GeographicLib 1.52
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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 (2009-2021) <charles@karney.com> 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
31
32#if defined(_MSC_VER)
33// Squelch warnings about potentially uninitialized local variables and
34// constant conditional expressions
35# pragma warning (disable: 4701 4127)
36#endif
37
38namespace GeographicLib {
39
40 using namespace std;
41
42 Geodesic::Geodesic(real a, real f)
43 : maxit2_(maxit1_ + Math::digits() + 10)
44 // Underflow guard. We require
45 // tiny_ * epsilon() > 0
46 // tiny_ + epsilon() == epsilon()
47 , tiny_(sqrt(numeric_limits<real>::min()))
48 , tol0_(numeric_limits<real>::epsilon())
49 // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
50 // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
51 // which otherwise failed for Visual Studio 10 (Release and Debug)
52 , tol1_(200 * tol0_)
53 , tol2_(sqrt(tol0_))
54 , tolb_(tol0_ * tol2_) // Check on bisection interval
55 , xthresh_(1000 * tol2_)
56 , _a(a)
57 , _f(f)
58 , _f1(1 - _f)
59 , _e2(_f * (2 - _f))
60 , _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
61 , _n(_f / ( 2 - _f))
62 , _b(_a * _f1)
63 , _c2((Math::sq(_a) + Math::sq(_b) *
64 (_e2 == 0 ? 1 :
65 Math::eatanhe(real(1), (_f < 0 ? -1 : 1) * sqrt(abs(_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( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
79 {
80 if (!(isfinite(_a) && _a > 0))
81 throw GeographicErr("Equatorial radius is not positive");
82 if (!(isfinite(_b) && _b > 0))
83 throw GeographicErr("Polar semi-axis is not positive");
84 A3coeff();
85 C3coeff();
86 C4coeff();
87 }
88
90 static const Geodesic wgs84(Constants::WGS84_a(), Constants::WGS84_f());
91 return wgs84;
92 }
93
94 Math::real Geodesic::SinCosSeries(bool sinp,
95 real sinx, real cosx,
96 const real c[], int n) {
97 // Evaluate
98 // y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
99 // sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
100 // using Clenshaw summation. N.B. c[0] is unused for sin series
101 // Approx operation count = (n + 5) mult and (2 * n + 2) add
102 c += (n + sinp); // Point to one beyond last element
103 real
104 ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
105 y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum
106 // Now n is even
107 n /= 2;
108 while (n--) {
109 // Unroll loop x 2, so accumulators return to their original role
110 y1 = ar * y0 - y1 + *--c;
111 y0 = ar * y1 - y0 + *--c;
112 }
113 return sinp
114 ? 2 * sinx * cosx * y0 // sin(2 * x) * y0
115 : cosx * (y0 - y1); // cos(x) * (y0 - y1)
116 }
117
118 GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1,
119 unsigned caps) const {
120 return GeodesicLine(*this, lat1, lon1, azi1, caps);
121 }
122
123 Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1,
124 bool arcmode, real s12_a12, unsigned outmask,
125 real& lat2, real& lon2, real& azi2,
126 real& s12, real& m12, real& M12, real& M21,
127 real& S12) const {
128 // Automatically supply DISTANCE_IN if necessary
129 if (!arcmode) outmask |= DISTANCE_IN;
130 return GeodesicLine(*this, lat1, lon1, azi1, outmask)
131 . // Note the dot!
132 GenPosition(arcmode, s12_a12, outmask,
133 lat2, lon2, azi2, s12, m12, M12, M21, S12);
134 }
135
136 GeodesicLine Geodesic::GenDirectLine(real lat1, real lon1, real azi1,
137 bool arcmode, real s12_a12,
138 unsigned caps) const {
139 azi1 = Math::AngNormalize(azi1);
140 real salp1, calp1;
141 // Guard against underflow in salp0. Also -0 is converted to +0.
142 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
143 // Automatically supply DISTANCE_IN if necessary
144 if (!arcmode) caps |= DISTANCE_IN;
145 return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1,
146 caps, arcmode, s12_a12);
147 }
148
149 GeodesicLine Geodesic::DirectLine(real lat1, real lon1, real azi1, real s12,
150 unsigned caps) const {
151 return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
152 }
153
154 GeodesicLine Geodesic::ArcDirectLine(real lat1, real lon1, real azi1,
155 real a12, unsigned caps) const {
156 return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
157 }
158
159 Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
160 unsigned outmask, real& s12,
161 real& salp1, real& calp1,
162 real& salp2, real& calp2,
163 real& m12, real& M12, real& M21,
164 real& S12) const {
165 // Compute longitude difference (AngDiff does this carefully). Result is
166 // in [-180, 180] but -180 is only for west-going geodesics. 180 is for
167 // east-going and meridional geodesics.
168 real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
169 // Make longitude difference positive.
170 int lonsign = lon12 >= 0 ? 1 : -1;
171 // If very close to being on the same half-meridian, then make it so.
172 lon12 = lonsign * Math::AngRound(lon12);
173 lon12s = Math::AngRound((180 - lon12) - lonsign * lon12s);
174 real
175 lam12 = lon12 * Math::degree(),
176 slam12, clam12;
177 if (lon12 > 90) {
178 Math::sincosd(lon12s, slam12, clam12);
179 clam12 = -clam12;
180 } else
181 Math::sincosd(lon12, slam12, clam12);
182
183 // If really close to the equator, treat as on equator.
184 lat1 = Math::AngRound(Math::LatFix(lat1));
185 lat2 = Math::AngRound(Math::LatFix(lat2));
186 // Swap points so that point with higher (abs) latitude is point 1.
187 // If one latitude is a nan, then it becomes lat1.
188 int swapp = abs(lat1) < abs(lat2) ? -1 : 1;
189 if (swapp < 0) {
190 lonsign *= -1;
191 swap(lat1, lat2);
192 }
193 // Make lat1 <= 0
194 int latsign = lat1 < 0 ? 1 : -1;
195 lat1 *= latsign;
196 lat2 *= latsign;
197 // Now we have
198 //
199 // 0 <= lon12 <= 180
200 // -90 <= lat1 <= 0
201 // lat1 <= lat2 <= -lat1
202 //
203 // longsign, swapp, latsign register the transformation to bring the
204 // coordinates to this canonical form. In all cases, 1 means no change was
205 // made. We make these transformations so that there are few cases to
206 // check, e.g., on verifying quadrants in atan2. In addition, this
207 // enforces some symmetries in the results returned.
208
209 real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
210
211 Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
212 // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
213 // will be <= 2*tiny for two points at the same pole.
214 Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
215
216 Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
217 // Ensure cbet2 = +epsilon at poles
218 Math::norm(sbet2, cbet2); cbet2 = max(tiny_, cbet2);
219
220 // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
221 // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
222 // a better measure. This logic is used in assigning calp2 in Lambda12.
223 // Sometimes these quantities vanish and in that case we force bet2 = +/-
224 // bet1 exactly. An example where is is necessary is the inverse problem
225 // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
226 // which failed with Visual Studio 10 (Release and Debug)
227
228 if (cbet1 < -sbet1) {
229 if (cbet2 == cbet1)
230 sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
231 } else {
232 if (abs(sbet2) == -sbet1)
233 cbet2 = cbet1;
234 }
235
236 real
237 dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
238 dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
239
240 real a12, sig12;
241 // index zero element of this array is unused
242 real Ca[nC_];
243
244 bool meridian = lat1 == -90 || slam12 == 0;
245
246 if (meridian) {
247
248 // Endpoints are on a single full meridian, so the geodesic might lie on
249 // a meridian.
250
251 calp1 = clam12; salp1 = slam12; // Head to the target longitude
252 calp2 = 1; salp2 = 0; // At the target we're heading north
253
254 real
255 // tan(bet) = tan(sig) * cos(alp)
256 ssig1 = sbet1, csig1 = calp1 * cbet1,
257 ssig2 = sbet2, csig2 = calp2 * cbet2;
258
259 // sig12 = sig2 - sig1
260 sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
261 csig1 * csig2 + ssig1 * ssig2);
262 {
263 real dummy;
264 Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
265 outmask | DISTANCE | REDUCEDLENGTH,
266 s12x, m12x, dummy, M12, M21, Ca);
267 }
268 // Add the check for sig12 since zero length geodesics might yield m12 <
269 // 0. Test case was
270 //
271 // echo 20.001 0 20.001 0 | GeodSolve -i
272 //
273 // In fact, we will have sig12 > pi/2 for meridional geodesic which is
274 // not a shortest path.
275 // TODO: investigate m12 < 0 result for aarch/ppc (with -f -p 20)
276 // 20.001000000000001 0.000000000000000 180.000000000000000
277 // 20.001000000000001 0.000000000000000 180.000000000000000
278 // 0.0000000002 0.000000000000001 -0.0000000001
279 // 0.99999999999999989 0.99999999999999989 0.000
280 if (sig12 < 1 || m12x >= 0) {
281 // Need at least 2, to handle 90 0 90 180
282 if (sig12 < 3 * tiny_ ||
283 // Prevent negative s12 or m12 for short lines
284 (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
285 sig12 = m12x = s12x = 0;
286 m12x *= _b;
287 s12x *= _b;
288 a12 = sig12 / Math::degree();
289 } else
290 // m12 < 0, i.e., prolate and too close to anti-podal
291 meridian = false;
292 }
293
294 // somg12 > 1 marks that it needs to be calculated
295 real omg12 = 0, somg12 = 2, comg12 = 0;
296 if (!meridian &&
297 sbet1 == 0 && // and sbet2 == 0
298 (_f <= 0 || lon12s >= _f * 180)) {
299
300 // Geodesic runs along equator
301 calp1 = calp2 = 0; salp1 = salp2 = 1;
302 s12x = _a * lam12;
303 sig12 = omg12 = lam12 / _f1;
304 m12x = _b * sin(sig12);
305 if (outmask & GEODESICSCALE)
306 M12 = M21 = cos(sig12);
307 a12 = lon12 / _f1;
308
309 } else if (!meridian) {
310
311 // Now point1 and point2 belong within a hemisphere bounded by a
312 // meridian and geodesic is neither meridional or equatorial.
313
314 // Figure a starting point for Newton's method
315 real dnm;
316 sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
317 lam12, slam12, clam12,
318 salp1, calp1, salp2, calp2, dnm,
319 Ca);
320
321 if (sig12 >= 0) {
322 // Short lines (InverseStart sets salp2, calp2, dnm)
323 s12x = sig12 * _b * dnm;
324 m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
325 if (outmask & GEODESICSCALE)
326 M12 = M21 = cos(sig12 / dnm);
327 a12 = sig12 / Math::degree();
328 omg12 = lam12 / (_f1 * dnm);
329 } else {
330
331 // Newton's method. This is a straightforward solution of f(alp1) =
332 // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
333 // root in the interval (0, pi) and its derivative is positive at the
334 // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
335 // alp1. During the course of the iteration, a range (alp1a, alp1b) is
336 // maintained which brackets the root and with each evaluation of
337 // f(alp) the range is shrunk, if possible. Newton's method is
338 // restarted whenever the derivative of f is negative (because the new
339 // value of alp1 is then further from the solution) or if the new
340 // estimate of alp1 lies outside (0,pi); in this case, the new starting
341 // guess is taken to be (alp1a + alp1b) / 2.
342 //
343 // initial values to suppress warnings (if loop is executed 0 times)
344 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
345 unsigned numit = 0;
346 // Bracketing range
347 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
348 for (bool tripn = false, tripb = false;
349 numit < maxit2_ || GEOGRAPHICLIB_PANIC;
350 ++numit) {
351 // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
352 // WGS84 and random input: mean = 2.85, sd = 0.60
353 real dv;
354 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
355 slam12, clam12,
356 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
357 eps, domg12, numit < maxit1_, dv, Ca);
358 // Reversed test to allow escape with NaNs
359 if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_)) break;
360 // Update bracketing values
361 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
362 { salp1b = salp1; calp1b = calp1; }
363 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
364 { salp1a = salp1; calp1a = calp1; }
365 if (numit < maxit1_ && dv > 0) {
366 real
367 dalp1 = -v/dv;
368 real
369 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
370 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
371 if (nsalp1 > 0 && abs(dalp1) < Math::pi()) {
372 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
373 salp1 = nsalp1;
374 Math::norm(salp1, calp1);
375 // In some regimes we don't get quadratic convergence because
376 // slope -> 0. So use convergence conditions based on epsilon
377 // instead of sqrt(epsilon).
378 tripn = abs(v) <= 16 * tol0_;
379 continue;
380 }
381 }
382 // Either dv was not positive or updated value was outside legal
383 // range. Use the midpoint of the bracket as the next estimate.
384 // This mechanism is not needed for the WGS84 ellipsoid, but it does
385 // catch problems with more eccentric ellipsoids. Its efficacy is
386 // such for the WGS84 test set with the starting guess set to alp1 =
387 // 90deg:
388 // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
389 // WGS84 and random input: mean = 4.74, sd = 0.99
390 salp1 = (salp1a + salp1b)/2;
391 calp1 = (calp1a + calp1b)/2;
392 Math::norm(salp1, calp1);
393 tripn = false;
394 tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
395 abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
396 }
397 {
398 real dummy;
399 // Ensure that the reduced length and geodesic scale are computed in
400 // a "canonical" way, with the I2 integral.
401 unsigned lengthmask = outmask |
402 (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE);
403 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
404 cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
405 }
406 m12x *= _b;
407 s12x *= _b;
408 a12 = sig12 / Math::degree();
409 if (outmask & AREA) {
410 // omg12 = lam12 - domg12
411 real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
412 somg12 = slam12 * cdomg12 - clam12 * sdomg12;
413 comg12 = clam12 * cdomg12 + slam12 * sdomg12;
414 }
415 }
416 }
417
418 if (outmask & DISTANCE)
419 s12 = 0 + s12x; // Convert -0 to 0
420
421 if (outmask & REDUCEDLENGTH)
422 m12 = 0 + m12x; // Convert -0 to 0
423
424 if (outmask & AREA) {
425 real
426 // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
427 salp0 = salp1 * cbet1,
428 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
429 real alp12;
430 if (calp0 != 0 && salp0 != 0) {
431 real
432 // From Lambda12: tan(bet) = tan(sig) * cos(alp)
433 ssig1 = sbet1, csig1 = calp1 * cbet1,
434 ssig2 = sbet2, csig2 = calp2 * cbet2,
435 k2 = Math::sq(calp0) * _ep2,
436 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
437 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
438 A4 = Math::sq(_a) * calp0 * salp0 * _e2;
439 Math::norm(ssig1, csig1);
440 Math::norm(ssig2, csig2);
441 C4f(eps, Ca);
442 real
443 B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
444 B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
445 S12 = A4 * (B42 - B41);
446 } else
447 // Avoid problems with indeterminate sig1, sig2 on equator
448 S12 = 0;
449
450 if (!meridian && somg12 > 1) {
451 somg12 = sin(omg12); comg12 = cos(omg12);
452 }
453
454 if (!meridian &&
455 // omg12 < 3/4 * pi
456 comg12 > -real(0.7071) && // Long difference not too big
457 sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
458 // Use tan(Gamma/2) = tan(omg12/2)
459 // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
460 // with tan(x/2) = sin(x)/(1+cos(x))
461 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
462 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
463 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
464 } else {
465 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
466 real
467 salp12 = salp2 * calp1 - calp2 * salp1,
468 calp12 = calp2 * calp1 + salp2 * salp1;
469 // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
470 // salp12 = -0 and alp12 = -180. However this depends on the sign
471 // being attached to 0 correctly. The following ensures the correct
472 // behavior.
473 if (salp12 == 0 && calp12 < 0) {
474 salp12 = tiny_ * calp1;
475 calp12 = -1;
476 }
477 alp12 = atan2(salp12, calp12);
478 }
479 S12 += _c2 * alp12;
480 S12 *= swapp * lonsign * latsign;
481 // Convert -0 to 0
482 S12 += 0;
483 }
484
485 // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
486 if (swapp < 0) {
487 swap(salp1, salp2);
488 swap(calp1, calp2);
489 if (outmask & GEODESICSCALE)
490 swap(M12, M21);
491 }
492
493 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
494 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
495
496 // Returned value in [0, 180]
497 return a12;
498 }
499
500 Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
501 unsigned outmask,
502 real& s12, real& azi1, real& azi2,
503 real& m12, real& M12, real& M21,
504 real& S12) const {
505 outmask &= OUT_MASK;
506 real salp1, calp1, salp2, calp2,
507 a12 = GenInverse(lat1, lon1, lat2, lon2,
508 outmask, s12, salp1, calp1, salp2, calp2,
509 m12, M12, M21, S12);
510 if (outmask & AZIMUTH) {
511 azi1 = Math::atan2d(salp1, calp1);
512 azi2 = Math::atan2d(salp2, calp2);
513 }
514 return a12;
515 }
516
518 real lat2, real lon2,
519 unsigned caps) const {
520 real t, salp1, calp1, salp2, calp2,
521 a12 = GenInverse(lat1, lon1, lat2, lon2,
522 // No need to specify AZIMUTH here
523 0u, t, salp1, calp1, salp2, calp2,
524 t, t, t, t),
525 azi1 = Math::atan2d(salp1, calp1);
526 // Ensure that a12 can be converted to a distance
527 if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
528 return
529 GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12);
530 }
531
532 void Geodesic::Lengths(real eps, real sig12,
533 real ssig1, real csig1, real dn1,
534 real ssig2, real csig2, real dn2,
535 real cbet1, real cbet2, unsigned outmask,
536 real& s12b, real& m12b, real& m0,
537 real& M12, real& M21,
538 // Scratch area of the right size
539 real Ca[]) const {
540 // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
541 // and m0 = coefficient of secular term in expression for reduced length.
542
543 outmask &= OUT_MASK;
544 // outmask & DISTANCE: set s12b
545 // outmask & REDUCEDLENGTH: set m12b & m0
546 // outmask & GEODESICSCALE: set M12 & M21
547
548 real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
549 real Cb[nC2_ + 1];
550 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
551 A1 = A1m1f(eps);
552 C1f(eps, Ca);
553 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
554 A2 = A2m1f(eps);
555 C2f(eps, Cb);
556 m0x = A1 - A2;
557 A2 = 1 + A2;
558 }
559 A1 = 1 + A1;
560 }
561 if (outmask & DISTANCE) {
562 real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
563 SinCosSeries(true, ssig1, csig1, Ca, nC1_);
564 // Missing a factor of _b
565 s12b = A1 * (sig12 + B1);
566 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
567 real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
568 SinCosSeries(true, ssig1, csig1, Cb, nC2_);
569 J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
570 }
571 } else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
572 // Assume here that nC1_ >= nC2_
573 for (int l = 1; l <= nC2_; ++l)
574 Cb[l] = A1 * Ca[l] - A2 * Cb[l];
575 J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
576 SinCosSeries(true, ssig1, csig1, Cb, nC2_));
577 }
578 if (outmask & REDUCEDLENGTH) {
579 m0 = m0x;
580 // Missing a factor of _b.
581 // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
582 // accurate cancellation in the case of coincident points.
583 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
584 csig1 * csig2 * J12;
585 }
586 if (outmask & GEODESICSCALE) {
587 real csig12 = csig1 * csig2 + ssig1 * ssig2;
588 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
589 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
590 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
591 }
592 }
593
594 Math::real Geodesic::Astroid(real x, real y) {
595 // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
596 // This solution is adapted from Geocentric::Reverse.
597 real k;
598 real
599 p = Math::sq(x),
600 q = Math::sq(y),
601 r = (p + q - 1) / 6;
602 if ( !(q == 0 && r <= 0) ) {
603 real
604 // Avoid possible division by zero when r = 0 by multiplying equations
605 // for s and t by r^3 and r, resp.
606 S = p * q / 4, // S = r^3 * s
607 r2 = Math::sq(r),
608 r3 = r * r2,
609 // The discriminant of the quadratic equation for T3. This is zero on
610 // the evolute curve p^(1/3)+q^(1/3) = 1
611 disc = S * (S + 2 * r3);
612 real u = r;
613 if (disc >= 0) {
614 real T3 = S + r3;
615 // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
616 // of precision due to cancellation. The result is unchanged because
617 // of the way the T is used in definition of u.
618 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
619 // N.B. cbrt always returns the real root. cbrt(-8) = -2.
620 real T = cbrt(T3); // T = r * t
621 // T can be zero; but then r2 / T -> 0.
622 u += T + (T != 0 ? r2 / T : 0);
623 } else {
624 // T is complex, but the way u is defined the result is real.
625 real ang = atan2(sqrt(-disc), -(S + r3));
626 // There are three possible cube roots. We choose the root which
627 // avoids cancellation. Note that disc < 0 implies that r < 0.
628 u += 2 * r * cos(ang / 3);
629 }
630 real
631 v = sqrt(Math::sq(u) + q), // guaranteed positive
632 // Avoid loss of accuracy when u < 0.
633 uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
634 w = (uv - q) / (2 * v); // positive?
635 // Rearrange expression for k to avoid loss of accuracy due to
636 // subtraction. Division by 0 not possible because uv > 0, w >= 0.
637 k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
638 } else { // q == 0 && r <= 0
639 // y = 0 with |x| <= 1. Handle this case directly.
640 // for y small, positive root is k = abs(y)/sqrt(1-x^2)
641 k = 0;
642 }
643 return k;
644 }
645
646 Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
647 real sbet2, real cbet2, real dn2,
648 real lam12, real slam12, real clam12,
649 real& salp1, real& calp1,
650 // Only updated if return val >= 0
651 real& salp2, real& calp2,
652 // Only updated for short lines
653 real& dnm,
654 // Scratch area of the right size
655 real Ca[]) const {
656 // Return a starting point for Newton's method in salp1 and calp1 (function
657 // value is -1). If Newton's method doesn't need to be used, return also
658 // salp2 and calp2 and function value is sig12.
659 real
660 sig12 = -1, // Return value
661 // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
662 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
663 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
664 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
665 bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
666 cbet2 * lam12 < real(0.5);
667 real somg12, comg12;
668 if (shortline) {
669 real sbetm2 = Math::sq(sbet1 + sbet2);
670 // sin((bet1+bet2)/2)^2
671 // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
672 sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
673 dnm = sqrt(1 + _ep2 * sbetm2);
674 real omg12 = lam12 / (_f1 * dnm);
675 somg12 = sin(omg12); comg12 = cos(omg12);
676 } else {
677 somg12 = slam12; comg12 = clam12;
678 }
679
680 salp1 = cbet2 * somg12;
681 calp1 = comg12 >= 0 ?
682 sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
683 sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
684
685 real
686 ssig12 = hypot(salp1, calp1),
687 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
688
689 if (shortline && ssig12 < _etol2) {
690 // really short lines
691 salp2 = cbet1 * somg12;
692 calp2 = sbet12 - cbet1 * sbet2 *
693 (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
694 Math::norm(salp2, calp2);
695 // Set return value
696 sig12 = atan2(ssig12, csig12);
697 } else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
698 csig12 >= 0 ||
699 ssig12 >= 6 * abs(_n) * Math::pi() * Math::sq(cbet1)) {
700 // Nothing to do, zeroth order spherical approximation is OK
701 } else {
702 // Scale lam12 and bet2 to x, y coordinate system where antipodal point
703 // is at origin and singular point is at y = 0, x = -1.
704 real y, lamscale, betscale;
705 // Volatile declaration needed to fix inverse case
706 // 56.320923501171 0 -56.320923501171 179.664747671772880215
707 // which otherwise fails with g++ 4.4.4 x86 -O3
709 real lam12x = atan2(-slam12, -clam12); // lam12 - pi
710 if (_f >= 0) { // In fact f == 0 does not get here
711 // x = dlong, y = dlat
712 {
713 real
714 k2 = Math::sq(sbet1) * _ep2,
715 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
716 lamscale = _f * cbet1 * A3f(eps) * Math::pi();
717 }
718 betscale = lamscale * cbet1;
719
720 x = lam12x / lamscale;
721 y = sbet12a / betscale;
722 } else { // _f < 0
723 // x = dlat, y = dlong
724 real
725 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
726 bet12a = atan2(sbet12a, cbet12a);
727 real m12b, m0, dummy;
728 // In the case of lon12 = 180, this repeats a calculation made in
729 // Inverse.
730 Lengths(_n, Math::pi() + bet12a,
731 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
732 cbet1, cbet2,
733 REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca);
734 x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
735 betscale = x < -real(0.01) ? sbet12a / x :
736 -_f * Math::sq(cbet1) * Math::pi();
737 lamscale = betscale / cbet1;
738 y = lam12x / lamscale;
739 }
740
741 if (y > -tol1_ && x > -1 - xthresh_) {
742 // strip near cut
743 // Need real(x) here to cast away the volatility of x for min/max
744 if (_f >= 0) {
745 salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 - Math::sq(salp1));
746 } else {
747 calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
748 salp1 = sqrt(1 - Math::sq(calp1));
749 }
750 } else {
751 // Estimate alp1, by solving the astroid problem.
752 //
753 // Could estimate alpha1 = theta + pi/2, directly, i.e.,
754 // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
755 // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
756 //
757 // However, it's better to estimate omg12 from astroid and use
758 // spherical formula to compute alp1. This reduces the mean number of
759 // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
760 // (min 0 max 5). The changes in the number of iterations are as
761 // follows:
762 //
763 // change percent
764 // 1 5
765 // 0 78
766 // -1 16
767 // -2 0.6
768 // -3 0.04
769 // -4 0.002
770 //
771 // The histogram of iterations is (m = number of iterations estimating
772 // alp1 directly, n = number of iterations estimating via omg12, total
773 // number of trials = 148605):
774 //
775 // iter m n
776 // 0 148 186
777 // 1 13046 13845
778 // 2 93315 102225
779 // 3 36189 32341
780 // 4 5396 7
781 // 5 455 1
782 // 6 56 0
783 //
784 // Because omg12 is near pi, estimate work with omg12a = pi - omg12
785 real k = Astroid(x, y);
786 real
787 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
788 somg12 = sin(omg12a); comg12 = -cos(omg12a);
789 // Update spherical estimate of alp1 using omg12 instead of lam12
790 salp1 = cbet2 * somg12;
791 calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
792 }
793 }
794 // Sanity check on starting guess. Backwards check allows NaN through.
795 if (!(salp1 <= 0))
796 Math::norm(salp1, calp1);
797 else {
798 salp1 = 1; calp1 = 0;
799 }
800 return sig12;
801 }
802
803 Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
804 real sbet2, real cbet2, real dn2,
805 real salp1, real calp1,
806 real slam120, real clam120,
807 real& salp2, real& calp2,
808 real& sig12,
809 real& ssig1, real& csig1,
810 real& ssig2, real& csig2,
811 real& eps, real& domg12,
812 bool diffp, real& dlam12,
813 // Scratch area of the right size
814 real Ca[]) const {
815
816 if (sbet1 == 0 && calp1 == 0)
817 // Break degeneracy of equatorial line. This case has already been
818 // handled.
819 calp1 = -tiny_;
820
821 real
822 // sin(alp1) * cos(bet1) = sin(alp0)
823 salp0 = salp1 * cbet1,
824 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
825
826 real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
827 // tan(bet1) = tan(sig1) * cos(alp1)
828 // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
829 ssig1 = sbet1; somg1 = salp0 * sbet1;
830 csig1 = comg1 = calp1 * cbet1;
831 Math::norm(ssig1, csig1);
832 // Math::norm(somg1, comg1); -- don't need to normalize!
833
834 // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
835 // about this case, since this can yield singularities in the Newton
836 // iteration.
837 // sin(alp2) * cos(bet2) = sin(alp0)
838 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
839 // calp2 = sqrt(1 - sq(salp2))
840 // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
841 // and subst for calp0 and rearrange to give (choose positive sqrt
842 // to give alp2 in [0, pi/2]).
843 calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
844 sqrt(Math::sq(calp1 * cbet1) +
845 (cbet1 < -sbet1 ?
846 (cbet2 - cbet1) * (cbet1 + cbet2) :
847 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
848 abs(calp1);
849 // tan(bet2) = tan(sig2) * cos(alp2)
850 // tan(omg2) = sin(alp0) * tan(sig2).
851 ssig2 = sbet2; somg2 = salp0 * sbet2;
852 csig2 = comg2 = calp2 * cbet2;
853 Math::norm(ssig2, csig2);
854 // Math::norm(somg2, comg2); -- don't need to normalize!
855
856 // sig12 = sig2 - sig1, limit to [0, pi]
857 sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
858 csig1 * csig2 + ssig1 * ssig2);
859
860 // omg12 = omg2 - omg1, limit to [0, pi]
861 somg12 = max(real(0), comg1 * somg2 - somg1 * comg2);
862 comg12 = comg1 * comg2 + somg1 * somg2;
863 // eta = omg12 - lam120
864 real eta = atan2(somg12 * clam120 - comg12 * slam120,
865 comg12 * clam120 + somg12 * slam120);
866 real B312;
867 real k2 = Math::sq(calp0) * _ep2;
868 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
869 C3f(eps, Ca);
870 B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) -
871 SinCosSeries(true, ssig1, csig1, Ca, nC3_-1));
872 domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
873 lam12 = eta + domg12;
874
875 if (diffp) {
876 if (calp2 == 0)
877 dlam12 = - 2 * _f1 * dn1 / sbet1;
878 else {
879 real dummy;
880 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
881 cbet1, cbet2, REDUCEDLENGTH,
882 dummy, dlam12, dummy, dummy, dummy, Ca);
883 dlam12 *= _f1 / (calp2 * cbet2);
884 }
885 }
886
887 return lam12;
888 }
889
890 Math::real Geodesic::A3f(real eps) const {
891 // Evaluate A3
892 return Math::polyval(nA3_ - 1, _A3x, eps);
893 }
894
895 void Geodesic::C3f(real eps, real c[]) const {
896 // Evaluate C3 coeffs
897 // Elements c[1] thru c[nC3_ - 1] are set
898 real mult = 1;
899 int o = 0;
900 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
901 int m = nC3_ - l - 1; // order of polynomial in eps
902 mult *= eps;
903 c[l] = mult * Math::polyval(m, _C3x + o, eps);
904 o += m + 1;
905 }
906 // Post condition: o == nC3x_
907 }
908
909 void Geodesic::C4f(real eps, real c[]) const {
910 // Evaluate C4 coeffs
911 // Elements c[0] thru c[nC4_ - 1] are set
912 real mult = 1;
913 int o = 0;
914 for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
915 int m = nC4_ - l - 1; // order of polynomial in eps
916 c[l] = mult * Math::polyval(m, _C4x + o, eps);
917 o += m + 1;
918 mult *= eps;
919 }
920 // Post condition: o == nC4x_
921 }
922
923 // The static const coefficient arrays in the following functions are
924 // generated by Maxima and give the coefficients of the Taylor expansions for
925 // the geodesics. The convention on the order of these coefficients is as
926 // follows:
927 //
928 // ascending order in the trigonometric expansion,
929 // then powers of eps in descending order,
930 // finally powers of n in descending order.
931 //
932 // (For some expansions, only a subset of levels occur.) For each polynomial
933 // of order n at the lowest level, the (n+1) coefficients of the polynomial
934 // are followed by a divisor which is applied to the whole polynomial. In
935 // this way, the coefficients are expressible with no round off error. The
936 // sizes of the coefficient arrays are:
937 //
938 // A1m1f, A2m1f = floor(N/2) + 2
939 // C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
940 // C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
941 // C4coeff = N * (N + 1) * (N + 5) / 6
942 //
943 // where N = GEOGRAPHICLIB_GEODESIC_ORDER
944 // = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
945
946 // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
947 Math::real Geodesic::A1m1f(real eps) {
948 // Generated by Maxima on 2015-05-05 18:08:12-04:00
949#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
950 static const real coeff[] = {
951 // (1-eps)*A1-1, polynomial in eps2 of order 1
952 1, 0, 4,
953 };
954#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
955 static const real coeff[] = {
956 // (1-eps)*A1-1, polynomial in eps2 of order 2
957 1, 16, 0, 64,
958 };
959#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
960 static const real coeff[] = {
961 // (1-eps)*A1-1, polynomial in eps2 of order 3
962 1, 4, 64, 0, 256,
963 };
964#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
965 static const real coeff[] = {
966 // (1-eps)*A1-1, polynomial in eps2 of order 4
967 25, 64, 256, 4096, 0, 16384,
968 };
969#else
970#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
971#endif
972 static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2,
973 "Coefficient array size mismatch in A1m1f");
974 int m = nA1_/2;
975 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
976 return (t + eps) / (1 - eps);
977 }
978
979 // The coefficients C1[l] in the Fourier expansion of B1
980 void Geodesic::C1f(real eps, real c[]) {
981 // Generated by Maxima on 2015-05-05 18:08:12-04:00
982#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
983 static const real coeff[] = {
984 // C1[1]/eps^1, polynomial in eps2 of order 1
985 3, -8, 16,
986 // C1[2]/eps^2, polynomial in eps2 of order 0
987 -1, 16,
988 // C1[3]/eps^3, polynomial in eps2 of order 0
989 -1, 48,
990 };
991#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
992 static const real coeff[] = {
993 // C1[1]/eps^1, polynomial in eps2 of order 1
994 3, -8, 16,
995 // C1[2]/eps^2, polynomial in eps2 of order 1
996 1, -2, 32,
997 // C1[3]/eps^3, polynomial in eps2 of order 0
998 -1, 48,
999 // C1[4]/eps^4, polynomial in eps2 of order 0
1000 -5, 512,
1001 };
1002#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1003 static const real coeff[] = {
1004 // C1[1]/eps^1, polynomial in eps2 of order 2
1005 -1, 6, -16, 32,
1006 // C1[2]/eps^2, polynomial in eps2 of order 1
1007 1, -2, 32,
1008 // C1[3]/eps^3, polynomial in eps2 of order 1
1009 9, -16, 768,
1010 // C1[4]/eps^4, polynomial in eps2 of order 0
1011 -5, 512,
1012 // C1[5]/eps^5, polynomial in eps2 of order 0
1013 -7, 1280,
1014 };
1015#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1016 static const real coeff[] = {
1017 // C1[1]/eps^1, polynomial in eps2 of order 2
1018 -1, 6, -16, 32,
1019 // C1[2]/eps^2, polynomial in eps2 of order 2
1020 -9, 64, -128, 2048,
1021 // C1[3]/eps^3, polynomial in eps2 of order 1
1022 9, -16, 768,
1023 // C1[4]/eps^4, polynomial in eps2 of order 1
1024 3, -5, 512,
1025 // C1[5]/eps^5, polynomial in eps2 of order 0
1026 -7, 1280,
1027 // C1[6]/eps^6, polynomial in eps2 of order 0
1028 -7, 2048,
1029 };
1030#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1031 static const real coeff[] = {
1032 // C1[1]/eps^1, polynomial in eps2 of order 3
1033 19, -64, 384, -1024, 2048,
1034 // C1[2]/eps^2, polynomial in eps2 of order 2
1035 -9, 64, -128, 2048,
1036 // C1[3]/eps^3, polynomial in eps2 of order 2
1037 -9, 72, -128, 6144,
1038 // C1[4]/eps^4, polynomial in eps2 of order 1
1039 3, -5, 512,
1040 // C1[5]/eps^5, polynomial in eps2 of order 1
1041 35, -56, 10240,
1042 // C1[6]/eps^6, polynomial in eps2 of order 0
1043 -7, 2048,
1044 // C1[7]/eps^7, polynomial in eps2 of order 0
1045 -33, 14336,
1046 };
1047#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1048 static const real coeff[] = {
1049 // C1[1]/eps^1, polynomial in eps2 of order 3
1050 19, -64, 384, -1024, 2048,
1051 // C1[2]/eps^2, polynomial in eps2 of order 3
1052 7, -18, 128, -256, 4096,
1053 // C1[3]/eps^3, polynomial in eps2 of order 2
1054 -9, 72, -128, 6144,
1055 // C1[4]/eps^4, polynomial in eps2 of order 2
1056 -11, 96, -160, 16384,
1057 // C1[5]/eps^5, polynomial in eps2 of order 1
1058 35, -56, 10240,
1059 // C1[6]/eps^6, polynomial in eps2 of order 1
1060 9, -14, 4096,
1061 // C1[7]/eps^7, polynomial in eps2 of order 0
1062 -33, 14336,
1063 // C1[8]/eps^8, polynomial in eps2 of order 0
1064 -429, 262144,
1065 };
1066#else
1067#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1068#endif
1069 static_assert(sizeof(coeff) / sizeof(real) ==
1070 (nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4,
1071 "Coefficient array size mismatch in C1f");
1072 real
1073 eps2 = Math::sq(eps),
1074 d = eps;
1075 int o = 0;
1076 for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
1077 int m = (nC1_ - l) / 2; // order of polynomial in eps^2
1078 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1079 o += m + 2;
1080 d *= eps;
1081 }
1082 // Post condition: o == sizeof(coeff) / sizeof(real)
1083 }
1084
1085 // The coefficients C1p[l] in the Fourier expansion of B1p
1086 void Geodesic::C1pf(real eps, real c[]) {
1087 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1088#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1089 static const real coeff[] = {
1090 // C1p[1]/eps^1, polynomial in eps2 of order 1
1091 -9, 16, 32,
1092 // C1p[2]/eps^2, polynomial in eps2 of order 0
1093 5, 16,
1094 // C1p[3]/eps^3, polynomial in eps2 of order 0
1095 29, 96,
1096 };
1097#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1098 static const real coeff[] = {
1099 // C1p[1]/eps^1, polynomial in eps2 of order 1
1100 -9, 16, 32,
1101 // C1p[2]/eps^2, polynomial in eps2 of order 1
1102 -37, 30, 96,
1103 // C1p[3]/eps^3, polynomial in eps2 of order 0
1104 29, 96,
1105 // C1p[4]/eps^4, polynomial in eps2 of order 0
1106 539, 1536,
1107 };
1108#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1109 static const real coeff[] = {
1110 // C1p[1]/eps^1, polynomial in eps2 of order 2
1111 205, -432, 768, 1536,
1112 // C1p[2]/eps^2, polynomial in eps2 of order 1
1113 -37, 30, 96,
1114 // C1p[3]/eps^3, polynomial in eps2 of order 1
1115 -225, 116, 384,
1116 // C1p[4]/eps^4, polynomial in eps2 of order 0
1117 539, 1536,
1118 // C1p[5]/eps^5, polynomial in eps2 of order 0
1119 3467, 7680,
1120 };
1121#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1122 static const real coeff[] = {
1123 // C1p[1]/eps^1, polynomial in eps2 of order 2
1124 205, -432, 768, 1536,
1125 // C1p[2]/eps^2, polynomial in eps2 of order 2
1126 4005, -4736, 3840, 12288,
1127 // C1p[3]/eps^3, polynomial in eps2 of order 1
1128 -225, 116, 384,
1129 // C1p[4]/eps^4, polynomial in eps2 of order 1
1130 -7173, 2695, 7680,
1131 // C1p[5]/eps^5, polynomial in eps2 of order 0
1132 3467, 7680,
1133 // C1p[6]/eps^6, polynomial in eps2 of order 0
1134 38081, 61440,
1135 };
1136#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1137 static const real coeff[] = {
1138 // C1p[1]/eps^1, polynomial in eps2 of order 3
1139 -4879, 9840, -20736, 36864, 73728,
1140 // C1p[2]/eps^2, polynomial in eps2 of order 2
1141 4005, -4736, 3840, 12288,
1142 // C1p[3]/eps^3, polynomial in eps2 of order 2
1143 8703, -7200, 3712, 12288,
1144 // C1p[4]/eps^4, polynomial in eps2 of order 1
1145 -7173, 2695, 7680,
1146 // C1p[5]/eps^5, polynomial in eps2 of order 1
1147 -141115, 41604, 92160,
1148 // C1p[6]/eps^6, polynomial in eps2 of order 0
1149 38081, 61440,
1150 // C1p[7]/eps^7, polynomial in eps2 of order 0
1151 459485, 516096,
1152 };
1153#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1154 static const real coeff[] = {
1155 // C1p[1]/eps^1, polynomial in eps2 of order 3
1156 -4879, 9840, -20736, 36864, 73728,
1157 // C1p[2]/eps^2, polynomial in eps2 of order 3
1158 -86171, 120150, -142080, 115200, 368640,
1159 // C1p[3]/eps^3, polynomial in eps2 of order 2
1160 8703, -7200, 3712, 12288,
1161 // C1p[4]/eps^4, polynomial in eps2 of order 2
1162 1082857, -688608, 258720, 737280,
1163 // C1p[5]/eps^5, polynomial in eps2 of order 1
1164 -141115, 41604, 92160,
1165 // C1p[6]/eps^6, polynomial in eps2 of order 1
1166 -2200311, 533134, 860160,
1167 // C1p[7]/eps^7, polynomial in eps2 of order 0
1168 459485, 516096,
1169 // C1p[8]/eps^8, polynomial in eps2 of order 0
1170 109167851, 82575360,
1171 };
1172#else
1173#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1174#endif
1175 static_assert(sizeof(coeff) / sizeof(real) ==
1176 (nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4,
1177 "Coefficient array size mismatch in C1pf");
1178 real
1179 eps2 = Math::sq(eps),
1180 d = eps;
1181 int o = 0;
1182 for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
1183 int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
1184 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1185 o += m + 2;
1186 d *= eps;
1187 }
1188 // Post condition: o == sizeof(coeff) / sizeof(real)
1189 }
1190
1191 // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
1192 Math::real Geodesic::A2m1f(real eps) {
1193 // Generated by Maxima on 2015-05-29 08:09:47-04:00
1194#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
1195 static const real coeff[] = {
1196 // (eps+1)*A2-1, polynomial in eps2 of order 1
1197 -3, 0, 4,
1198 }; // count = 3
1199#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
1200 static const real coeff[] = {
1201 // (eps+1)*A2-1, polynomial in eps2 of order 2
1202 -7, -48, 0, 64,
1203 }; // count = 4
1204#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
1205 static const real coeff[] = {
1206 // (eps+1)*A2-1, polynomial in eps2 of order 3
1207 -11, -28, -192, 0, 256,
1208 }; // count = 5
1209#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
1210 static const real coeff[] = {
1211 // (eps+1)*A2-1, polynomial in eps2 of order 4
1212 -375, -704, -1792, -12288, 0, 16384,
1213 }; // count = 6
1214#else
1215#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1216#endif
1217 static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2,
1218 "Coefficient array size mismatch in A2m1f");
1219 int m = nA2_/2;
1220 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
1221 return (t - eps) / (1 + eps);
1222 }
1223
1224 // The coefficients C2[l] in the Fourier expansion of B2
1225 void Geodesic::C2f(real eps, real c[]) {
1226 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1227#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1228 static const real coeff[] = {
1229 // C2[1]/eps^1, polynomial in eps2 of order 1
1230 1, 8, 16,
1231 // C2[2]/eps^2, polynomial in eps2 of order 0
1232 3, 16,
1233 // C2[3]/eps^3, polynomial in eps2 of order 0
1234 5, 48,
1235 };
1236#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1237 static const real coeff[] = {
1238 // C2[1]/eps^1, polynomial in eps2 of order 1
1239 1, 8, 16,
1240 // C2[2]/eps^2, polynomial in eps2 of order 1
1241 1, 6, 32,
1242 // C2[3]/eps^3, polynomial in eps2 of order 0
1243 5, 48,
1244 // C2[4]/eps^4, polynomial in eps2 of order 0
1245 35, 512,
1246 };
1247#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1248 static const real coeff[] = {
1249 // C2[1]/eps^1, polynomial in eps2 of order 2
1250 1, 2, 16, 32,
1251 // C2[2]/eps^2, polynomial in eps2 of order 1
1252 1, 6, 32,
1253 // C2[3]/eps^3, polynomial in eps2 of order 1
1254 15, 80, 768,
1255 // C2[4]/eps^4, polynomial in eps2 of order 0
1256 35, 512,
1257 // C2[5]/eps^5, polynomial in eps2 of order 0
1258 63, 1280,
1259 };
1260#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1261 static const real coeff[] = {
1262 // C2[1]/eps^1, polynomial in eps2 of order 2
1263 1, 2, 16, 32,
1264 // C2[2]/eps^2, polynomial in eps2 of order 2
1265 35, 64, 384, 2048,
1266 // C2[3]/eps^3, polynomial in eps2 of order 1
1267 15, 80, 768,
1268 // C2[4]/eps^4, polynomial in eps2 of order 1
1269 7, 35, 512,
1270 // C2[5]/eps^5, polynomial in eps2 of order 0
1271 63, 1280,
1272 // C2[6]/eps^6, polynomial in eps2 of order 0
1273 77, 2048,
1274 };
1275#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1276 static const real coeff[] = {
1277 // C2[1]/eps^1, polynomial in eps2 of order 3
1278 41, 64, 128, 1024, 2048,
1279 // C2[2]/eps^2, polynomial in eps2 of order 2
1280 35, 64, 384, 2048,
1281 // C2[3]/eps^3, polynomial in eps2 of order 2
1282 69, 120, 640, 6144,
1283 // C2[4]/eps^4, polynomial in eps2 of order 1
1284 7, 35, 512,
1285 // C2[5]/eps^5, polynomial in eps2 of order 1
1286 105, 504, 10240,
1287 // C2[6]/eps^6, polynomial in eps2 of order 0
1288 77, 2048,
1289 // C2[7]/eps^7, polynomial in eps2 of order 0
1290 429, 14336,
1291 };
1292#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1293 static const real coeff[] = {
1294 // C2[1]/eps^1, polynomial in eps2 of order 3
1295 41, 64, 128, 1024, 2048,
1296 // C2[2]/eps^2, polynomial in eps2 of order 3
1297 47, 70, 128, 768, 4096,
1298 // C2[3]/eps^3, polynomial in eps2 of order 2
1299 69, 120, 640, 6144,
1300 // C2[4]/eps^4, polynomial in eps2 of order 2
1301 133, 224, 1120, 16384,
1302 // C2[5]/eps^5, polynomial in eps2 of order 1
1303 105, 504, 10240,
1304 // C2[6]/eps^6, polynomial in eps2 of order 1
1305 33, 154, 4096,
1306 // C2[7]/eps^7, polynomial in eps2 of order 0
1307 429, 14336,
1308 // C2[8]/eps^8, polynomial in eps2 of order 0
1309 6435, 262144,
1310 };
1311#else
1312#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1313#endif
1314 static_assert(sizeof(coeff) / sizeof(real) ==
1315 (nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4,
1316 "Coefficient array size mismatch in C2f");
1317 real
1318 eps2 = Math::sq(eps),
1319 d = eps;
1320 int o = 0;
1321 for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
1322 int m = (nC2_ - l) / 2; // order of polynomial in eps^2
1323 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1324 o += m + 2;
1325 d *= eps;
1326 }
1327 // Post condition: o == sizeof(coeff) / sizeof(real)
1328 }
1329
1330 // The scale factor A3 = mean value of (d/dsigma)I3
1331 void Geodesic::A3coeff() {
1332 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1333#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1334 static const real coeff[] = {
1335 // A3, coeff of eps^2, polynomial in n of order 0
1336 -1, 4,
1337 // A3, coeff of eps^1, polynomial in n of order 1
1338 1, -1, 2,
1339 // A3, coeff of eps^0, polynomial in n of order 0
1340 1, 1,
1341 };
1342#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1343 static const real coeff[] = {
1344 // A3, coeff of eps^3, polynomial in n of order 0
1345 -1, 16,
1346 // A3, coeff of eps^2, polynomial in n of order 1
1347 -1, -2, 8,
1348 // A3, coeff of eps^1, polynomial in n of order 1
1349 1, -1, 2,
1350 // A3, coeff of eps^0, polynomial in n of order 0
1351 1, 1,
1352 };
1353#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1354 static const real coeff[] = {
1355 // A3, coeff of eps^4, polynomial in n of order 0
1356 -3, 64,
1357 // A3, coeff of eps^3, polynomial in n of order 1
1358 -3, -1, 16,
1359 // A3, coeff of eps^2, polynomial in n of order 2
1360 3, -1, -2, 8,
1361 // A3, coeff of eps^1, polynomial in n of order 1
1362 1, -1, 2,
1363 // A3, coeff of eps^0, polynomial in n of order 0
1364 1, 1,
1365 };
1366#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1367 static const real coeff[] = {
1368 // A3, coeff of eps^5, polynomial in n of order 0
1369 -3, 128,
1370 // A3, coeff of eps^4, polynomial in n of order 1
1371 -2, -3, 64,
1372 // A3, coeff of eps^3, polynomial in n of order 2
1373 -1, -3, -1, 16,
1374 // A3, coeff of eps^2, polynomial in n of order 2
1375 3, -1, -2, 8,
1376 // A3, coeff of eps^1, polynomial in n of order 1
1377 1, -1, 2,
1378 // A3, coeff of eps^0, polynomial in n of order 0
1379 1, 1,
1380 };
1381#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1382 static const real coeff[] = {
1383 // A3, coeff of eps^6, polynomial in n of order 0
1384 -5, 256,
1385 // A3, coeff of eps^5, polynomial in n of order 1
1386 -5, -3, 128,
1387 // A3, coeff of eps^4, polynomial in n of order 2
1388 -10, -2, -3, 64,
1389 // A3, coeff of eps^3, polynomial in n of order 3
1390 5, -1, -3, -1, 16,
1391 // A3, coeff of eps^2, polynomial in n of order 2
1392 3, -1, -2, 8,
1393 // A3, coeff of eps^1, polynomial in n of order 1
1394 1, -1, 2,
1395 // A3, coeff of eps^0, polynomial in n of order 0
1396 1, 1,
1397 };
1398#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1399 static const real coeff[] = {
1400 // A3, coeff of eps^7, polynomial in n of order 0
1401 -25, 2048,
1402 // A3, coeff of eps^6, polynomial in n of order 1
1403 -15, -20, 1024,
1404 // A3, coeff of eps^5, polynomial in n of order 2
1405 -5, -10, -6, 256,
1406 // A3, coeff of eps^4, polynomial in n of order 3
1407 -5, -20, -4, -6, 128,
1408 // A3, coeff of eps^3, polynomial in n of order 3
1409 5, -1, -3, -1, 16,
1410 // A3, coeff of eps^2, polynomial in n of order 2
1411 3, -1, -2, 8,
1412 // A3, coeff of eps^1, polynomial in n of order 1
1413 1, -1, 2,
1414 // A3, coeff of eps^0, polynomial in n of order 0
1415 1, 1,
1416 };
1417#else
1418#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1419#endif
1420 static_assert(sizeof(coeff) / sizeof(real) ==
1421 (nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4,
1422 "Coefficient array size mismatch in A3f");
1423 int o = 0, k = 0;
1424 for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
1425 int m = min(nA3_ - j - 1, j); // order of polynomial in n
1426 _A3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1427 o += m + 2;
1428 }
1429 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
1430 }
1431
1432 // The coefficients C3[l] in the Fourier expansion of B3
1433 void Geodesic::C3coeff() {
1434 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1435#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1436 static const real coeff[] = {
1437 // C3[1], coeff of eps^2, polynomial in n of order 0
1438 1, 8,
1439 // C3[1], coeff of eps^1, polynomial in n of order 1
1440 -1, 1, 4,
1441 // C3[2], coeff of eps^2, polynomial in n of order 0
1442 1, 16,
1443 };
1444#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1445 static const real coeff[] = {
1446 // C3[1], coeff of eps^3, polynomial in n of order 0
1447 3, 64,
1448 // C3[1], coeff of eps^2, polynomial in n of order 1
1449 // This is a case where a leading 0 term has been inserted to maintain the
1450 // pattern in the orders of the polynomials.
1451 0, 1, 8,
1452 // C3[1], coeff of eps^1, polynomial in n of order 1
1453 -1, 1, 4,
1454 // C3[2], coeff of eps^3, polynomial in n of order 0
1455 3, 64,
1456 // C3[2], coeff of eps^2, polynomial in n of order 1
1457 -3, 2, 32,
1458 // C3[3], coeff of eps^3, polynomial in n of order 0
1459 5, 192,
1460 };
1461#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1462 static const real coeff[] = {
1463 // C3[1], coeff of eps^4, polynomial in n of order 0
1464 5, 128,
1465 // C3[1], coeff of eps^3, polynomial in n of order 1
1466 3, 3, 64,
1467 // C3[1], coeff of eps^2, polynomial in n of order 2
1468 -1, 0, 1, 8,
1469 // C3[1], coeff of eps^1, polynomial in n of order 1
1470 -1, 1, 4,
1471 // C3[2], coeff of eps^4, polynomial in n of order 0
1472 3, 128,
1473 // C3[2], coeff of eps^3, polynomial in n of order 1
1474 -2, 3, 64,
1475 // C3[2], coeff of eps^2, polynomial in n of order 2
1476 1, -3, 2, 32,
1477 // C3[3], coeff of eps^4, polynomial in n of order 0
1478 3, 128,
1479 // C3[3], coeff of eps^3, polynomial in n of order 1
1480 -9, 5, 192,
1481 // C3[4], coeff of eps^4, polynomial in n of order 0
1482 7, 512,
1483 };
1484#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1485 static const real coeff[] = {
1486 // C3[1], coeff of eps^5, polynomial in n of order 0
1487 3, 128,
1488 // C3[1], coeff of eps^4, polynomial in n of order 1
1489 2, 5, 128,
1490 // C3[1], coeff of eps^3, polynomial in n of order 2
1491 -1, 3, 3, 64,
1492 // C3[1], coeff of eps^2, polynomial in n of order 2
1493 -1, 0, 1, 8,
1494 // C3[1], coeff of eps^1, polynomial in n of order 1
1495 -1, 1, 4,
1496 // C3[2], coeff of eps^5, polynomial in n of order 0
1497 5, 256,
1498 // C3[2], coeff of eps^4, polynomial in n of order 1
1499 1, 3, 128,
1500 // C3[2], coeff of eps^3, polynomial in n of order 2
1501 -3, -2, 3, 64,
1502 // C3[2], coeff of eps^2, polynomial in n of order 2
1503 1, -3, 2, 32,
1504 // C3[3], coeff of eps^5, polynomial in n of order 0
1505 7, 512,
1506 // C3[3], coeff of eps^4, polynomial in n of order 1
1507 -10, 9, 384,
1508 // C3[3], coeff of eps^3, polynomial in n of order 2
1509 5, -9, 5, 192,
1510 // C3[4], coeff of eps^5, polynomial in n of order 0
1511 7, 512,
1512 // C3[4], coeff of eps^4, polynomial in n of order 1
1513 -14, 7, 512,
1514 // C3[5], coeff of eps^5, polynomial in n of order 0
1515 21, 2560,
1516 };
1517#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1518 static const real coeff[] = {
1519 // C3[1], coeff of eps^6, polynomial in n of order 0
1520 21, 1024,
1521 // C3[1], coeff of eps^5, polynomial in n of order 1
1522 11, 12, 512,
1523 // C3[1], coeff of eps^4, polynomial in n of order 2
1524 2, 2, 5, 128,
1525 // C3[1], coeff of eps^3, polynomial in n of order 3
1526 -5, -1, 3, 3, 64,
1527 // C3[1], coeff of eps^2, polynomial in n of order 2
1528 -1, 0, 1, 8,
1529 // C3[1], coeff of eps^1, polynomial in n of order 1
1530 -1, 1, 4,
1531 // C3[2], coeff of eps^6, polynomial in n of order 0
1532 27, 2048,
1533 // C3[2], coeff of eps^5, polynomial in n of order 1
1534 1, 5, 256,
1535 // C3[2], coeff of eps^4, polynomial in n of order 2
1536 -9, 2, 6, 256,
1537 // C3[2], coeff of eps^3, polynomial in n of order 3
1538 2, -3, -2, 3, 64,
1539 // C3[2], coeff of eps^2, polynomial in n of order 2
1540 1, -3, 2, 32,
1541 // C3[3], coeff of eps^6, polynomial in n of order 0
1542 3, 256,
1543 // C3[3], coeff of eps^5, polynomial in n of order 1
1544 -4, 21, 1536,
1545 // C3[3], coeff of eps^4, polynomial in n of order 2
1546 -6, -10, 9, 384,
1547 // C3[3], coeff of eps^3, polynomial in n of order 3
1548 -1, 5, -9, 5, 192,
1549 // C3[4], coeff of eps^6, polynomial in n of order 0
1550 9, 1024,
1551 // C3[4], coeff of eps^5, polynomial in n of order 1
1552 -10, 7, 512,
1553 // C3[4], coeff of eps^4, polynomial in n of order 2
1554 10, -14, 7, 512,
1555 // C3[5], coeff of eps^6, polynomial in n of order 0
1556 9, 1024,
1557 // C3[5], coeff of eps^5, polynomial in n of order 1
1558 -45, 21, 2560,
1559 // C3[6], coeff of eps^6, polynomial in n of order 0
1560 11, 2048,
1561 };
1562#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1563 static const real coeff[] = {
1564 // C3[1], coeff of eps^7, polynomial in n of order 0
1565 243, 16384,
1566 // C3[1], coeff of eps^6, polynomial in n of order 1
1567 10, 21, 1024,
1568 // C3[1], coeff of eps^5, polynomial in n of order 2
1569 3, 11, 12, 512,
1570 // C3[1], coeff of eps^4, polynomial in n of order 3
1571 -2, 2, 2, 5, 128,
1572 // C3[1], coeff of eps^3, polynomial in n of order 3
1573 -5, -1, 3, 3, 64,
1574 // C3[1], coeff of eps^2, polynomial in n of order 2
1575 -1, 0, 1, 8,
1576 // C3[1], coeff of eps^1, polynomial in n of order 1
1577 -1, 1, 4,
1578 // C3[2], coeff of eps^7, polynomial in n of order 0
1579 187, 16384,
1580 // C3[2], coeff of eps^6, polynomial in n of order 1
1581 69, 108, 8192,
1582 // C3[2], coeff of eps^5, polynomial in n of order 2
1583 -2, 1, 5, 256,
1584 // C3[2], coeff of eps^4, polynomial in n of order 3
1585 -6, -9, 2, 6, 256,
1586 // C3[2], coeff of eps^3, polynomial in n of order 3
1587 2, -3, -2, 3, 64,
1588 // C3[2], coeff of eps^2, polynomial in n of order 2
1589 1, -3, 2, 32,
1590 // C3[3], coeff of eps^7, polynomial in n of order 0
1591 139, 16384,
1592 // C3[3], coeff of eps^6, polynomial in n of order 1
1593 -1, 12, 1024,
1594 // C3[3], coeff of eps^5, polynomial in n of order 2
1595 -77, -8, 42, 3072,
1596 // C3[3], coeff of eps^4, polynomial in n of order 3
1597 10, -6, -10, 9, 384,
1598 // C3[3], coeff of eps^3, polynomial in n of order 3
1599 -1, 5, -9, 5, 192,
1600 // C3[4], coeff of eps^7, polynomial in n of order 0
1601 127, 16384,
1602 // C3[4], coeff of eps^6, polynomial in n of order 1
1603 -43, 72, 8192,
1604 // C3[4], coeff of eps^5, polynomial in n of order 2
1605 -7, -40, 28, 2048,
1606 // C3[4], coeff of eps^4, polynomial in n of order 3
1607 -7, 20, -28, 14, 1024,
1608 // C3[5], coeff of eps^7, polynomial in n of order 0
1609 99, 16384,
1610 // C3[5], coeff of eps^6, polynomial in n of order 1
1611 -15, 9, 1024,
1612 // C3[5], coeff of eps^5, polynomial in n of order 2
1613 75, -90, 42, 5120,
1614 // C3[6], coeff of eps^7, polynomial in n of order 0
1615 99, 16384,
1616 // C3[6], coeff of eps^6, polynomial in n of order 1
1617 -99, 44, 8192,
1618 // C3[7], coeff of eps^7, polynomial in n of order 0
1619 429, 114688,
1620 };
1621#else
1622#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1623#endif
1624 static_assert(sizeof(coeff) / sizeof(real) ==
1625 ((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8,
1626 "Coefficient array size mismatch in C3coeff");
1627 int o = 0, k = 0;
1628 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
1629 for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
1630 int m = min(nC3_ - j - 1, j); // order of polynomial in n
1631 _C3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1632 o += m + 2;
1633 }
1634 }
1635 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
1636 }
1637
1638 void Geodesic::C4coeff() {
1639 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1640#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1641 static const real coeff[] = {
1642 // C4[0], coeff of eps^2, polynomial in n of order 0
1643 -2, 105,
1644 // C4[0], coeff of eps^1, polynomial in n of order 1
1645 16, -7, 35,
1646 // C4[0], coeff of eps^0, polynomial in n of order 2
1647 8, -28, 70, 105,
1648 // C4[1], coeff of eps^2, polynomial in n of order 0
1649 -2, 105,
1650 // C4[1], coeff of eps^1, polynomial in n of order 1
1651 -16, 7, 315,
1652 // C4[2], coeff of eps^2, polynomial in n of order 0
1653 4, 525,
1654 };
1655#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1656 static const real coeff[] = {
1657 // C4[0], coeff of eps^3, polynomial in n of order 0
1658 11, 315,
1659 // C4[0], coeff of eps^2, polynomial in n of order 1
1660 -32, -6, 315,
1661 // C4[0], coeff of eps^1, polynomial in n of order 2
1662 -32, 48, -21, 105,
1663 // C4[0], coeff of eps^0, polynomial in n of order 3
1664 4, 24, -84, 210, 315,
1665 // C4[1], coeff of eps^3, polynomial in n of order 0
1666 -1, 105,
1667 // C4[1], coeff of eps^2, polynomial in n of order 1
1668 64, -18, 945,
1669 // C4[1], coeff of eps^1, polynomial in n of order 2
1670 32, -48, 21, 945,
1671 // C4[2], coeff of eps^3, polynomial in n of order 0
1672 -8, 1575,
1673 // C4[2], coeff of eps^2, polynomial in n of order 1
1674 -32, 12, 1575,
1675 // C4[3], coeff of eps^3, polynomial in n of order 0
1676 8, 2205,
1677 };
1678#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1679 static const real coeff[] = {
1680 // C4[0], coeff of eps^4, polynomial in n of order 0
1681 4, 1155,
1682 // C4[0], coeff of eps^3, polynomial in n of order 1
1683 -368, 121, 3465,
1684 // C4[0], coeff of eps^2, polynomial in n of order 2
1685 1088, -352, -66, 3465,
1686 // C4[0], coeff of eps^1, polynomial in n of order 3
1687 48, -352, 528, -231, 1155,
1688 // C4[0], coeff of eps^0, polynomial in n of order 4
1689 16, 44, 264, -924, 2310, 3465,
1690 // C4[1], coeff of eps^4, polynomial in n of order 0
1691 4, 1155,
1692 // C4[1], coeff of eps^3, polynomial in n of order 1
1693 80, -99, 10395,
1694 // C4[1], coeff of eps^2, polynomial in n of order 2
1695 -896, 704, -198, 10395,
1696 // C4[1], coeff of eps^1, polynomial in n of order 3
1697 -48, 352, -528, 231, 10395,
1698 // C4[2], coeff of eps^4, polynomial in n of order 0
1699 -8, 1925,
1700 // C4[2], coeff of eps^3, polynomial in n of order 1
1701 384, -88, 17325,
1702 // C4[2], coeff of eps^2, polynomial in n of order 2
1703 320, -352, 132, 17325,
1704 // C4[3], coeff of eps^4, polynomial in n of order 0
1705 -16, 8085,
1706 // C4[3], coeff of eps^3, polynomial in n of order 1
1707 -256, 88, 24255,
1708 // C4[4], coeff of eps^4, polynomial in n of order 0
1709 64, 31185,
1710 };
1711#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1712 static const real coeff[] = {
1713 // C4[0], coeff of eps^5, polynomial in n of order 0
1714 97, 15015,
1715 // C4[0], coeff of eps^4, polynomial in n of order 1
1716 1088, 156, 45045,
1717 // C4[0], coeff of eps^3, polynomial in n of order 2
1718 -224, -4784, 1573, 45045,
1719 // C4[0], coeff of eps^2, polynomial in n of order 3
1720 -10656, 14144, -4576, -858, 45045,
1721 // C4[0], coeff of eps^1, polynomial in n of order 4
1722 64, 624, -4576, 6864, -3003, 15015,
1723 // C4[0], coeff of eps^0, polynomial in n of order 5
1724 100, 208, 572, 3432, -12012, 30030, 45045,
1725 // C4[1], coeff of eps^5, polynomial in n of order 0
1726 1, 9009,
1727 // C4[1], coeff of eps^4, polynomial in n of order 1
1728 -2944, 468, 135135,
1729 // C4[1], coeff of eps^3, polynomial in n of order 2
1730 5792, 1040, -1287, 135135,
1731 // C4[1], coeff of eps^2, polynomial in n of order 3
1732 5952, -11648, 9152, -2574, 135135,
1733 // C4[1], coeff of eps^1, polynomial in n of order 4
1734 -64, -624, 4576, -6864, 3003, 135135,
1735 // C4[2], coeff of eps^5, polynomial in n of order 0
1736 8, 10725,
1737 // C4[2], coeff of eps^4, polynomial in n of order 1
1738 1856, -936, 225225,
1739 // C4[2], coeff of eps^3, polynomial in n of order 2
1740 -8448, 4992, -1144, 225225,
1741 // C4[2], coeff of eps^2, polynomial in n of order 3
1742 -1440, 4160, -4576, 1716, 225225,
1743 // C4[3], coeff of eps^5, polynomial in n of order 0
1744 -136, 63063,
1745 // C4[3], coeff of eps^4, polynomial in n of order 1
1746 1024, -208, 105105,
1747 // C4[3], coeff of eps^3, polynomial in n of order 2
1748 3584, -3328, 1144, 315315,
1749 // C4[4], coeff of eps^5, polynomial in n of order 0
1750 -128, 135135,
1751 // C4[4], coeff of eps^4, polynomial in n of order 1
1752 -2560, 832, 405405,
1753 // C4[5], coeff of eps^5, polynomial in n of order 0
1754 128, 99099,
1755 };
1756#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1757 static const real coeff[] = {
1758 // C4[0], coeff of eps^6, polynomial in n of order 0
1759 10, 9009,
1760 // C4[0], coeff of eps^5, polynomial in n of order 1
1761 -464, 291, 45045,
1762 // C4[0], coeff of eps^4, polynomial in n of order 2
1763 -4480, 1088, 156, 45045,
1764 // C4[0], coeff of eps^3, polynomial in n of order 3
1765 10736, -224, -4784, 1573, 45045,
1766 // C4[0], coeff of eps^2, polynomial in n of order 4
1767 1664, -10656, 14144, -4576, -858, 45045,
1768 // C4[0], coeff of eps^1, polynomial in n of order 5
1769 16, 64, 624, -4576, 6864, -3003, 15015,
1770 // C4[0], coeff of eps^0, polynomial in n of order 6
1771 56, 100, 208, 572, 3432, -12012, 30030, 45045,
1772 // C4[1], coeff of eps^6, polynomial in n of order 0
1773 10, 9009,
1774 // C4[1], coeff of eps^5, polynomial in n of order 1
1775 112, 15, 135135,
1776 // C4[1], coeff of eps^4, polynomial in n of order 2
1777 3840, -2944, 468, 135135,
1778 // C4[1], coeff of eps^3, polynomial in n of order 3
1779 -10704, 5792, 1040, -1287, 135135,
1780 // C4[1], coeff of eps^2, polynomial in n of order 4
1781 -768, 5952, -11648, 9152, -2574, 135135,
1782 // C4[1], coeff of eps^1, polynomial in n of order 5
1783 -16, -64, -624, 4576, -6864, 3003, 135135,
1784 // C4[2], coeff of eps^6, polynomial in n of order 0
1785 -4, 25025,
1786 // C4[2], coeff of eps^5, polynomial in n of order 1
1787 -1664, 168, 225225,
1788 // C4[2], coeff of eps^4, polynomial in n of order 2
1789 1664, 1856, -936, 225225,
1790 // C4[2], coeff of eps^3, polynomial in n of order 3
1791 6784, -8448, 4992, -1144, 225225,
1792 // C4[2], coeff of eps^2, polynomial in n of order 4
1793 128, -1440, 4160, -4576, 1716, 225225,
1794 // C4[3], coeff of eps^6, polynomial in n of order 0
1795 64, 315315,
1796 // C4[3], coeff of eps^5, polynomial in n of order 1
1797 1792, -680, 315315,
1798 // C4[3], coeff of eps^4, polynomial in n of order 2
1799 -2048, 1024, -208, 105105,
1800 // C4[3], coeff of eps^3, polynomial in n of order 3
1801 -1792, 3584, -3328, 1144, 315315,
1802 // C4[4], coeff of eps^6, polynomial in n of order 0
1803 -512, 405405,
1804 // C4[4], coeff of eps^5, polynomial in n of order 1
1805 2048, -384, 405405,
1806 // C4[4], coeff of eps^4, polynomial in n of order 2
1807 3072, -2560, 832, 405405,
1808 // C4[5], coeff of eps^6, polynomial in n of order 0
1809 -256, 495495,
1810 // C4[5], coeff of eps^5, polynomial in n of order 1
1811 -2048, 640, 495495,
1812 // C4[6], coeff of eps^6, polynomial in n of order 0
1813 512, 585585,
1814 };
1815#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1816 static const real coeff[] = {
1817 // C4[0], coeff of eps^7, polynomial in n of order 0
1818 193, 85085,
1819 // C4[0], coeff of eps^6, polynomial in n of order 1
1820 4192, 850, 765765,
1821 // C4[0], coeff of eps^5, polynomial in n of order 2
1822 20960, -7888, 4947, 765765,
1823 // C4[0], coeff of eps^4, polynomial in n of order 3
1824 12480, -76160, 18496, 2652, 765765,
1825 // C4[0], coeff of eps^3, polynomial in n of order 4
1826 -154048, 182512, -3808, -81328, 26741, 765765,
1827 // C4[0], coeff of eps^2, polynomial in n of order 5
1828 3232, 28288, -181152, 240448, -77792, -14586, 765765,
1829 // C4[0], coeff of eps^1, polynomial in n of order 6
1830 96, 272, 1088, 10608, -77792, 116688, -51051, 255255,
1831 // C4[0], coeff of eps^0, polynomial in n of order 7
1832 588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765,
1833 // C4[1], coeff of eps^7, polynomial in n of order 0
1834 349, 2297295,
1835 // C4[1], coeff of eps^6, polynomial in n of order 1
1836 -1472, 510, 459459,
1837 // C4[1], coeff of eps^5, polynomial in n of order 2
1838 -39840, 1904, 255, 2297295,
1839 // C4[1], coeff of eps^4, polynomial in n of order 3
1840 52608, 65280, -50048, 7956, 2297295,
1841 // C4[1], coeff of eps^3, polynomial in n of order 4
1842 103744, -181968, 98464, 17680, -21879, 2297295,
1843 // C4[1], coeff of eps^2, polynomial in n of order 5
1844 -1344, -13056, 101184, -198016, 155584, -43758, 2297295,
1845 // C4[1], coeff of eps^1, polynomial in n of order 6
1846 -96, -272, -1088, -10608, 77792, -116688, 51051, 2297295,
1847 // C4[2], coeff of eps^7, polynomial in n of order 0
1848 464, 1276275,
1849 // C4[2], coeff of eps^6, polynomial in n of order 1
1850 -928, -612, 3828825,
1851 // C4[2], coeff of eps^5, polynomial in n of order 2
1852 64256, -28288, 2856, 3828825,
1853 // C4[2], coeff of eps^4, polynomial in n of order 3
1854 -126528, 28288, 31552, -15912, 3828825,
1855 // C4[2], coeff of eps^3, polynomial in n of order 4
1856 -41472, 115328, -143616, 84864, -19448, 3828825,
1857 // C4[2], coeff of eps^2, polynomial in n of order 5
1858 160, 2176, -24480, 70720, -77792, 29172, 3828825,
1859 // C4[3], coeff of eps^7, polynomial in n of order 0
1860 -16, 97461,
1861 // C4[3], coeff of eps^6, polynomial in n of order 1
1862 -16384, 1088, 5360355,
1863 // C4[3], coeff of eps^5, polynomial in n of order 2
1864 -2560, 30464, -11560, 5360355,
1865 // C4[3], coeff of eps^4, polynomial in n of order 3
1866 35840, -34816, 17408, -3536, 1786785,
1867 // C4[3], coeff of eps^3, polynomial in n of order 4
1868 7168, -30464, 60928, -56576, 19448, 5360355,
1869 // C4[4], coeff of eps^7, polynomial in n of order 0
1870 128, 2297295,
1871 // C4[4], coeff of eps^6, polynomial in n of order 1
1872 26624, -8704, 6891885,
1873 // C4[4], coeff of eps^5, polynomial in n of order 2
1874 -77824, 34816, -6528, 6891885,
1875 // C4[4], coeff of eps^4, polynomial in n of order 3
1876 -32256, 52224, -43520, 14144, 6891885,
1877 // C4[5], coeff of eps^7, polynomial in n of order 0
1878 -6784, 8423415,
1879 // C4[5], coeff of eps^6, polynomial in n of order 1
1880 24576, -4352, 8423415,
1881 // C4[5], coeff of eps^5, polynomial in n of order 2
1882 45056, -34816, 10880, 8423415,
1883 // C4[6], coeff of eps^7, polynomial in n of order 0
1884 -1024, 3318315,
1885 // C4[6], coeff of eps^6, polynomial in n of order 1
1886 -28672, 8704, 9954945,
1887 // C4[7], coeff of eps^7, polynomial in n of order 0
1888 1024, 1640925,
1889 };
1890#else
1891#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1892#endif
1893 static_assert(sizeof(coeff) / sizeof(real) ==
1894 (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
1895 "Coefficient array size mismatch in C4coeff");
1896 int o = 0, k = 0;
1897 for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
1898 for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
1899 int m = nC4_ - j - 1; // order of polynomial in n
1900 _C4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1901 o += m + 2;
1902 }
1903 }
1904 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
1905 }
1906
1907} // namespace GeographicLib
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::GeodesicLine class.
Header for GeographicLib::Geodesic class.
#define GEOGRAPHICLIB_VOLATILE
Definition: Math.hpp:58
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:61
Geodesic calculations
Definition: Geodesic.hpp:172
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition: Geodesic.cpp:517
Geodesic(real a, real f)
Definition: Geodesic.cpp:42
static const Geodesic & WGS84()
Definition: Geodesic.cpp:89
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition: Geodesic.cpp:154
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition: Geodesic.cpp:118
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition: Geodesic.cpp:136
friend class GeodesicLine
Definition: Geodesic.hpp:175
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:123
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition: Geodesic.cpp:149
Exception handling for GeographicLib.
Definition: Constants.hpp:315
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T AngNormalize(T x)
Definition: Math.hpp:420
static T degree()
Definition: Math.hpp:159
static T LatFix(T x)
Definition: Math.hpp:433
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T atan2d(T y, T x)
Definition: Math.cpp:180
static void norm(T &x, T &y)
Definition: Math.hpp:355
static T AngRound(T x)
Definition: Math.cpp:117
static T sq(T x)
Definition: Math.hpp:171
static T pi()
Definition: Math.hpp:149
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:402
static T AngDiff(T x, T y, T &e)
Definition: Math.hpp:452
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)