GeographicLib 2.1.2
Geodesic.cpp
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1/**
2 * \file Geodesic.cpp
3 * \brief Implementation for GeographicLib::Geodesic class
4 *
5 * Copyright (c) Charles Karney (2009-2022) <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
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// constant conditional and enum-float expressions and mixing enums
35# pragma warning (disable: 4701 4127 5055 5054)
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_) // 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(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 {
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
89 const Geodesic& Geodesic::WGS84() {
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).
166 using std::isnan; // Needed for Centos 7, ubuntu 14
167 real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
168 // Make longitude difference positive.
169 int lonsign = signbit(lon12) ? -1 : 1;
170 lon12 *= lonsign; lon12s *= lonsign;
171 real
172 lam12 = lon12 * Math::degree(),
173 slam12, clam12;
174 // Calculate sincos of lon12 + error (this applies AngRound internally).
175 Math::sincosde(lon12, lon12s, slam12, clam12);
176 // the supplementary longitude difference
177 lon12s = (Math::hd - lon12) - lon12s;
178
179 // If really close to the equator, treat as on equator.
180 lat1 = Math::AngRound(Math::LatFix(lat1));
181 lat2 = Math::AngRound(Math::LatFix(lat2));
182 // Swap points so that point with higher (abs) latitude is point 1.
183 // If one latitude is a nan, then it becomes lat1.
184 int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1;
185 if (swapp < 0) {
186 lonsign *= -1;
187 swap(lat1, lat2);
188 }
189 // Make lat1 <= -0
190 int latsign = signbit(lat1) ? 1 : -1;
191 lat1 *= latsign;
192 lat2 *= latsign;
193 // Now we have
194 //
195 // 0 <= lon12 <= 180
196 // -90 <= lat1 <= -0
197 // lat1 <= lat2 <= -lat1
198 //
199 // longsign, swapp, latsign register the transformation to bring the
200 // coordinates to this canonical form. In all cases, 1 means no change was
201 // made. We make these transformations so that there are few cases to
202 // check, e.g., on verifying quadrants in atan2. In addition, this
203 // enforces some symmetries in the results returned.
204
205 real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
206
207 Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
208 // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
209 // will be <= 2*tiny for two points at the same pole.
210 Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
211
212 Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
213 // Ensure cbet2 = +epsilon at poles
214 Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2);
215
216 // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
217 // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
218 // a better measure. This logic is used in assigning calp2 in Lambda12.
219 // Sometimes these quantities vanish and in that case we force bet2 = +/-
220 // bet1 exactly. An example where is is necessary is the inverse problem
221 // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
222 // which failed with Visual Studio 10 (Release and Debug)
223
224 if (cbet1 < -sbet1) {
225 if (cbet2 == cbet1)
226 sbet2 = copysign(sbet1, sbet2);
227 } else {
228 if (fabs(sbet2) == -sbet1)
229 cbet2 = cbet1;
230 }
231
232 real
233 dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
234 dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
235
236 real a12, sig12;
237 // index zero element of this array is unused
238 real Ca[nC_];
239
240 bool meridian = lat1 == -Math::qd || slam12 == 0;
241
242 if (meridian) {
243
244 // Endpoints are on a single full meridian, so the geodesic might lie on
245 // a meridian.
246
247 calp1 = clam12; salp1 = slam12; // Head to the target longitude
248 calp2 = 1; salp2 = 0; // At the target we're heading north
249
250 real
251 // tan(bet) = tan(sig) * cos(alp)
252 ssig1 = sbet1, csig1 = calp1 * cbet1,
253 ssig2 = sbet2, csig2 = calp2 * cbet2;
254
255 // sig12 = sig2 - sig1
256 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
257 csig1 * csig2 + ssig1 * ssig2);
258 {
259 real dummy;
260 Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
261 outmask | DISTANCE | REDUCEDLENGTH,
262 s12x, m12x, dummy, M12, M21, Ca);
263 }
264 // Add the check for sig12 since zero length geodesics might yield m12 <
265 // 0. Test case was
266 //
267 // echo 20.001 0 20.001 0 | GeodSolve -i
268 //
269 // In fact, we will have sig12 > pi/2 for meridional geodesic which is
270 // not a shortest path.
271 // TODO: investigate m12 < 0 result for aarch/ppc (with -f -p 20)
272 // 20.001000000000001 0.000000000000000 180.000000000000000
273 // 20.001000000000001 0.000000000000000 180.000000000000000
274 // 0.0000000002 0.000000000000001 -0.0000000001
275 // 0.99999999999999989 0.99999999999999989 0.000
276 if (sig12 < 1 || m12x >= 0) {
277 // Need at least 2, to handle 90 0 90 180
278 if (sig12 < 3 * tiny_ ||
279 // Prevent negative s12 or m12 for short lines
280 (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
281 sig12 = m12x = s12x = 0;
282 m12x *= _b;
283 s12x *= _b;
284 a12 = sig12 / Math::degree();
285 } else
286 // m12 < 0, i.e., prolate and too close to anti-podal
287 meridian = false;
288 }
289
290 // somg12 == 2 marks that it needs to be calculated
291 real omg12 = 0, somg12 = 2, comg12 = 0;
292 if (!meridian &&
293 sbet1 == 0 && // and sbet2 == 0
294 (_f <= 0 || lon12s >= _f * Math::hd)) {
295
296 // Geodesic runs along equator
297 calp1 = calp2 = 0; salp1 = salp2 = 1;
298 s12x = _a * lam12;
299 sig12 = omg12 = lam12 / _f1;
300 m12x = _b * sin(sig12);
301 if (outmask & GEODESICSCALE)
302 M12 = M21 = cos(sig12);
303 a12 = lon12 / _f1;
304
305 } else if (!meridian) {
306
307 // Now point1 and point2 belong within a hemisphere bounded by a
308 // meridian and geodesic is neither meridional or equatorial.
309
310 // Figure a starting point for Newton's method
311 real dnm;
312 sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
313 lam12, slam12, clam12,
314 salp1, calp1, salp2, calp2, dnm,
315 Ca);
316
317 if (sig12 >= 0) {
318 // Short lines (InverseStart sets salp2, calp2, dnm)
319 s12x = sig12 * _b * dnm;
320 m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
321 if (outmask & GEODESICSCALE)
322 M12 = M21 = cos(sig12 / dnm);
323 a12 = sig12 / Math::degree();
324 omg12 = lam12 / (_f1 * dnm);
325 } else {
326
327 // Newton's method. This is a straightforward solution of f(alp1) =
328 // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
329 // root in the interval (0, pi) and its derivative is positive at the
330 // root. Thus f(alp) is positive for alp > alp1 and negative for alp <
331 // alp1. During the course of the iteration, a range (alp1a, alp1b) is
332 // maintained which brackets the root and with each evaluation of
333 // f(alp) the range is shrunk, if possible. Newton's method is
334 // restarted whenever the derivative of f is negative (because the new
335 // value of alp1 is then further from the solution) or if the new
336 // estimate of alp1 lies outside (0,pi); in this case, the new starting
337 // guess is taken to be (alp1a + alp1b) / 2.
338 //
339 // initial values to suppress warnings (if loop is executed 0 times)
340 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
341 unsigned numit = 0;
342 // Bracketing range
343 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
344 for (bool tripn = false, tripb = false;; ++numit) {
345 // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
346 // WGS84 and random input: mean = 2.85, sd = 0.60
347 real dv;
348 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
349 slam12, clam12,
350 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
351 eps, domg12, numit < maxit1_, dv, Ca);
352 if (tripb ||
353 // Reversed test to allow escape with NaNs
354 !(fabs(v) >= (tripn ? 8 : 1) * tol0_) ||
355 // Enough bisections to get accurate result
356 numit == maxit2_)
357 break;
358 // Update bracketing values
359 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
360 { salp1b = salp1; calp1b = calp1; }
361 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
362 { salp1a = salp1; calp1a = calp1; }
363 if (numit < maxit1_ && dv > 0) {
364 real
365 dalp1 = -v/dv;
366 // |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION
367 // = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages
368 // (because of the need to do accurate range reduction).
369 if (fabs(dalp1) < Math::pi()) {
370 real
371 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
372 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
373 if (nsalp1 > 0) {
374 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
375 salp1 = nsalp1;
376 Math::norm(salp1, calp1);
377 // In some regimes we don't get quadratic convergence because
378 // slope -> 0. So use convergence conditions based on epsilon
379 // instead of sqrt(epsilon).
380 tripn = fabs(v) <= 16 * tol0_;
381 continue;
382 }
383 }
384 }
385 // Either dv was not positive or updated value was outside legal
386 // range. Use the midpoint of the bracket as the next estimate.
387 // This mechanism is not needed for the WGS84 ellipsoid, but it does
388 // catch problems with more eccentric ellipsoids. Its efficacy is
389 // such for the WGS84 test set with the starting guess set to alp1 =
390 // 90deg:
391 // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
392 // WGS84 and random input: mean = 4.74, sd = 0.99
393 salp1 = (salp1a + salp1b)/2;
394 calp1 = (calp1a + calp1b)/2;
395 Math::norm(salp1, calp1);
396 tripn = false;
397 tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
398 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
399 }
400 {
401 real dummy;
402 // Ensure that the reduced length and geodesic scale are computed in
403 // a "canonical" way, with the I2 integral.
404 unsigned lengthmask = outmask |
405 (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE);
406 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
407 cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
408 }
409 m12x *= _b;
410 s12x *= _b;
411 a12 = sig12 / Math::degree();
412 if (outmask & AREA) {
413 // omg12 = lam12 - domg12
414 real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
415 somg12 = slam12 * cdomg12 - clam12 * sdomg12;
416 comg12 = clam12 * cdomg12 + slam12 * sdomg12;
417 }
418 }
419 }
420
421 if (outmask & DISTANCE)
422 s12 = real(0) + s12x; // Convert -0 to 0
423
424 if (outmask & REDUCEDLENGTH)
425 m12 = real(0) + m12x; // Convert -0 to 0
426
427 if (outmask & AREA) {
428 real
429 // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
430 salp0 = salp1 * cbet1,
431 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
432 real alp12;
433 if (calp0 != 0 && salp0 != 0) {
434 real
435 // From Lambda12: tan(bet) = tan(sig) * cos(alp)
436 ssig1 = sbet1, csig1 = calp1 * cbet1,
437 ssig2 = sbet2, csig2 = calp2 * cbet2,
438 k2 = Math::sq(calp0) * _ep2,
439 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
440 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
441 A4 = Math::sq(_a) * calp0 * salp0 * _e2;
442 Math::norm(ssig1, csig1);
443 Math::norm(ssig2, csig2);
444 C4f(eps, Ca);
445 real
446 B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
447 B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
448 S12 = A4 * (B42 - B41);
449 } else
450 // Avoid problems with indeterminate sig1, sig2 on equator
451 S12 = 0;
452 if (!meridian && somg12 == 2) {
453 somg12 = sin(omg12); comg12 = cos(omg12);
454 }
455
456 if (!meridian &&
457 // omg12 < 3/4 * pi
458 comg12 > -real(0.7071) && // Long difference not too big
459 sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
460 // Use tan(Gamma/2) = tan(omg12/2)
461 // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
462 // with tan(x/2) = sin(x)/(1+cos(x))
463 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
464 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
465 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
466 } else {
467 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
468 real
469 salp12 = salp2 * calp1 - calp2 * salp1,
470 calp12 = calp2 * calp1 + salp2 * salp1;
471 // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
472 // salp12 = -0 and alp12 = -180. However this depends on the sign
473 // being attached to 0 correctly. The following ensures the correct
474 // behavior.
475 if (salp12 == 0 && calp12 < 0) {
476 salp12 = tiny_ * calp1;
477 calp12 = -1;
478 }
479 alp12 = atan2(salp12, calp12);
480 }
481 S12 += _c2 * alp12;
482 S12 *= swapp * lonsign * latsign;
483 // Convert -0 to 0
484 S12 += 0;
485 }
486
487 // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
488 if (swapp < 0) {
489 swap(salp1, salp2);
490 swap(calp1, calp2);
491 if (outmask & GEODESICSCALE)
492 swap(M12, M21);
493 }
494
495 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
496 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
497 // Returned value in [0, 180]
498 return a12;
499 }
500
501 Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
502 unsigned outmask,
503 real& s12, real& azi1, real& azi2,
504 real& m12, real& M12, real& M21,
505 real& S12) const {
506 outmask &= OUT_MASK;
507 real salp1, calp1, salp2, calp2,
508 a12 = GenInverse(lat1, lon1, lat2, lon2,
509 outmask, s12, salp1, calp1, salp2, calp2,
510 m12, M12, M21, S12);
511 if (outmask & AZIMUTH) {
512 azi1 = Math::atan2d(salp1, calp1);
513 azi2 = Math::atan2d(salp2, calp2);
514 }
515 return a12;
516 }
517
518 GeodesicLine Geodesic::InverseLine(real lat1, real lon1,
519 real lat2, real lon2,
520 unsigned caps) const {
521 real t, salp1, calp1, salp2, calp2,
522 a12 = GenInverse(lat1, lon1, lat2, lon2,
523 // No need to specify AZIMUTH here
524 0u, t, salp1, calp1, salp2, calp2,
525 t, t, t, t),
526 azi1 = Math::atan2d(salp1, calp1);
527 // Ensure that a12 can be converted to a distance
528 if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
529 return
530 GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12);
531 }
532
533 void Geodesic::Lengths(real eps, real sig12,
534 real ssig1, real csig1, real dn1,
535 real ssig2, real csig2, real dn2,
536 real cbet1, real cbet2, unsigned outmask,
537 real& s12b, real& m12b, real& m0,
538 real& M12, real& M21,
539 // Scratch area of the right size
540 real Ca[]) const {
541 // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
542 // and m0 = coefficient of secular term in expression for reduced length.
543
544 outmask &= OUT_MASK;
545 // outmask & DISTANCE: set s12b
546 // outmask & REDUCEDLENGTH: set m12b & m0
547 // outmask & GEODESICSCALE: set M12 & M21
548
549 real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
550 real Cb[nC2_ + 1];
551 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
552 A1 = A1m1f(eps);
553 C1f(eps, Ca);
554 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
555 A2 = A2m1f(eps);
556 C2f(eps, Cb);
557 m0x = A1 - A2;
558 A2 = 1 + A2;
559 }
560 A1 = 1 + A1;
561 }
562 if (outmask & DISTANCE) {
563 real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
564 SinCosSeries(true, ssig1, csig1, Ca, nC1_);
565 // Missing a factor of _b
566 s12b = A1 * (sig12 + B1);
567 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
568 real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
569 SinCosSeries(true, ssig1, csig1, Cb, nC2_);
570 J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
571 }
572 } else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
573 // Assume here that nC1_ >= nC2_
574 for (int l = 1; l <= nC2_; ++l)
575 Cb[l] = A1 * Ca[l] - A2 * Cb[l];
576 J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
577 SinCosSeries(true, ssig1, csig1, Cb, nC2_));
578 }
579 if (outmask & REDUCEDLENGTH) {
580 m0 = m0x;
581 // Missing a factor of _b.
582 // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
583 // accurate cancellation in the case of coincident points.
584 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
585 csig1 * csig2 * J12;
586 }
587 if (outmask & GEODESICSCALE) {
588 real csig12 = csig1 * csig2 + ssig1 * ssig2;
589 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
590 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
591 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
592 }
593 }
594
595 Math::real Geodesic::Astroid(real x, real y) {
596 // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
597 // This solution is adapted from Geocentric::Reverse.
598 real k;
599 real
600 p = Math::sq(x),
601 q = Math::sq(y),
602 r = (p + q - 1) / 6;
603 if ( !(q == 0 && r <= 0) ) {
604 real
605 // Avoid possible division by zero when r = 0 by multiplying equations
606 // for s and t by r^3 and r, resp.
607 S = p * q / 4, // S = r^3 * s
608 r2 = Math::sq(r),
609 r3 = r * r2,
610 // The discriminant of the quadratic equation for T3. This is zero on
611 // the evolute curve p^(1/3)+q^(1/3) = 1
612 disc = S * (S + 2 * r3);
613 real u = r;
614 if (disc >= 0) {
615 real T3 = S + r3;
616 // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
617 // of precision due to cancellation. The result is unchanged because
618 // of the way the T is used in definition of u.
619 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
620 // N.B. cbrt always returns the real root. cbrt(-8) = -2.
621 real T = cbrt(T3); // T = r * t
622 // T can be zero; but then r2 / T -> 0.
623 u += T + (T != 0 ? r2 / T : 0);
624 } else {
625 // T is complex, but the way u is defined the result is real.
626 real ang = atan2(sqrt(-disc), -(S + r3));
627 // There are three possible cube roots. We choose the root which
628 // avoids cancellation. Note that disc < 0 implies that r < 0.
629 u += 2 * r * cos(ang / 3);
630 }
631 real
632 v = sqrt(Math::sq(u) + q), // guaranteed positive
633 // Avoid loss of accuracy when u < 0.
634 uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
635 w = (uv - q) / (2 * v); // positive?
636 // Rearrange expression for k to avoid loss of accuracy due to
637 // subtraction. Division by 0 not possible because uv > 0, w >= 0.
638 k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive
639 } else { // q == 0 && r <= 0
640 // y = 0 with |x| <= 1. Handle this case directly.
641 // for y small, positive root is k = abs(y)/sqrt(1-x^2)
642 k = 0;
643 }
644 return k;
645 }
646
647 Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
648 real sbet2, real cbet2, real dn2,
649 real lam12, real slam12, real clam12,
650 real& salp1, real& calp1,
651 // Only updated if return val >= 0
652 real& salp2, real& calp2,
653 // Only updated for short lines
654 real& dnm,
655 // Scratch area of the right size
656 real Ca[]) const {
657 // Return a starting point for Newton's method in salp1 and calp1 (function
658 // value is -1). If Newton's method doesn't need to be used, return also
659 // salp2 and calp2 and function value is sig12.
660 real
661 sig12 = -1, // Return value
662 // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
663 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
664 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
665 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
666 bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
667 cbet2 * lam12 < real(0.5);
668 real somg12, comg12;
669 if (shortline) {
670 real sbetm2 = Math::sq(sbet1 + sbet2);
671 // sin((bet1+bet2)/2)^2
672 // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
673 sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
674 dnm = sqrt(1 + _ep2 * sbetm2);
675 real omg12 = lam12 / (_f1 * dnm);
676 somg12 = sin(omg12); comg12 = cos(omg12);
677 } else {
678 somg12 = slam12; comg12 = clam12;
679 }
680
681 salp1 = cbet2 * somg12;
682 calp1 = comg12 >= 0 ?
683 sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
684 sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
685
686 real
687 ssig12 = hypot(salp1, calp1),
688 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
689
690 if (shortline && ssig12 < _etol2) {
691 // really short lines
692 salp2 = cbet1 * somg12;
693 calp2 = sbet12 - cbet1 * sbet2 *
694 (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
695 Math::norm(salp2, calp2);
696 // Set return value
697 sig12 = atan2(ssig12, csig12);
698 } else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric
699 csig12 >= 0 ||
700 ssig12 >= 6 * fabs(_n) * Math::pi() * Math::sq(cbet1)) {
701 // Nothing to do, zeroth order spherical approximation is OK
702 } else {
703 // Scale lam12 and bet2 to x, y coordinate system where antipodal point
704 // is at origin and singular point is at y = 0, x = -1.
705 real x, y, lamscale, betscale;
706 real lam12x = atan2(-slam12, -clam12); // lam12 - pi
707 if (_f >= 0) { // In fact f == 0 does not get here
708 // x = dlong, y = dlat
709 {
710 real
711 k2 = Math::sq(sbet1) * _ep2,
712 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
713 lamscale = _f * cbet1 * A3f(eps) * Math::pi();
714 }
715 betscale = lamscale * cbet1;
716
717 x = lam12x / lamscale;
718 y = sbet12a / betscale;
719 } else { // _f < 0
720 // x = dlat, y = dlong
721 real
722 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
723 bet12a = atan2(sbet12a, cbet12a);
724 real m12b, m0, dummy;
725 // In the case of lon12 = 180, this repeats a calculation made in
726 // Inverse.
727 Lengths(_n, Math::pi() + bet12a,
728 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
729 cbet1, cbet2,
730 REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca);
731 x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
732 betscale = x < -real(0.01) ? sbet12a / x :
733 -_f * Math::sq(cbet1) * Math::pi();
734 lamscale = betscale / cbet1;
735 y = lam12x / lamscale;
736 }
737
738 if (y > -tol1_ && x > -1 - xthresh_) {
739 // strip near cut
740 // Need real(x) here to cast away the volatility of x for min/max
741 if (_f >= 0) {
742 salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1));
743 } else {
744 calp1 = fmax(real(x > -tol1_ ? 0 : -1), x);
745 salp1 = sqrt(1 - Math::sq(calp1));
746 }
747 } else {
748 // Estimate alp1, by solving the astroid problem.
749 //
750 // Could estimate alpha1 = theta + pi/2, directly, i.e.,
751 // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
752 // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
753 //
754 // However, it's better to estimate omg12 from astroid and use
755 // spherical formula to compute alp1. This reduces the mean number of
756 // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
757 // (min 0 max 5). The changes in the number of iterations are as
758 // follows:
759 //
760 // change percent
761 // 1 5
762 // 0 78
763 // -1 16
764 // -2 0.6
765 // -3 0.04
766 // -4 0.002
767 //
768 // The histogram of iterations is (m = number of iterations estimating
769 // alp1 directly, n = number of iterations estimating via omg12, total
770 // number of trials = 148605):
771 //
772 // iter m n
773 // 0 148 186
774 // 1 13046 13845
775 // 2 93315 102225
776 // 3 36189 32341
777 // 4 5396 7
778 // 5 455 1
779 // 6 56 0
780 //
781 // Because omg12 is near pi, estimate work with omg12a = pi - omg12
782 real k = Astroid(x, y);
783 real
784 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
785 somg12 = sin(omg12a); comg12 = -cos(omg12a);
786 // Update spherical estimate of alp1 using omg12 instead of lam12
787 salp1 = cbet2 * somg12;
788 calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
789 }
790 }
791 // Sanity check on starting guess. Backwards check allows NaN through.
792 if (!(salp1 <= 0))
793 Math::norm(salp1, calp1);
794 else {
795 salp1 = 1; calp1 = 0;
796 }
797 return sig12;
798 }
799
800 Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
801 real sbet2, real cbet2, real dn2,
802 real salp1, real calp1,
803 real slam120, real clam120,
804 real& salp2, real& calp2,
805 real& sig12,
806 real& ssig1, real& csig1,
807 real& ssig2, real& csig2,
808 real& eps, real& domg12,
809 bool diffp, real& dlam12,
810 // Scratch area of the right size
811 real Ca[]) const {
812
813 if (sbet1 == 0 && calp1 == 0)
814 // Break degeneracy of equatorial line. This case has already been
815 // handled.
816 calp1 = -tiny_;
817
818 real
819 // sin(alp1) * cos(bet1) = sin(alp0)
820 salp0 = salp1 * cbet1,
821 calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
822
823 real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
824 // tan(bet1) = tan(sig1) * cos(alp1)
825 // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
826 ssig1 = sbet1; somg1 = salp0 * sbet1;
827 csig1 = comg1 = calp1 * cbet1;
828 Math::norm(ssig1, csig1);
829 // Math::norm(somg1, comg1); -- don't need to normalize!
830
831 // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
832 // about this case, since this can yield singularities in the Newton
833 // iteration.
834 // sin(alp2) * cos(bet2) = sin(alp0)
835 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
836 // calp2 = sqrt(1 - sq(salp2))
837 // = sqrt(sq(calp0) - sq(sbet2)) / cbet2
838 // and subst for calp0 and rearrange to give (choose positive sqrt
839 // to give alp2 in [0, pi/2]).
840 calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
841 sqrt(Math::sq(calp1 * cbet1) +
842 (cbet1 < -sbet1 ?
843 (cbet2 - cbet1) * (cbet1 + cbet2) :
844 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
845 fabs(calp1);
846 // tan(bet2) = tan(sig2) * cos(alp2)
847 // tan(omg2) = sin(alp0) * tan(sig2).
848 ssig2 = sbet2; somg2 = salp0 * sbet2;
849 csig2 = comg2 = calp2 * cbet2;
850 Math::norm(ssig2, csig2);
851 // Math::norm(somg2, comg2); -- don't need to normalize!
852
853 // sig12 = sig2 - sig1, limit to [0, pi]
854 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
855 csig1 * csig2 + ssig1 * ssig2);
856
857 // omg12 = omg2 - omg1, limit to [0, pi]
858 somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2) + real(0);
859 comg12 = comg1 * comg2 + somg1 * somg2;
860 // eta = omg12 - lam120
861 real eta = atan2(somg12 * clam120 - comg12 * slam120,
862 comg12 * clam120 + somg12 * slam120);
863 real B312;
864 real k2 = Math::sq(calp0) * _ep2;
865 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
866 C3f(eps, Ca);
867 B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) -
868 SinCosSeries(true, ssig1, csig1, Ca, nC3_-1));
869 domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
870 lam12 = eta + domg12;
871
872 if (diffp) {
873 if (calp2 == 0)
874 dlam12 = - 2 * _f1 * dn1 / sbet1;
875 else {
876 real dummy;
877 Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
878 cbet1, cbet2, REDUCEDLENGTH,
879 dummy, dlam12, dummy, dummy, dummy, Ca);
880 dlam12 *= _f1 / (calp2 * cbet2);
881 }
882 }
883
884 return lam12;
885 }
886
887 Math::real Geodesic::A3f(real eps) const {
888 // Evaluate A3
889 return Math::polyval(nA3_ - 1, _aA3x, eps);
890 }
891
892 void Geodesic::C3f(real eps, real c[]) const {
893 // Evaluate C3 coeffs
894 // Elements c[1] thru c[nC3_ - 1] are set
895 real mult = 1;
896 int o = 0;
897 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
898 int m = nC3_ - l - 1; // order of polynomial in eps
899 mult *= eps;
900 c[l] = mult * Math::polyval(m, _cC3x + o, eps);
901 o += m + 1;
902 }
903 // Post condition: o == nC3x_
904 }
905
906 void Geodesic::C4f(real eps, real c[]) const {
907 // Evaluate C4 coeffs
908 // Elements c[0] thru c[nC4_ - 1] are set
909 real mult = 1;
910 int o = 0;
911 for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
912 int m = nC4_ - l - 1; // order of polynomial in eps
913 c[l] = mult * Math::polyval(m, _cC4x + o, eps);
914 o += m + 1;
915 mult *= eps;
916 }
917 // Post condition: o == nC4x_
918 }
919
920 // The static const coefficient arrays in the following functions are
921 // generated by Maxima and give the coefficients of the Taylor expansions for
922 // the geodesics. The convention on the order of these coefficients is as
923 // follows:
924 //
925 // ascending order in the trigonometric expansion,
926 // then powers of eps in descending order,
927 // finally powers of n in descending order.
928 //
929 // (For some expansions, only a subset of levels occur.) For each polynomial
930 // of order n at the lowest level, the (n+1) coefficients of the polynomial
931 // are followed by a divisor which is applied to the whole polynomial. In
932 // this way, the coefficients are expressible with no round off error. The
933 // sizes of the coefficient arrays are:
934 //
935 // A1m1f, A2m1f = floor(N/2) + 2
936 // C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
937 // C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
938 // C4coeff = N * (N + 1) * (N + 5) / 6
939 //
940 // where N = GEOGRAPHICLIB_GEODESIC_ORDER
941 // = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
942
943 // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
944 Math::real Geodesic::A1m1f(real eps) {
945 // Generated by Maxima on 2015-05-05 18:08:12-04:00
946#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
947 static const real coeff[] = {
948 // (1-eps)*A1-1, polynomial in eps2 of order 1
949 1, 0, 4,
950 };
951#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
952 static const real coeff[] = {
953 // (1-eps)*A1-1, polynomial in eps2 of order 2
954 1, 16, 0, 64,
955 };
956#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
957 static const real coeff[] = {
958 // (1-eps)*A1-1, polynomial in eps2 of order 3
959 1, 4, 64, 0, 256,
960 };
961#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
962 static const real coeff[] = {
963 // (1-eps)*A1-1, polynomial in eps2 of order 4
964 25, 64, 256, 4096, 0, 16384,
965 };
966#else
967#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
968#endif
969 static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2,
970 "Coefficient array size mismatch in A1m1f");
971 int m = nA1_/2;
972 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
973 return (t + eps) / (1 - eps);
974 }
975
976 // The coefficients C1[l] in the Fourier expansion of B1
977 void Geodesic::C1f(real eps, real c[]) {
978 // Generated by Maxima on 2015-05-05 18:08:12-04:00
979#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
980 static const real coeff[] = {
981 // C1[1]/eps^1, polynomial in eps2 of order 1
982 3, -8, 16,
983 // C1[2]/eps^2, polynomial in eps2 of order 0
984 -1, 16,
985 // C1[3]/eps^3, polynomial in eps2 of order 0
986 -1, 48,
987 };
988#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
989 static const real coeff[] = {
990 // C1[1]/eps^1, polynomial in eps2 of order 1
991 3, -8, 16,
992 // C1[2]/eps^2, polynomial in eps2 of order 1
993 1, -2, 32,
994 // C1[3]/eps^3, polynomial in eps2 of order 0
995 -1, 48,
996 // C1[4]/eps^4, polynomial in eps2 of order 0
997 -5, 512,
998 };
999#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1000 static const real coeff[] = {
1001 // C1[1]/eps^1, polynomial in eps2 of order 2
1002 -1, 6, -16, 32,
1003 // C1[2]/eps^2, polynomial in eps2 of order 1
1004 1, -2, 32,
1005 // C1[3]/eps^3, polynomial in eps2 of order 1
1006 9, -16, 768,
1007 // C1[4]/eps^4, polynomial in eps2 of order 0
1008 -5, 512,
1009 // C1[5]/eps^5, polynomial in eps2 of order 0
1010 -7, 1280,
1011 };
1012#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1013 static const real coeff[] = {
1014 // C1[1]/eps^1, polynomial in eps2 of order 2
1015 -1, 6, -16, 32,
1016 // C1[2]/eps^2, polynomial in eps2 of order 2
1017 -9, 64, -128, 2048,
1018 // C1[3]/eps^3, polynomial in eps2 of order 1
1019 9, -16, 768,
1020 // C1[4]/eps^4, polynomial in eps2 of order 1
1021 3, -5, 512,
1022 // C1[5]/eps^5, polynomial in eps2 of order 0
1023 -7, 1280,
1024 // C1[6]/eps^6, polynomial in eps2 of order 0
1025 -7, 2048,
1026 };
1027#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1028 static const real coeff[] = {
1029 // C1[1]/eps^1, polynomial in eps2 of order 3
1030 19, -64, 384, -1024, 2048,
1031 // C1[2]/eps^2, polynomial in eps2 of order 2
1032 -9, 64, -128, 2048,
1033 // C1[3]/eps^3, polynomial in eps2 of order 2
1034 -9, 72, -128, 6144,
1035 // C1[4]/eps^4, polynomial in eps2 of order 1
1036 3, -5, 512,
1037 // C1[5]/eps^5, polynomial in eps2 of order 1
1038 35, -56, 10240,
1039 // C1[6]/eps^6, polynomial in eps2 of order 0
1040 -7, 2048,
1041 // C1[7]/eps^7, polynomial in eps2 of order 0
1042 -33, 14336,
1043 };
1044#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1045 static const real coeff[] = {
1046 // C1[1]/eps^1, polynomial in eps2 of order 3
1047 19, -64, 384, -1024, 2048,
1048 // C1[2]/eps^2, polynomial in eps2 of order 3
1049 7, -18, 128, -256, 4096,
1050 // C1[3]/eps^3, polynomial in eps2 of order 2
1051 -9, 72, -128, 6144,
1052 // C1[4]/eps^4, polynomial in eps2 of order 2
1053 -11, 96, -160, 16384,
1054 // C1[5]/eps^5, polynomial in eps2 of order 1
1055 35, -56, 10240,
1056 // C1[6]/eps^6, polynomial in eps2 of order 1
1057 9, -14, 4096,
1058 // C1[7]/eps^7, polynomial in eps2 of order 0
1059 -33, 14336,
1060 // C1[8]/eps^8, polynomial in eps2 of order 0
1061 -429, 262144,
1062 };
1063#else
1064#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1065#endif
1066 static_assert(sizeof(coeff) / sizeof(real) ==
1067 (nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4,
1068 "Coefficient array size mismatch in C1f");
1069 real
1070 eps2 = Math::sq(eps),
1071 d = eps;
1072 int o = 0;
1073 for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
1074 int m = (nC1_ - l) / 2; // order of polynomial in eps^2
1075 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1076 o += m + 2;
1077 d *= eps;
1078 }
1079 // Post condition: o == sizeof(coeff) / sizeof(real)
1080 }
1081
1082 // The coefficients C1p[l] in the Fourier expansion of B1p
1083 void Geodesic::C1pf(real eps, real c[]) {
1084 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1085#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1086 static const real coeff[] = {
1087 // C1p[1]/eps^1, polynomial in eps2 of order 1
1088 -9, 16, 32,
1089 // C1p[2]/eps^2, polynomial in eps2 of order 0
1090 5, 16,
1091 // C1p[3]/eps^3, polynomial in eps2 of order 0
1092 29, 96,
1093 };
1094#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1095 static const real coeff[] = {
1096 // C1p[1]/eps^1, polynomial in eps2 of order 1
1097 -9, 16, 32,
1098 // C1p[2]/eps^2, polynomial in eps2 of order 1
1099 -37, 30, 96,
1100 // C1p[3]/eps^3, polynomial in eps2 of order 0
1101 29, 96,
1102 // C1p[4]/eps^4, polynomial in eps2 of order 0
1103 539, 1536,
1104 };
1105#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1106 static const real coeff[] = {
1107 // C1p[1]/eps^1, polynomial in eps2 of order 2
1108 205, -432, 768, 1536,
1109 // C1p[2]/eps^2, polynomial in eps2 of order 1
1110 -37, 30, 96,
1111 // C1p[3]/eps^3, polynomial in eps2 of order 1
1112 -225, 116, 384,
1113 // C1p[4]/eps^4, polynomial in eps2 of order 0
1114 539, 1536,
1115 // C1p[5]/eps^5, polynomial in eps2 of order 0
1116 3467, 7680,
1117 };
1118#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1119 static const real coeff[] = {
1120 // C1p[1]/eps^1, polynomial in eps2 of order 2
1121 205, -432, 768, 1536,
1122 // C1p[2]/eps^2, polynomial in eps2 of order 2
1123 4005, -4736, 3840, 12288,
1124 // C1p[3]/eps^3, polynomial in eps2 of order 1
1125 -225, 116, 384,
1126 // C1p[4]/eps^4, polynomial in eps2 of order 1
1127 -7173, 2695, 7680,
1128 // C1p[5]/eps^5, polynomial in eps2 of order 0
1129 3467, 7680,
1130 // C1p[6]/eps^6, polynomial in eps2 of order 0
1131 38081, 61440,
1132 };
1133#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1134 static const real coeff[] = {
1135 // C1p[1]/eps^1, polynomial in eps2 of order 3
1136 -4879, 9840, -20736, 36864, 73728,
1137 // C1p[2]/eps^2, polynomial in eps2 of order 2
1138 4005, -4736, 3840, 12288,
1139 // C1p[3]/eps^3, polynomial in eps2 of order 2
1140 8703, -7200, 3712, 12288,
1141 // C1p[4]/eps^4, polynomial in eps2 of order 1
1142 -7173, 2695, 7680,
1143 // C1p[5]/eps^5, polynomial in eps2 of order 1
1144 -141115, 41604, 92160,
1145 // C1p[6]/eps^6, polynomial in eps2 of order 0
1146 38081, 61440,
1147 // C1p[7]/eps^7, polynomial in eps2 of order 0
1148 459485, 516096,
1149 };
1150#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1151 static const real coeff[] = {
1152 // C1p[1]/eps^1, polynomial in eps2 of order 3
1153 -4879, 9840, -20736, 36864, 73728,
1154 // C1p[2]/eps^2, polynomial in eps2 of order 3
1155 -86171, 120150, -142080, 115200, 368640,
1156 // C1p[3]/eps^3, polynomial in eps2 of order 2
1157 8703, -7200, 3712, 12288,
1158 // C1p[4]/eps^4, polynomial in eps2 of order 2
1159 1082857, -688608, 258720, 737280,
1160 // C1p[5]/eps^5, polynomial in eps2 of order 1
1161 -141115, 41604, 92160,
1162 // C1p[6]/eps^6, polynomial in eps2 of order 1
1163 -2200311, 533134, 860160,
1164 // C1p[7]/eps^7, polynomial in eps2 of order 0
1165 459485, 516096,
1166 // C1p[8]/eps^8, polynomial in eps2 of order 0
1167 109167851, 82575360,
1168 };
1169#else
1170#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1171#endif
1172 static_assert(sizeof(coeff) / sizeof(real) ==
1173 (nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4,
1174 "Coefficient array size mismatch in C1pf");
1175 real
1176 eps2 = Math::sq(eps),
1177 d = eps;
1178 int o = 0;
1179 for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
1180 int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
1181 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1182 o += m + 2;
1183 d *= eps;
1184 }
1185 // Post condition: o == sizeof(coeff) / sizeof(real)
1186 }
1187
1188 // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
1189 Math::real Geodesic::A2m1f(real eps) {
1190 // Generated by Maxima on 2015-05-29 08:09:47-04:00
1191#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
1192 static const real coeff[] = {
1193 // (eps+1)*A2-1, polynomial in eps2 of order 1
1194 -3, 0, 4,
1195 }; // count = 3
1196#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
1197 static const real coeff[] = {
1198 // (eps+1)*A2-1, polynomial in eps2 of order 2
1199 -7, -48, 0, 64,
1200 }; // count = 4
1201#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
1202 static const real coeff[] = {
1203 // (eps+1)*A2-1, polynomial in eps2 of order 3
1204 -11, -28, -192, 0, 256,
1205 }; // count = 5
1206#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
1207 static const real coeff[] = {
1208 // (eps+1)*A2-1, polynomial in eps2 of order 4
1209 -375, -704, -1792, -12288, 0, 16384,
1210 }; // count = 6
1211#else
1212#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1213#endif
1214 static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2,
1215 "Coefficient array size mismatch in A2m1f");
1216 int m = nA2_/2;
1217 real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
1218 return (t - eps) / (1 + eps);
1219 }
1220
1221 // The coefficients C2[l] in the Fourier expansion of B2
1222 void Geodesic::C2f(real eps, real c[]) {
1223 // Generated by Maxima on 2015-05-05 18:08:12-04:00
1224#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1225 static const real coeff[] = {
1226 // C2[1]/eps^1, polynomial in eps2 of order 1
1227 1, 8, 16,
1228 // C2[2]/eps^2, polynomial in eps2 of order 0
1229 3, 16,
1230 // C2[3]/eps^3, polynomial in eps2 of order 0
1231 5, 48,
1232 };
1233#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1234 static const real coeff[] = {
1235 // C2[1]/eps^1, polynomial in eps2 of order 1
1236 1, 8, 16,
1237 // C2[2]/eps^2, polynomial in eps2 of order 1
1238 1, 6, 32,
1239 // C2[3]/eps^3, polynomial in eps2 of order 0
1240 5, 48,
1241 // C2[4]/eps^4, polynomial in eps2 of order 0
1242 35, 512,
1243 };
1244#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1245 static const real coeff[] = {
1246 // C2[1]/eps^1, polynomial in eps2 of order 2
1247 1, 2, 16, 32,
1248 // C2[2]/eps^2, polynomial in eps2 of order 1
1249 1, 6, 32,
1250 // C2[3]/eps^3, polynomial in eps2 of order 1
1251 15, 80, 768,
1252 // C2[4]/eps^4, polynomial in eps2 of order 0
1253 35, 512,
1254 // C2[5]/eps^5, polynomial in eps2 of order 0
1255 63, 1280,
1256 };
1257#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1258 static const real coeff[] = {
1259 // C2[1]/eps^1, polynomial in eps2 of order 2
1260 1, 2, 16, 32,
1261 // C2[2]/eps^2, polynomial in eps2 of order 2
1262 35, 64, 384, 2048,
1263 // C2[3]/eps^3, polynomial in eps2 of order 1
1264 15, 80, 768,
1265 // C2[4]/eps^4, polynomial in eps2 of order 1
1266 7, 35, 512,
1267 // C2[5]/eps^5, polynomial in eps2 of order 0
1268 63, 1280,
1269 // C2[6]/eps^6, polynomial in eps2 of order 0
1270 77, 2048,
1271 };
1272#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1273 static const real coeff[] = {
1274 // C2[1]/eps^1, polynomial in eps2 of order 3
1275 41, 64, 128, 1024, 2048,
1276 // C2[2]/eps^2, polynomial in eps2 of order 2
1277 35, 64, 384, 2048,
1278 // C2[3]/eps^3, polynomial in eps2 of order 2
1279 69, 120, 640, 6144,
1280 // C2[4]/eps^4, polynomial in eps2 of order 1
1281 7, 35, 512,
1282 // C2[5]/eps^5, polynomial in eps2 of order 1
1283 105, 504, 10240,
1284 // C2[6]/eps^6, polynomial in eps2 of order 0
1285 77, 2048,
1286 // C2[7]/eps^7, polynomial in eps2 of order 0
1287 429, 14336,
1288 };
1289#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1290 static const real coeff[] = {
1291 // C2[1]/eps^1, polynomial in eps2 of order 3
1292 41, 64, 128, 1024, 2048,
1293 // C2[2]/eps^2, polynomial in eps2 of order 3
1294 47, 70, 128, 768, 4096,
1295 // C2[3]/eps^3, polynomial in eps2 of order 2
1296 69, 120, 640, 6144,
1297 // C2[4]/eps^4, polynomial in eps2 of order 2
1298 133, 224, 1120, 16384,
1299 // C2[5]/eps^5, polynomial in eps2 of order 1
1300 105, 504, 10240,
1301 // C2[6]/eps^6, polynomial in eps2 of order 1
1302 33, 154, 4096,
1303 // C2[7]/eps^7, polynomial in eps2 of order 0
1304 429, 14336,
1305 // C2[8]/eps^8, polynomial in eps2 of order 0
1306 6435, 262144,
1307 };
1308#else
1309#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1310#endif
1311 static_assert(sizeof(coeff) / sizeof(real) ==
1312 (nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4,
1313 "Coefficient array size mismatch in C2f");
1314 real
1315 eps2 = Math::sq(eps),
1316 d = eps;
1317 int o = 0;
1318 for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
1319 int m = (nC2_ - l) / 2; // order of polynomial in eps^2
1320 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1321 o += m + 2;
1322 d *= eps;
1323 }
1324 // Post condition: o == sizeof(coeff) / sizeof(real)
1325 }
1326
1327 // The scale factor A3 = mean value of (d/dsigma)I3
1328 void Geodesic::A3coeff() {
1329 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1330#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1331 static const real coeff[] = {
1332 // A3, coeff of eps^2, polynomial in n of order 0
1333 -1, 4,
1334 // A3, coeff of eps^1, polynomial in n of order 1
1335 1, -1, 2,
1336 // A3, coeff of eps^0, polynomial in n of order 0
1337 1, 1,
1338 };
1339#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1340 static const real coeff[] = {
1341 // A3, coeff of eps^3, polynomial in n of order 0
1342 -1, 16,
1343 // A3, coeff of eps^2, polynomial in n of order 1
1344 -1, -2, 8,
1345 // A3, coeff of eps^1, polynomial in n of order 1
1346 1, -1, 2,
1347 // A3, coeff of eps^0, polynomial in n of order 0
1348 1, 1,
1349 };
1350#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1351 static const real coeff[] = {
1352 // A3, coeff of eps^4, polynomial in n of order 0
1353 -3, 64,
1354 // A3, coeff of eps^3, polynomial in n of order 1
1355 -3, -1, 16,
1356 // A3, coeff of eps^2, polynomial in n of order 2
1357 3, -1, -2, 8,
1358 // A3, coeff of eps^1, polynomial in n of order 1
1359 1, -1, 2,
1360 // A3, coeff of eps^0, polynomial in n of order 0
1361 1, 1,
1362 };
1363#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1364 static const real coeff[] = {
1365 // A3, coeff of eps^5, polynomial in n of order 0
1366 -3, 128,
1367 // A3, coeff of eps^4, polynomial in n of order 1
1368 -2, -3, 64,
1369 // A3, coeff of eps^3, polynomial in n of order 2
1370 -1, -3, -1, 16,
1371 // A3, coeff of eps^2, polynomial in n of order 2
1372 3, -1, -2, 8,
1373 // A3, coeff of eps^1, polynomial in n of order 1
1374 1, -1, 2,
1375 // A3, coeff of eps^0, polynomial in n of order 0
1376 1, 1,
1377 };
1378#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1379 static const real coeff[] = {
1380 // A3, coeff of eps^6, polynomial in n of order 0
1381 -5, 256,
1382 // A3, coeff of eps^5, polynomial in n of order 1
1383 -5, -3, 128,
1384 // A3, coeff of eps^4, polynomial in n of order 2
1385 -10, -2, -3, 64,
1386 // A3, coeff of eps^3, polynomial in n of order 3
1387 5, -1, -3, -1, 16,
1388 // A3, coeff of eps^2, polynomial in n of order 2
1389 3, -1, -2, 8,
1390 // A3, coeff of eps^1, polynomial in n of order 1
1391 1, -1, 2,
1392 // A3, coeff of eps^0, polynomial in n of order 0
1393 1, 1,
1394 };
1395#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1396 static const real coeff[] = {
1397 // A3, coeff of eps^7, polynomial in n of order 0
1398 -25, 2048,
1399 // A3, coeff of eps^6, polynomial in n of order 1
1400 -15, -20, 1024,
1401 // A3, coeff of eps^5, polynomial in n of order 2
1402 -5, -10, -6, 256,
1403 // A3, coeff of eps^4, polynomial in n of order 3
1404 -5, -20, -4, -6, 128,
1405 // A3, coeff of eps^3, polynomial in n of order 3
1406 5, -1, -3, -1, 16,
1407 // A3, coeff of eps^2, polynomial in n of order 2
1408 3, -1, -2, 8,
1409 // A3, coeff of eps^1, polynomial in n of order 1
1410 1, -1, 2,
1411 // A3, coeff of eps^0, polynomial in n of order 0
1412 1, 1,
1413 };
1414#else
1415#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1416#endif
1417 static_assert(sizeof(coeff) / sizeof(real) ==
1418 (nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4,
1419 "Coefficient array size mismatch in A3f");
1420 int o = 0, k = 0;
1421 for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
1422 int m = min(nA3_ - j - 1, j); // order of polynomial in n
1423 _aA3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1424 o += m + 2;
1425 }
1426 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
1427 }
1428
1429 // The coefficients C3[l] in the Fourier expansion of B3
1430 void Geodesic::C3coeff() {
1431 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1432#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1433 static const real coeff[] = {
1434 // C3[1], coeff of eps^2, polynomial in n of order 0
1435 1, 8,
1436 // C3[1], coeff of eps^1, polynomial in n of order 1
1437 -1, 1, 4,
1438 // C3[2], coeff of eps^2, polynomial in n of order 0
1439 1, 16,
1440 };
1441#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1442 static const real coeff[] = {
1443 // C3[1], coeff of eps^3, polynomial in n of order 0
1444 3, 64,
1445 // C3[1], coeff of eps^2, polynomial in n of order 1
1446 // This is a case where a leading 0 term has been inserted to maintain the
1447 // pattern in the orders of the polynomials.
1448 0, 1, 8,
1449 // C3[1], coeff of eps^1, polynomial in n of order 1
1450 -1, 1, 4,
1451 // C3[2], coeff of eps^3, polynomial in n of order 0
1452 3, 64,
1453 // C3[2], coeff of eps^2, polynomial in n of order 1
1454 -3, 2, 32,
1455 // C3[3], coeff of eps^3, polynomial in n of order 0
1456 5, 192,
1457 };
1458#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1459 static const real coeff[] = {
1460 // C3[1], coeff of eps^4, polynomial in n of order 0
1461 5, 128,
1462 // C3[1], coeff of eps^3, polynomial in n of order 1
1463 3, 3, 64,
1464 // C3[1], coeff of eps^2, polynomial in n of order 2
1465 -1, 0, 1, 8,
1466 // C3[1], coeff of eps^1, polynomial in n of order 1
1467 -1, 1, 4,
1468 // C3[2], coeff of eps^4, polynomial in n of order 0
1469 3, 128,
1470 // C3[2], coeff of eps^3, polynomial in n of order 1
1471 -2, 3, 64,
1472 // C3[2], coeff of eps^2, polynomial in n of order 2
1473 1, -3, 2, 32,
1474 // C3[3], coeff of eps^4, polynomial in n of order 0
1475 3, 128,
1476 // C3[3], coeff of eps^3, polynomial in n of order 1
1477 -9, 5, 192,
1478 // C3[4], coeff of eps^4, polynomial in n of order 0
1479 7, 512,
1480 };
1481#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1482 static const real coeff[] = {
1483 // C3[1], coeff of eps^5, polynomial in n of order 0
1484 3, 128,
1485 // C3[1], coeff of eps^4, polynomial in n of order 1
1486 2, 5, 128,
1487 // C3[1], coeff of eps^3, polynomial in n of order 2
1488 -1, 3, 3, 64,
1489 // C3[1], coeff of eps^2, polynomial in n of order 2
1490 -1, 0, 1, 8,
1491 // C3[1], coeff of eps^1, polynomial in n of order 1
1492 -1, 1, 4,
1493 // C3[2], coeff of eps^5, polynomial in n of order 0
1494 5, 256,
1495 // C3[2], coeff of eps^4, polynomial in n of order 1
1496 1, 3, 128,
1497 // C3[2], coeff of eps^3, polynomial in n of order 2
1498 -3, -2, 3, 64,
1499 // C3[2], coeff of eps^2, polynomial in n of order 2
1500 1, -3, 2, 32,
1501 // C3[3], coeff of eps^5, polynomial in n of order 0
1502 7, 512,
1503 // C3[3], coeff of eps^4, polynomial in n of order 1
1504 -10, 9, 384,
1505 // C3[3], coeff of eps^3, polynomial in n of order 2
1506 5, -9, 5, 192,
1507 // C3[4], coeff of eps^5, polynomial in n of order 0
1508 7, 512,
1509 // C3[4], coeff of eps^4, polynomial in n of order 1
1510 -14, 7, 512,
1511 // C3[5], coeff of eps^5, polynomial in n of order 0
1512 21, 2560,
1513 };
1514#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1515 static const real coeff[] = {
1516 // C3[1], coeff of eps^6, polynomial in n of order 0
1517 21, 1024,
1518 // C3[1], coeff of eps^5, polynomial in n of order 1
1519 11, 12, 512,
1520 // C3[1], coeff of eps^4, polynomial in n of order 2
1521 2, 2, 5, 128,
1522 // C3[1], coeff of eps^3, polynomial in n of order 3
1523 -5, -1, 3, 3, 64,
1524 // C3[1], coeff of eps^2, polynomial in n of order 2
1525 -1, 0, 1, 8,
1526 // C3[1], coeff of eps^1, polynomial in n of order 1
1527 -1, 1, 4,
1528 // C3[2], coeff of eps^6, polynomial in n of order 0
1529 27, 2048,
1530 // C3[2], coeff of eps^5, polynomial in n of order 1
1531 1, 5, 256,
1532 // C3[2], coeff of eps^4, polynomial in n of order 2
1533 -9, 2, 6, 256,
1534 // C3[2], coeff of eps^3, polynomial in n of order 3
1535 2, -3, -2, 3, 64,
1536 // C3[2], coeff of eps^2, polynomial in n of order 2
1537 1, -3, 2, 32,
1538 // C3[3], coeff of eps^6, polynomial in n of order 0
1539 3, 256,
1540 // C3[3], coeff of eps^5, polynomial in n of order 1
1541 -4, 21, 1536,
1542 // C3[3], coeff of eps^4, polynomial in n of order 2
1543 -6, -10, 9, 384,
1544 // C3[3], coeff of eps^3, polynomial in n of order 3
1545 -1, 5, -9, 5, 192,
1546 // C3[4], coeff of eps^6, polynomial in n of order 0
1547 9, 1024,
1548 // C3[4], coeff of eps^5, polynomial in n of order 1
1549 -10, 7, 512,
1550 // C3[4], coeff of eps^4, polynomial in n of order 2
1551 10, -14, 7, 512,
1552 // C3[5], coeff of eps^6, polynomial in n of order 0
1553 9, 1024,
1554 // C3[5], coeff of eps^5, polynomial in n of order 1
1555 -45, 21, 2560,
1556 // C3[6], coeff of eps^6, polynomial in n of order 0
1557 11, 2048,
1558 };
1559#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1560 static const real coeff[] = {
1561 // C3[1], coeff of eps^7, polynomial in n of order 0
1562 243, 16384,
1563 // C3[1], coeff of eps^6, polynomial in n of order 1
1564 10, 21, 1024,
1565 // C3[1], coeff of eps^5, polynomial in n of order 2
1566 3, 11, 12, 512,
1567 // C3[1], coeff of eps^4, polynomial in n of order 3
1568 -2, 2, 2, 5, 128,
1569 // C3[1], coeff of eps^3, polynomial in n of order 3
1570 -5, -1, 3, 3, 64,
1571 // C3[1], coeff of eps^2, polynomial in n of order 2
1572 -1, 0, 1, 8,
1573 // C3[1], coeff of eps^1, polynomial in n of order 1
1574 -1, 1, 4,
1575 // C3[2], coeff of eps^7, polynomial in n of order 0
1576 187, 16384,
1577 // C3[2], coeff of eps^6, polynomial in n of order 1
1578 69, 108, 8192,
1579 // C3[2], coeff of eps^5, polynomial in n of order 2
1580 -2, 1, 5, 256,
1581 // C3[2], coeff of eps^4, polynomial in n of order 3
1582 -6, -9, 2, 6, 256,
1583 // C3[2], coeff of eps^3, polynomial in n of order 3
1584 2, -3, -2, 3, 64,
1585 // C3[2], coeff of eps^2, polynomial in n of order 2
1586 1, -3, 2, 32,
1587 // C3[3], coeff of eps^7, polynomial in n of order 0
1588 139, 16384,
1589 // C3[3], coeff of eps^6, polynomial in n of order 1
1590 -1, 12, 1024,
1591 // C3[3], coeff of eps^5, polynomial in n of order 2
1592 -77, -8, 42, 3072,
1593 // C3[3], coeff of eps^4, polynomial in n of order 3
1594 10, -6, -10, 9, 384,
1595 // C3[3], coeff of eps^3, polynomial in n of order 3
1596 -1, 5, -9, 5, 192,
1597 // C3[4], coeff of eps^7, polynomial in n of order 0
1598 127, 16384,
1599 // C3[4], coeff of eps^6, polynomial in n of order 1
1600 -43, 72, 8192,
1601 // C3[4], coeff of eps^5, polynomial in n of order 2
1602 -7, -40, 28, 2048,
1603 // C3[4], coeff of eps^4, polynomial in n of order 3
1604 -7, 20, -28, 14, 1024,
1605 // C3[5], coeff of eps^7, polynomial in n of order 0
1606 99, 16384,
1607 // C3[5], coeff of eps^6, polynomial in n of order 1
1608 -15, 9, 1024,
1609 // C3[5], coeff of eps^5, polynomial in n of order 2
1610 75, -90, 42, 5120,
1611 // C3[6], coeff of eps^7, polynomial in n of order 0
1612 99, 16384,
1613 // C3[6], coeff of eps^6, polynomial in n of order 1
1614 -99, 44, 8192,
1615 // C3[7], coeff of eps^7, polynomial in n of order 0
1616 429, 114688,
1617 };
1618#else
1619#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1620#endif
1621 static_assert(sizeof(coeff) / sizeof(real) ==
1622 ((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8,
1623 "Coefficient array size mismatch in C3coeff");
1624 int o = 0, k = 0;
1625 for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
1626 for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
1627 int m = min(nC3_ - j - 1, j); // order of polynomial in n
1628 _cC3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1629 o += m + 2;
1630 }
1631 }
1632 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
1633 }
1634
1635 void Geodesic::C4coeff() {
1636 // Generated by Maxima on 2015-05-05 18:08:13-04:00
1637#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
1638 static const real coeff[] = {
1639 // C4[0], coeff of eps^2, polynomial in n of order 0
1640 -2, 105,
1641 // C4[0], coeff of eps^1, polynomial in n of order 1
1642 16, -7, 35,
1643 // C4[0], coeff of eps^0, polynomial in n of order 2
1644 8, -28, 70, 105,
1645 // C4[1], coeff of eps^2, polynomial in n of order 0
1646 -2, 105,
1647 // C4[1], coeff of eps^1, polynomial in n of order 1
1648 -16, 7, 315,
1649 // C4[2], coeff of eps^2, polynomial in n of order 0
1650 4, 525,
1651 };
1652#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
1653 static const real coeff[] = {
1654 // C4[0], coeff of eps^3, polynomial in n of order 0
1655 11, 315,
1656 // C4[0], coeff of eps^2, polynomial in n of order 1
1657 -32, -6, 315,
1658 // C4[0], coeff of eps^1, polynomial in n of order 2
1659 -32, 48, -21, 105,
1660 // C4[0], coeff of eps^0, polynomial in n of order 3
1661 4, 24, -84, 210, 315,
1662 // C4[1], coeff of eps^3, polynomial in n of order 0
1663 -1, 105,
1664 // C4[1], coeff of eps^2, polynomial in n of order 1
1665 64, -18, 945,
1666 // C4[1], coeff of eps^1, polynomial in n of order 2
1667 32, -48, 21, 945,
1668 // C4[2], coeff of eps^3, polynomial in n of order 0
1669 -8, 1575,
1670 // C4[2], coeff of eps^2, polynomial in n of order 1
1671 -32, 12, 1575,
1672 // C4[3], coeff of eps^3, polynomial in n of order 0
1673 8, 2205,
1674 };
1675#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
1676 static const real coeff[] = {
1677 // C4[0], coeff of eps^4, polynomial in n of order 0
1678 4, 1155,
1679 // C4[0], coeff of eps^3, polynomial in n of order 1
1680 -368, 121, 3465,
1681 // C4[0], coeff of eps^2, polynomial in n of order 2
1682 1088, -352, -66, 3465,
1683 // C4[0], coeff of eps^1, polynomial in n of order 3
1684 48, -352, 528, -231, 1155,
1685 // C4[0], coeff of eps^0, polynomial in n of order 4
1686 16, 44, 264, -924, 2310, 3465,
1687 // C4[1], coeff of eps^4, polynomial in n of order 0
1688 4, 1155,
1689 // C4[1], coeff of eps^3, polynomial in n of order 1
1690 80, -99, 10395,
1691 // C4[1], coeff of eps^2, polynomial in n of order 2
1692 -896, 704, -198, 10395,
1693 // C4[1], coeff of eps^1, polynomial in n of order 3
1694 -48, 352, -528, 231, 10395,
1695 // C4[2], coeff of eps^4, polynomial in n of order 0
1696 -8, 1925,
1697 // C4[2], coeff of eps^3, polynomial in n of order 1
1698 384, -88, 17325,
1699 // C4[2], coeff of eps^2, polynomial in n of order 2
1700 320, -352, 132, 17325,
1701 // C4[3], coeff of eps^4, polynomial in n of order 0
1702 -16, 8085,
1703 // C4[3], coeff of eps^3, polynomial in n of order 1
1704 -256, 88, 24255,
1705 // C4[4], coeff of eps^4, polynomial in n of order 0
1706 64, 31185,
1707 };
1708#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
1709 static const real coeff[] = {
1710 // C4[0], coeff of eps^5, polynomial in n of order 0
1711 97, 15015,
1712 // C4[0], coeff of eps^4, polynomial in n of order 1
1713 1088, 156, 45045,
1714 // C4[0], coeff of eps^3, polynomial in n of order 2
1715 -224, -4784, 1573, 45045,
1716 // C4[0], coeff of eps^2, polynomial in n of order 3
1717 -10656, 14144, -4576, -858, 45045,
1718 // C4[0], coeff of eps^1, polynomial in n of order 4
1719 64, 624, -4576, 6864, -3003, 15015,
1720 // C4[0], coeff of eps^0, polynomial in n of order 5
1721 100, 208, 572, 3432, -12012, 30030, 45045,
1722 // C4[1], coeff of eps^5, polynomial in n of order 0
1723 1, 9009,
1724 // C4[1], coeff of eps^4, polynomial in n of order 1
1725 -2944, 468, 135135,
1726 // C4[1], coeff of eps^3, polynomial in n of order 2
1727 5792, 1040, -1287, 135135,
1728 // C4[1], coeff of eps^2, polynomial in n of order 3
1729 5952, -11648, 9152, -2574, 135135,
1730 // C4[1], coeff of eps^1, polynomial in n of order 4
1731 -64, -624, 4576, -6864, 3003, 135135,
1732 // C4[2], coeff of eps^5, polynomial in n of order 0
1733 8, 10725,
1734 // C4[2], coeff of eps^4, polynomial in n of order 1
1735 1856, -936, 225225,
1736 // C4[2], coeff of eps^3, polynomial in n of order 2
1737 -8448, 4992, -1144, 225225,
1738 // C4[2], coeff of eps^2, polynomial in n of order 3
1739 -1440, 4160, -4576, 1716, 225225,
1740 // C4[3], coeff of eps^5, polynomial in n of order 0
1741 -136, 63063,
1742 // C4[3], coeff of eps^4, polynomial in n of order 1
1743 1024, -208, 105105,
1744 // C4[3], coeff of eps^3, polynomial in n of order 2
1745 3584, -3328, 1144, 315315,
1746 // C4[4], coeff of eps^5, polynomial in n of order 0
1747 -128, 135135,
1748 // C4[4], coeff of eps^4, polynomial in n of order 1
1749 -2560, 832, 405405,
1750 // C4[5], coeff of eps^5, polynomial in n of order 0
1751 128, 99099,
1752 };
1753#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
1754 static const real coeff[] = {
1755 // C4[0], coeff of eps^6, polynomial in n of order 0
1756 10, 9009,
1757 // C4[0], coeff of eps^5, polynomial in n of order 1
1758 -464, 291, 45045,
1759 // C4[0], coeff of eps^4, polynomial in n of order 2
1760 -4480, 1088, 156, 45045,
1761 // C4[0], coeff of eps^3, polynomial in n of order 3
1762 10736, -224, -4784, 1573, 45045,
1763 // C4[0], coeff of eps^2, polynomial in n of order 4
1764 1664, -10656, 14144, -4576, -858, 45045,
1765 // C4[0], coeff of eps^1, polynomial in n of order 5
1766 16, 64, 624, -4576, 6864, -3003, 15015,
1767 // C4[0], coeff of eps^0, polynomial in n of order 6
1768 56, 100, 208, 572, 3432, -12012, 30030, 45045,
1769 // C4[1], coeff of eps^6, polynomial in n of order 0
1770 10, 9009,
1771 // C4[1], coeff of eps^5, polynomial in n of order 1
1772 112, 15, 135135,
1773 // C4[1], coeff of eps^4, polynomial in n of order 2
1774 3840, -2944, 468, 135135,
1775 // C4[1], coeff of eps^3, polynomial in n of order 3
1776 -10704, 5792, 1040, -1287, 135135,
1777 // C4[1], coeff of eps^2, polynomial in n of order 4
1778 -768, 5952, -11648, 9152, -2574, 135135,
1779 // C4[1], coeff of eps^1, polynomial in n of order 5
1780 -16, -64, -624, 4576, -6864, 3003, 135135,
1781 // C4[2], coeff of eps^6, polynomial in n of order 0
1782 -4, 25025,
1783 // C4[2], coeff of eps^5, polynomial in n of order 1
1784 -1664, 168, 225225,
1785 // C4[2], coeff of eps^4, polynomial in n of order 2
1786 1664, 1856, -936, 225225,
1787 // C4[2], coeff of eps^3, polynomial in n of order 3
1788 6784, -8448, 4992, -1144, 225225,
1789 // C4[2], coeff of eps^2, polynomial in n of order 4
1790 128, -1440, 4160, -4576, 1716, 225225,
1791 // C4[3], coeff of eps^6, polynomial in n of order 0
1792 64, 315315,
1793 // C4[3], coeff of eps^5, polynomial in n of order 1
1794 1792, -680, 315315,
1795 // C4[3], coeff of eps^4, polynomial in n of order 2
1796 -2048, 1024, -208, 105105,
1797 // C4[3], coeff of eps^3, polynomial in n of order 3
1798 -1792, 3584, -3328, 1144, 315315,
1799 // C4[4], coeff of eps^6, polynomial in n of order 0
1800 -512, 405405,
1801 // C4[4], coeff of eps^5, polynomial in n of order 1
1802 2048, -384, 405405,
1803 // C4[4], coeff of eps^4, polynomial in n of order 2
1804 3072, -2560, 832, 405405,
1805 // C4[5], coeff of eps^6, polynomial in n of order 0
1806 -256, 495495,
1807 // C4[5], coeff of eps^5, polynomial in n of order 1
1808 -2048, 640, 495495,
1809 // C4[6], coeff of eps^6, polynomial in n of order 0
1810 512, 585585,
1811 };
1812#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
1813 static const real coeff[] = {
1814 // C4[0], coeff of eps^7, polynomial in n of order 0
1815 193, 85085,
1816 // C4[0], coeff of eps^6, polynomial in n of order 1
1817 4192, 850, 765765,
1818 // C4[0], coeff of eps^5, polynomial in n of order 2
1819 20960, -7888, 4947, 765765,
1820 // C4[0], coeff of eps^4, polynomial in n of order 3
1821 12480, -76160, 18496, 2652, 765765,
1822 // C4[0], coeff of eps^3, polynomial in n of order 4
1823 -154048, 182512, -3808, -81328, 26741, 765765,
1824 // C4[0], coeff of eps^2, polynomial in n of order 5
1825 3232, 28288, -181152, 240448, -77792, -14586, 765765,
1826 // C4[0], coeff of eps^1, polynomial in n of order 6
1827 96, 272, 1088, 10608, -77792, 116688, -51051, 255255,
1828 // C4[0], coeff of eps^0, polynomial in n of order 7
1829 588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765,
1830 // C4[1], coeff of eps^7, polynomial in n of order 0
1831 349, 2297295,
1832 // C4[1], coeff of eps^6, polynomial in n of order 1
1833 -1472, 510, 459459,
1834 // C4[1], coeff of eps^5, polynomial in n of order 2
1835 -39840, 1904, 255, 2297295,
1836 // C4[1], coeff of eps^4, polynomial in n of order 3
1837 52608, 65280, -50048, 7956, 2297295,
1838 // C4[1], coeff of eps^3, polynomial in n of order 4
1839 103744, -181968, 98464, 17680, -21879, 2297295,
1840 // C4[1], coeff of eps^2, polynomial in n of order 5
1841 -1344, -13056, 101184, -198016, 155584, -43758, 2297295,
1842 // C4[1], coeff of eps^1, polynomial in n of order 6
1843 -96, -272, -1088, -10608, 77792, -116688, 51051, 2297295,
1844 // C4[2], coeff of eps^7, polynomial in n of order 0
1845 464, 1276275,
1846 // C4[2], coeff of eps^6, polynomial in n of order 1
1847 -928, -612, 3828825,
1848 // C4[2], coeff of eps^5, polynomial in n of order 2
1849 64256, -28288, 2856, 3828825,
1850 // C4[2], coeff of eps^4, polynomial in n of order 3
1851 -126528, 28288, 31552, -15912, 3828825,
1852 // C4[2], coeff of eps^3, polynomial in n of order 4
1853 -41472, 115328, -143616, 84864, -19448, 3828825,
1854 // C4[2], coeff of eps^2, polynomial in n of order 5
1855 160, 2176, -24480, 70720, -77792, 29172, 3828825,
1856 // C4[3], coeff of eps^7, polynomial in n of order 0
1857 -16, 97461,
1858 // C4[3], coeff of eps^6, polynomial in n of order 1
1859 -16384, 1088, 5360355,
1860 // C4[3], coeff of eps^5, polynomial in n of order 2
1861 -2560, 30464, -11560, 5360355,
1862 // C4[3], coeff of eps^4, polynomial in n of order 3
1863 35840, -34816, 17408, -3536, 1786785,
1864 // C4[3], coeff of eps^3, polynomial in n of order 4
1865 7168, -30464, 60928, -56576, 19448, 5360355,
1866 // C4[4], coeff of eps^7, polynomial in n of order 0
1867 128, 2297295,
1868 // C4[4], coeff of eps^6, polynomial in n of order 1
1869 26624, -8704, 6891885,
1870 // C4[4], coeff of eps^5, polynomial in n of order 2
1871 -77824, 34816, -6528, 6891885,
1872 // C4[4], coeff of eps^4, polynomial in n of order 3
1873 -32256, 52224, -43520, 14144, 6891885,
1874 // C4[5], coeff of eps^7, polynomial in n of order 0
1875 -6784, 8423415,
1876 // C4[5], coeff of eps^6, polynomial in n of order 1
1877 24576, -4352, 8423415,
1878 // C4[5], coeff of eps^5, polynomial in n of order 2
1879 45056, -34816, 10880, 8423415,
1880 // C4[6], coeff of eps^7, polynomial in n of order 0
1881 -1024, 3318315,
1882 // C4[6], coeff of eps^6, polynomial in n of order 1
1883 -28672, 8704, 9954945,
1884 // C4[7], coeff of eps^7, polynomial in n of order 0
1885 1024, 1640925,
1886 };
1887#else
1888#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
1889#endif
1890 static_assert(sizeof(coeff) / sizeof(real) ==
1891 (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
1892 "Coefficient array size mismatch in C4coeff");
1893 int o = 0, k = 0;
1894 for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
1895 for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
1896 int m = nC4_ - j - 1; // order of polynomial in n
1897 _cC4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
1898 o += m + 2;
1899 }
1900 }
1901 // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
1902 }
1903
1904} // namespace GeographicLib
real
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
Geodesic.hpp
Header for GeographicLib::Geodesic class.
GeographicLib::Geodesic
Geodesic calculations
Definition: Geodesic.hpp:172
GeographicLib::Geodesic::InverseLine
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition: Geodesic.cpp:518
GeographicLib::Geodesic::Geodesic
Geodesic(real a, real f)
Definition: Geodesic.cpp:42
GeographicLib::Geodesic::WGS84
static const Geodesic & WGS84()
Definition: Geodesic.cpp:89
GeographicLib::Geodesic::ArcDirectLine
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition: Geodesic.cpp:154
GeographicLib::Geodesic::Line
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition: Geodesic.cpp:118
GeographicLib::Geodesic::GenDirectLine
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition: Geodesic.cpp:136
GeographicLib::Geodesic::GeodesicLine
friend class GeodesicLine
Definition: Geodesic.hpp:175
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:123
GeographicLib::Geodesic::DirectLine
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition: Geodesic.cpp:149
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition: Constants.hpp:316
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
GeographicLib::Math::degree
static T degree()
Definition: Math.hpp:200
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition: Math.hpp:300
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:106
GeographicLib::Math::atan2d
static T atan2d(T y, T x)
Definition: Math.cpp:183
GeographicLib::Math::norm
static void norm(T &x, T &y)
Definition: Math.hpp:222
GeographicLib::Math::AngRound
static T AngRound(T x)
Definition: Math.cpp:97
GeographicLib::Math::sq
static T sq(T x)
Definition: Math.hpp:212
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition: Math.cpp:71
GeographicLib::Math::sincosde
static void sincosde(T x, T t, T &sinx, T &cosx)
Definition: Math.cpp:126
GeographicLib::Math::pi
static T pi()
Definition: Math.hpp:190
GeographicLib::Math::polyval
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:271
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition: Math.cpp:82
GeographicLib::Math::hd
@ hd
degrees per half turn
Definition: Math.hpp:144
GeographicLib::Math::qd
@ qd
degrees per quarter turn
Definition: Math.hpp:141
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:826