 0
1728 1, 9009,
1729 // C4[1], coeff of eps^4, polynomial in n of order 1
1730 -2944, 468, 135135,
1731 // C4[1], coeff of eps^3, polynomial in n of order 2
1732 5792, 1040, -1287, 135135,
1733 // C4[1], coeff of eps^2, polynomial in n of order 3
1734 5952, -11648, 9152, -2574, 135135,
1735 // C4[1], coeff of eps^1, polynomial in n of order 4
1736 -64, -624, 4576, -6864, 3003, 135135,
1737 // C4[2], coeff of eps^5, polynomial in n of order 0
1738 8, 10725,
1739 // C4[2], coeff of eps^4, polynomial in n of order 1
1740 1856, -936, 225225,
1741 // C4[2], coeff of eps^3, polynomial in n of order 2
1742 -8448, 4992, -1144, 225225,
1743 // C4[2], coeff of eps^2, polynomial in n of order 3
1744 -1440, 4160, -4576, 1716, 225225,
1745 // C4[3], coeff of eps^5, polynomial in n of order 0
1746 -136, 63063,
1747 // C4[3], coeff of eps^4, polynomial in n of order 1
1748 1024, -208, 105105,
1749 // C4[3], coeff of eps^3, polynomial in n of order 2
1750 3584, -3328, 1144, 315315,
1751 // C4[4], coeff of eps^5, polynomial in n of order 0
1752 -128, 135135,
1753 // C4[4], coeff of eps^4, polynomial in n of order 1
1754 -2560, 832, 405405,
1755 // C4[5], coeff of eps^5, polynomial in n of order 0
1756 128, 99099,
1757 };
1758#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1759 static const real coeff[] = {
1760 // C4[0], coeff of eps^6, polynomial in n of order 0
1761 10, 9009,
1762 // C4[0], coeff of eps^5, polynomial in n of order 1
1763 -464, 291, 45045,
1764 // C4[0], coeff of eps^4, polynomial in n of order 2
1765 -4480, 1088, 156, 45045,
1766 // C4[0], coeff of eps^3, polynomial in n of order 3
1767 10736, -224, -4784, 1573, 45045,
1768 // C4[0], coeff of eps^2, polynomial in n of order 4
1769 1664, -10656, 14144, -4576, -858, 45045,
1770 // C4[0], coeff of eps^1, polynomial in n of order 5
1771 16, 64, 624, -4576, 6864, -3003, 15015,
1772 // C4[0], coeff of eps^0, polynomial in n of order 6
1773 56, 100, 208, 572, 3432, -12012, 30030, 45045,
1774 // C4[1], coeff of eps^6, polynomial in n of order 0
1775 10, 9009,
1776 // C4[1], coeff of eps^5, polynomial in n of order 1
1777 112, 15, 135135,
1778 // C4[1], coeff of eps^4, polynomial in n of order 2
1779 3840, -2944, 468, 135135,
1780 // C4[1], coeff of eps^3, polynomial in n of order 3
1781 -10704, 5792, 1040, -1287, 135135,
1782 // C4[1], coeff of eps^2, polynomial in n of order 4
1783 -768, 5952, -11648, 9152, -2574, 135135,
1784 // C4[1], coeff of eps^1, polynomial in n of order 5
1785 -16, -64, -624, 4576, -6864, 3003, 135135,
1786 // C4[2], coeff of eps^6, polynomial in n of order 0
1787 -4, 25025,
1788 // C4[2], coeff of eps^5, polynomial in n of order 1
1789 -1664, 168, 225225,
1790 // C4[2], coeff of eps^4, polynomial in n of order 2
1791 1664, 1856, -936, 225225,
1792 // C4[2], coeff of eps^3, polynomial in n of order 3
1793 6784, -8448, 4992, -1144, 225225,
1794 // C4[2], coeff of eps^2, polynomial in n of order 4
1795 128, -1440, 4160, -4576, 1716, 225225,
1796 // C4[3], coeff of eps^6, polynomial in n of order 0
1797 64, 315315,
1798 // C4[3], coeff of eps^5, polynomial in n of order 1
1799 1792, -680, 315315,
1800 // C4[3], coeff of eps^4, polynomial in n of order 2
1801 -2048, 1024, -208, 105105,
1802 // C4[3], coeff of eps^3, polynomial in n of order 3
1803 -1792, 3584, -3328, 1144, 315315,
1804 // C4[4], coeff of eps^6, polynomial in n of order 0
1805 -512, 405405,
1806 // C4[4], coeff of eps^5, polynomial in n of order 1
1807 2048, -384, 405405,
1808 // C4[4], coeff of eps^4, polynomial in n of order 2
1809 3072, -2560, 832, 405405,
1810 // C4[5], coeff of eps^6, polynomial in n of order 0
1811 -256, 495495,
1812 // C4[5], coeff of eps^5, polynomial in n of order 1
1813 -2048, 640, 495495,
1814 // C4[6], coeff of eps^6, polynomial in n of order 0
1815 512, 585585,
1816 };
1817#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1818 static const real coeff[] = {
1819 // C4[0], coeff of eps^7, polynomial in n of order 0
1820 193, 85085,
1821 // C4[0], coeff of eps^6, polynomial in n of order 1
1822 4192, 850, 765765,
1823 // C4[0], coeff of eps^5, polynomial in n of order 2
1824 20960, -7888, 4947, 765765,
1825 // C4[0], coeff of eps^4, polynomial in n of order 3
1826 12480, -76160, 18496, 2652, 765765,
1827 // C4[0], coeff of eps^3, polynomial in n of order 4
1828 -154048, 182512, -3808, -81328, 26741, 765765,
1829 // C4[0], coeff of eps^2, polynomial in n of order 5
1830 3232, 28288, -181152, 240448, -77792, -14586, 765765,
1831 // C4[0], coeff of eps^1, polynomial in n of order 6
1832 96, 272, 1088, 10608, -77792, 116688, -51051, 255255,
1833 // C4[0], coeff of eps^0, polynomial in n of order 7
1834 588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765,
1835 // C4[1], coeff of eps^7, polynomial in n of order 0
1836 349, 2297295,
1837 // C4[1], coeff of eps^6, polynomial in n of order 1
1838 -1472, 510, 459459,
1839 // C4[1], coeff of eps^5, polynomial in n of order 2
1840 -39840, 1904, 255, 2297295,
1841 // C4[1], coeff of eps^4, polynomial in n of order 3
1842 52608, 65280, -50048, 7956, 2297295,
1843 // C4[1], coeff of eps^3, polynomial in n of order 4
1844 103744, -181968, 98464, 17680, -21879, 2297295,
1845 // C4[1], coeff of eps^2, polynomial in n of order 5
1846 -1344, -13056, 101184, -198016, 155584, -43758, 2297295,
1847 // C4[1], coeff of eps^1, polynomial in n of order 6
1848 -96, -272, -1088, -10608, 77792, -116688, 51051, 2297295,
1849 // C4[2], coeff of eps^7, polynomial in n of order 0
1850 464, 1276275,
1851 // C4[2], coeff of eps^6, polynomial in n of order 1
1852 -928, -612, 3828825,
1853 // C4[2], coeff of eps^5, polynomial in n of order 2
1854 64256, -28288, 2856, 3828825,
1855 // C4[2], coeff of eps^4, polynomial in n of order 3
1856 -126528, 28288, 31552, -15912, 3828825,
1857 // C4[2], coeff of eps^3, polynomial in n of order 4
1858 -41472, 115328, -143616, 84864, -19448, 3828825,
1859 // C4[2], coeff of eps^2, polynomial in n of order 5
1860 160, 2176, -24480, 70720, -77792, 29172, 3828825,
1861 // C4[3], coeff of eps^7, polynomial in n of order 0
1862 -16, 97461,
1863 // C4[3], coeff of eps^6, polynomial in n of order 1
1864 -16384, 1088, 5360355,
1865 // C4[3], coeff of eps^5, polynomial in n of order 2
1866 -2560, 30464, -11560, 5360355,
1867 // C4[3], coeff of eps^4, polynomial in n of order 3
1868 35840, -34816, 17408, -3536, 1786785,
1869 // C4[3], coeff of eps^3, polynomial in n of order 4
1870 7168, -30464, 60928, -56576, 19448, 5360355,
1871 // C4[4], coeff of eps^7, polynomial in n of order 0
1872 128, 2297295,
1873 // C4[4], coeff of eps^6, polynomial in n of order 1
1874 26624, -8704, 6891885,
1875 // C4[4], coeff of eps^5, polynomial in n of order 2
1876 -77824, 34816, -6528, 6891885,
1877 // C4[4], coeff of eps^4, polynomial in n of order 3
1878 -32256, 52224, -43520, 14144, 6891885,
1879 // C4[5], coeff of eps^7, polynomial in n of order 0
1880 -6784, 8423415,
1881 // C4[5], coeff of eps^6, polynomial in n of order 1
1882 24576, -4352, 8423415,
1883 // C4[5], coeff of eps^5, polynomial in n of order 2
1884 45056, -34816, 10880, 8423415,
1885 // C4[6], coeff of eps^7, polynomial in n of order 0
1886 -1024, 3318315,
1887 // C4[6], coeff of eps^6, polynomial in n of order 1
1888 -28672, 8704, 9954945,
1889 // C4[7], coeff of eps^7, polynomial in n of order 0
1890 1024, 1640925,
1891 };
1892#else
1893#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1894#endif
1895 static_assert(sizeof(coeff) / sizeof(real) ==
1896 (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
1897 "Coefficient array size mismatch in C4coeff");
1898 int o = 0, k = 0;
1899 for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
1900 for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
1901 int m = nC4_ - j - 1; // order of polynomial in n
1902 _cC4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1903 o += m + 2;
1904 }
1905 }
1906 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
1907 }
1908
1909} // namespace GeographicLib
ang
GeographicLib::Angle ang
Definition Geod3Solve.cpp:26
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
Geodesic.hpp
Header for GeographicLib::Geodesic class.
GeographicLib::Constants::WGS84_f
static T WGS84_f()
Definition Constants.hpp:118
GeographicLib::Constants::WGS84_a
static T WGS84_a()
Definition Constants.hpp:112
GeographicLib::GeodesicExact
Exact geodesic calculations.
Definition GeodesicExact.hpp:88
GeographicLib::Geodesic::InverseLine
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition Geodesic.cpp:523
GeographicLib::Geodesic::WGS84
static const Geodesic & WGS84()
Definition Geodesic.cpp:94
GeographicLib::Geodesic::ArcDirectLine
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition Geodesic.cpp:163
GeographicLib::Geodesic::Line
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition Geodesic.cpp:123
GeographicLib::Geodesic::GenDirectLine
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition Geodesic.cpp:145
GeographicLib::Geodesic::DISTANCE
@ DISTANCE
Definition Geodesic.hpp:294
GeographicLib::Geodesic::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition Geodesic.hpp:311
GeographicLib::Geodesic::AREA
@ AREA
Definition Geodesic.hpp:321
GeographicLib::Geodesic::AZIMUTH
@ AZIMUTH
Definition Geodesic.hpp:289
GeographicLib::Geodesic::NONE
@ NONE
Definition Geodesic.hpp:271
GeographicLib::Geodesic::DISTANCE_IN
@ DISTANCE_IN
Definition Geodesic.hpp:306
GeographicLib::Geodesic::GEODESICSCALE
@ GEODESICSCALE
Definition Geodesic.hpp:316
GeographicLib::Geodesic::GeodesicLine
friend class GeodesicLine
Definition Geodesic.hpp:178
GeographicLib::Geodesic::GenDirect
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
Definition Geodesic.cpp:128
GeographicLib::Geodesic::DirectLine
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition Geodesic.cpp:158
GeographicLib::Geodesic::Geodesic
Geodesic(real a, real f, bool exact=false)
Definition Geodesic.cpp:41
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition Math.hpp:82
GeographicLib::Math::degree
static T degree()
Definition Math.hpp:197
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition Math.hpp:303
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition Math.cpp:104
GeographicLib::Math::atan2d
static T atan2d(T y, T x)
Definition Math.cpp:212
GeographicLib::Math::norm
static void norm(T &x, T &y)
Definition Math.hpp:219
GeographicLib::Math::AngRound
static T AngRound(T x)
Definition Math.cpp:95
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:142
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::sincosde
static void sincosde(T x, T t, T &sinx, T &cosx)
Definition Math.cpp:131
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:187
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:301
GeographicLib::Math::polyval
static T polyval(int N, const T p[], T x)
Definition Math.hpp:270
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::hd
static constexpr int hd
degrees per half turn
Definition Math.hpp:145
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition NearestNeighbor.hpp:819