C library for Geodesics  1.38
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geodesic.c
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1 /**
2  * \file geodesic.c
3  * \brief Implementation of the geodesic routines in C
4  *
5  * For the full documentation see geodesic.h.
6  **********************************************************************/
7 
8 /** @cond SKIP */
9 
10 /*
11  * This is a C implementation of the geodesic algorithms described in
12  *
13  * C. F. F. Karney,
14  * Algorithms for geodesics,
15  * J. Geodesy <b>87</b>, 43--55 (2013);
16  * http://dx.doi.org/10.1007/s00190-012-0578-z
17  * Addenda: http://geographiclib.sf.net/geod-addenda.html
18  *
19  * See the comments in geodesic.h for documentation.
20  *
21  * Copyright (c) Charles Karney (2012-2013) <charles@karney.com> and licensed
22  * under the MIT/X11 License. For more information, see
23  * http://geographiclib.sourceforge.net/
24  */
25 
26 #include "geodesic.h"
27 #include <math.h>
28 
29 #define GEOGRAPHICLIB_GEODESIC_ORDER 6
30 #define nC1 GEOGRAPHICLIB_GEODESIC_ORDER
31 #define nC1p GEOGRAPHICLIB_GEODESIC_ORDER
32 #define nC2 GEOGRAPHICLIB_GEODESIC_ORDER
33 #define nA3 GEOGRAPHICLIB_GEODESIC_ORDER
34 #define nA3x nA3
35 #define nC3 GEOGRAPHICLIB_GEODESIC_ORDER
36 #define nC3x ((nC3 * (nC3 - 1)) / 2)
37 #define nC4 GEOGRAPHICLIB_GEODESIC_ORDER
38 #define nC4x ((nC4 * (nC4 + 1)) / 2)
39 
40 typedef double real;
41 typedef int boolx;
42 
43 static unsigned init = 0;
44 static const int FALSE = 0;
45 static const int TRUE = 1;
46 static unsigned digits, maxit1, maxit2;
47 static real epsilon, realmin, pi, degree, NaN,
48  tiny, tol0, tol1, tol2, tolb, xthresh;
49 
50 static void Init() {
51  if (!init) {
52 #if defined(__DBL_MANT_DIG__)
53  digits = __DBL_MANT_DIG__;
54 #else
55  digits = 53;
56 #endif
57 #if defined(__DBL_EPSILON__)
58  epsilon = __DBL_EPSILON__;
59 #else
60  epsilon = pow(0.5, digits - 1);
61 #endif
62 #if defined(__DBL_MIN__)
63  realmin = __DBL_MIN__;
64 #else
65  realmin = pow(0.5, 1022);
66 #endif
67 #if defined(M_PI)
68  pi = M_PI;
69 #else
70  pi = atan2(0.0, -1.0);
71 #endif
72  maxit1 = 20;
73  maxit2 = maxit1 + digits + 10;
74  tiny = sqrt(realmin);
75  tol0 = epsilon;
76  /* Increase multiplier in defn of tol1 from 100 to 200 to fix inverse case
77  * 52.784459512564 0 -52.784459512563990912 179.634407464943777557
78  * which otherwise failed for Visual Studio 10 (Release and Debug) */
79  tol1 = 200 * tol0;
80  tol2 = sqrt(tol0);
81  /* Check on bisection interval */
82  tolb = tol0 * tol2;
83  xthresh = 1000 * tol2;
84  degree = pi/180;
85  NaN = sqrt(-1.0);
86  init = 1;
87  }
88 }
89 
90 enum captype {
91  CAP_NONE = 0U,
92  CAP_C1 = 1U<<0,
93  CAP_C1p = 1U<<1,
94  CAP_C2 = 1U<<2,
95  CAP_C3 = 1U<<3,
96  CAP_C4 = 1U<<4,
97  CAP_ALL = 0x1FU,
98  OUT_ALL = 0x7F80U
99 };
100 
101 static real sq(real x) { return x * x; }
102 static real log1px(real x) {
103  volatile real
104  y = 1 + x,
105  z = y - 1;
106  /* Here's the explanation for this magic: y = 1 + z, exactly, and z
107  * approx x, thus log(y)/z (which is nearly constant near z = 0) returns
108  * a good approximation to the true log(1 + x)/x. The multiplication x *
109  * (log(y)/z) introduces little additional error. */
110  return z == 0 ? x : x * log(y) / z;
111 }
112 
113 static real atanhx(real x) {
114  real y = fabs(x); /* Enforce odd parity */
115  y = log1px(2 * y/(1 - y))/2;
116  return x < 0 ? -y : y;
117 }
118 
119 static real hypotx(real x, real y)
120 { return sqrt(x * x + y * y); }
121 
122 static real cbrtx(real x) {
123  real y = pow(fabs(x), 1/(real)(3)); /* Return the real cube root */
124  return x < 0 ? -y : y;
125 }
126 
127 static real sumx(real u, real v, real* t) {
128  volatile real s = u + v;
129  volatile real up = s - v;
130  volatile real vpp = s - up;
131  up -= u;
132  vpp -= v;
133  *t = -(up + vpp);
134  /* error-free sum:
135  * u + v = s + t
136  * = round(u + v) + t */
137  return s;
138 }
139 
140 static real minx(real x, real y)
141 { return x < y ? x : y; }
142 
143 static real maxx(real x, real y)
144 { return x > y ? x : y; }
145 
146 static void swapx(real* x, real* y)
147 { real t = *x; *x = *y; *y = t; }
148 
149 static void SinCosNorm(real* sinx, real* cosx) {
150  real r = hypotx(*sinx, *cosx);
151  *sinx /= r;
152  *cosx /= r;
153 }
154 
155 static real AngNormalize(real x)
156 { return x >= 180 ? x - 360 : (x < -180 ? x + 360 : x); }
157 static real AngNormalize2(real x)
158 { return AngNormalize(fmod(x, (real)(360))); }
159 
160 static real AngDiff(real x, real y) {
161  real t, d = sumx(-x, y, &t);
162  if ((d - (real)(180)) + t > (real)(0)) /* y - x > 180 */
163  d -= (real)(360); /* exact */
164  else if ((d + (real)(180)) + t <= (real)(0)) /* y - x <= -180 */
165  d += (real)(360); /* exact */
166  return d + t;
167 }
168 
169 static real AngRound(real x) {
170  const real z = 1/(real)(16);
171  volatile real y = fabs(x);
172  /* The compiler mustn't "simplify" z - (z - y) to y */
173  y = y < z ? z - (z - y) : y;
174  return x < 0 ? -y : y;
175 }
176 
177 static void A3coeff(struct geod_geodesic* g);
178 static void C3coeff(struct geod_geodesic* g);
179 static void C4coeff(struct geod_geodesic* g);
180 static real SinCosSeries(boolx sinp,
181  real sinx, real cosx,
182  const real c[], int n);
183 static void Lengths(const struct geod_geodesic* g,
184  real eps, real sig12,
185  real ssig1, real csig1, real dn1,
186  real ssig2, real csig2, real dn2,
187  real cbet1, real cbet2,
188  real* ps12b, real* pm12b, real* pm0,
189  boolx scalep, real* pM12, real* pM21,
190  /* Scratch areas of the right size */
191  real C1a[], real C2a[]);
192 static real Astroid(real x, real y);
193 static real InverseStart(const struct geod_geodesic* g,
194  real sbet1, real cbet1, real dn1,
195  real sbet2, real cbet2, real dn2,
196  real lam12,
197  real* psalp1, real* pcalp1,
198  /* Only updated if return val >= 0 */
199  real* psalp2, real* pcalp2,
200  /* Only updated for short lines */
201  real* pdnm,
202  /* Scratch areas of the right size */
203  real C1a[], real C2a[]);
204 static real Lambda12(const struct geod_geodesic* g,
205  real sbet1, real cbet1, real dn1,
206  real sbet2, real cbet2, real dn2,
207  real salp1, real calp1,
208  real* psalp2, real* pcalp2,
209  real* psig12,
210  real* pssig1, real* pcsig1,
211  real* pssig2, real* pcsig2,
212  real* peps, real* pdomg12,
213  boolx diffp, real* pdlam12,
214  /* Scratch areas of the right size */
215  real C1a[], real C2a[], real C3a[]);
216 static real A3f(const struct geod_geodesic* g, real eps);
217 static void C3f(const struct geod_geodesic* g, real eps, real c[]);
218 static void C4f(const struct geod_geodesic* g, real eps, real c[]);
219 static real A1m1f(real eps);
220 static void C1f(real eps, real c[]);
221 static void C1pf(real eps, real c[]);
222 static real A2m1f(real eps);
223 static void C2f(real eps, real c[]);
224 static int transit(real lon1, real lon2);
225 static void accini(real s[]);
226 static void acccopy(const real s[], real t[]);
227 static void accadd(real s[], real y);
228 static real accsum(const real s[], real y);
229 static void accneg(real s[]);
230 
231 void geod_init(struct geod_geodesic* g, real a, real f) {
232  if (!init) Init();
233  g->a = a;
234  g->f = f <= 1 ? f : 1/f;
235  g->f1 = 1 - g->f;
236  g->e2 = g->f * (2 - g->f);
237  g->ep2 = g->e2 / sq(g->f1); /* e2 / (1 - e2) */
238  g->n = g->f / ( 2 - g->f);
239  g->b = g->a * g->f1;
240  g->c2 = (sq(g->a) + sq(g->b) *
241  (g->e2 == 0 ? 1 :
242  (g->e2 > 0 ? atanhx(sqrt(g->e2)) : atan(sqrt(-g->e2))) /
243  sqrt(fabs(g->e2))))/2; /* authalic radius squared */
244  /* The sig12 threshold for "really short". Using the auxiliary sphere
245  * solution with dnm computed at (bet1 + bet2) / 2, the relative error in the
246  * azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2. (Error
247  * measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a given f and
248  * sig12, the max error occurs for lines near the pole. If the old rule for
249  * computing dnm = (dn1 + dn2)/2 is used, then the error increases by a
250  * factor of 2.) Setting this equal to epsilon gives sig12 = etol2. Here
251  * 0.1 is a safety factor (error decreased by 100) and max(0.001, abs(f))
252  * stops etol2 getting too large in the nearly spherical case. */
253  g->etol2 = 0.1 * tol2 /
254  sqrt( maxx((real)(0.001), fabs(g->f)) * minx((real)(1), 1 - g->f/2) / 2 );
255 
256  A3coeff(g);
257  C3coeff(g);
258  C4coeff(g);
259 }
260 
261 void geod_lineinit(struct geod_geodesicline* l,
262  const struct geod_geodesic* g,
263  real lat1, real lon1, real azi1, unsigned caps) {
264  real alp1, cbet1, sbet1, phi, eps;
265  l->a = g->a;
266  l->f = g->f;
267  l->b = g->b;
268  l->c2 = g->c2;
269  l->f1 = g->f1;
270  /* If caps is 0 assume the standard direct calculation */
271  l->caps = (caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE) |
272  GEOD_LATITUDE | GEOD_AZIMUTH; /* Always allow latitude and azimuth */
273 
274  /* Guard against underflow in salp0 */
275  azi1 = AngRound(AngNormalize(azi1));
276  lon1 = AngNormalize(lon1);
277  l->lat1 = lat1;
278  l->lon1 = lon1;
279  l->azi1 = azi1;
280  /* alp1 is in [0, pi] */
281  alp1 = azi1 * degree;
282  /* Enforce sin(pi) == 0 and cos(pi/2) == 0. Better to face the ensuing
283  * problems directly than to skirt them. */
284  l->salp1 = azi1 == -180 ? 0 : sin(alp1);
285  l->calp1 = fabs(azi1) == 90 ? 0 : cos(alp1);
286  phi = lat1 * degree;
287  /* Ensure cbet1 = +epsilon at poles */
288  sbet1 = l->f1 * sin(phi);
289  cbet1 = fabs(lat1) == 90 ? tiny : cos(phi);
290  SinCosNorm(&sbet1, &cbet1);
291  l->dn1 = sqrt(1 + g->ep2 * sq(sbet1));
292 
293  /* Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), */
294  l->salp0 = l->salp1 * cbet1; /* alp0 in [0, pi/2 - |bet1|] */
295  /* Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
296  * is slightly better (consider the case salp1 = 0). */
297  l->calp0 = hypotx(l->calp1, l->salp1 * sbet1);
298  /* Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
299  * sig = 0 is nearest northward crossing of equator.
300  * With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
301  * With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
302  * With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
303  * Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
304  * With alp0 in (0, pi/2], quadrants for sig and omg coincide.
305  * No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
306  * With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. */
307  l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1;
308  l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1;
309  SinCosNorm(&l->ssig1, &l->csig1); /* sig1 in (-pi, pi] */
310  /* SinCosNorm(somg1, comg1); -- don't need to normalize! */
311 
312  l->k2 = sq(l->calp0) * g->ep2;
313  eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2);
314 
315  if (l->caps & CAP_C1) {
316  real s, c;
317  l->A1m1 = A1m1f(eps);
318  C1f(eps, l->C1a);
319  l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1);
320  s = sin(l->B11); c = cos(l->B11);
321  /* tau1 = sig1 + B11 */
322  l->stau1 = l->ssig1 * c + l->csig1 * s;
323  l->ctau1 = l->csig1 * c - l->ssig1 * s;
324  /* Not necessary because C1pa reverts C1a
325  * B11 = -SinCosSeries(TRUE, stau1, ctau1, C1pa, nC1p); */
326  }
327 
328  if (l->caps & CAP_C1p)
329  C1pf(eps, l->C1pa);
330 
331  if (l->caps & CAP_C2) {
332  l->A2m1 = A2m1f(eps);
333  C2f(eps, l->C2a);
334  l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2);
335  }
336 
337  if (l->caps & CAP_C3) {
338  C3f(g, eps, l->C3a);
339  l->A3c = -l->f * l->salp0 * A3f(g, eps);
340  l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1);
341  }
342 
343  if (l->caps & CAP_C4) {
344  C4f(g, eps, l->C4a);
345  /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) */
346  l->A4 = sq(l->a) * l->calp0 * l->salp0 * g->e2;
347  l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4);
348  }
349 }
350 
351 real geod_genposition(const struct geod_geodesicline* l,
352  boolx arcmode, real s12_a12,
353  real* plat2, real* plon2, real* pazi2,
354  real* ps12, real* pm12,
355  real* pM12, real* pM21,
356  real* pS12) {
357  real lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0,
358  m12 = 0, M12 = 0, M21 = 0, S12 = 0;
359  /* Avoid warning about uninitialized B12. */
360  real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
361  real omg12, lam12, lon12;
362  real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2;
363  unsigned outmask =
364  (plat2 ? GEOD_LATITUDE : 0U) |
365  (plon2 ? GEOD_LONGITUDE : 0U) |
366  (pazi2 ? GEOD_AZIMUTH : 0U) |
367  (ps12 ? GEOD_DISTANCE : 0U) |
368  (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
369  (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
370  (pS12 ? GEOD_AREA : 0U);
371 
372  outmask &= l->caps & OUT_ALL;
373  if (!( TRUE /*Init()*/ &&
374  (arcmode || (l->caps & GEOD_DISTANCE_IN & OUT_ALL)) ))
375  /* Uninitialized or impossible distance calculation requested */
376  return NaN;
377 
378  if (arcmode) {
379  real s12a;
380  /* Interpret s12_a12 as spherical arc length */
381  sig12 = s12_a12 * degree;
382  s12a = fabs(s12_a12);
383  s12a -= 180 * floor(s12a / 180);
384  ssig12 = s12a == 0 ? 0 : sin(sig12);
385  csig12 = s12a == 90 ? 0 : cos(sig12);
386  } else {
387  /* Interpret s12_a12 as distance */
388  real
389  tau12 = s12_a12 / (l->b * (1 + l->A1m1)),
390  s = sin(tau12),
391  c = cos(tau12);
392  /* tau2 = tau1 + tau12 */
393  B12 = - SinCosSeries(TRUE,
394  l->stau1 * c + l->ctau1 * s,
395  l->ctau1 * c - l->stau1 * s,
396  l->C1pa, nC1p);
397  sig12 = tau12 - (B12 - l->B11);
398  ssig12 = sin(sig12); csig12 = cos(sig12);
399  if (fabs(l->f) > 0.01) {
400  /* Reverted distance series is inaccurate for |f| > 1/100, so correct
401  * sig12 with 1 Newton iteration. The following table shows the
402  * approximate maximum error for a = WGS_a() and various f relative to
403  * GeodesicExact.
404  * erri = the error in the inverse solution (nm)
405  * errd = the error in the direct solution (series only) (nm)
406  * errda = the error in the direct solution (series + 1 Newton) (nm)
407  *
408  * f erri errd errda
409  * -1/5 12e6 1.2e9 69e6
410  * -1/10 123e3 12e6 765e3
411  * -1/20 1110 108e3 7155
412  * -1/50 18.63 200.9 27.12
413  * -1/100 18.63 23.78 23.37
414  * -1/150 18.63 21.05 20.26
415  * 1/150 22.35 24.73 25.83
416  * 1/100 22.35 25.03 25.31
417  * 1/50 29.80 231.9 30.44
418  * 1/20 5376 146e3 10e3
419  * 1/10 829e3 22e6 1.5e6
420  * 1/5 157e6 3.8e9 280e6 */
421  real
422  ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12,
423  csig2 = l->csig1 * csig12 - l->ssig1 * ssig12,
424  serr;
425  B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
426  serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b;
427  sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2));
428  ssig12 = sin(sig12); csig12 = cos(sig12);
429  /* Update B12 below */
430  }
431  }
432 
433  /* sig2 = sig1 + sig12 */
434  ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
435  csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
436  dn2 = sqrt(1 + l->k2 * sq(ssig2));
438  if (arcmode || fabs(l->f) > 0.01)
439  B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
440  AB1 = (1 + l->A1m1) * (B12 - l->B11);
441  }
442  /* sin(bet2) = cos(alp0) * sin(sig2) */
443  sbet2 = l->calp0 * ssig2;
444  /* Alt: cbet2 = hypot(csig2, salp0 * ssig2); */
445  cbet2 = hypotx(l->salp0, l->calp0 * csig2);
446  if (cbet2 == 0)
447  /* I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case */
448  cbet2 = csig2 = tiny;
449  /* tan(omg2) = sin(alp0) * tan(sig2) */
450  somg2 = l->salp0 * ssig2; comg2 = csig2; /* No need to normalize */
451  /* tan(alp0) = cos(sig2)*tan(alp2) */
452  salp2 = l->salp0; calp2 = l->calp0 * csig2; /* No need to normalize */
453  /* omg12 = omg2 - omg1 */
454  omg12 = atan2(somg2 * l->comg1 - comg2 * l->somg1,
455  comg2 * l->comg1 + somg2 * l->somg1);
456 
457  if (outmask & GEOD_DISTANCE)
458  s12 = arcmode ? l->b * ((1 + l->A1m1) * sig12 + AB1) : s12_a12;
459 
460  if (outmask & GEOD_LONGITUDE) {
461  lam12 = omg12 + l->A3c *
462  ( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1)
463  - l->B31));
464  lon12 = lam12 / degree;
465  /* Use AngNormalize2 because longitude might have wrapped multiple
466  * times. */
467  lon12 = AngNormalize2(lon12);
468  lon2 = AngNormalize(l->lon1 + lon12);
469  }
470 
471  if (outmask & GEOD_LATITUDE)
472  lat2 = atan2(sbet2, l->f1 * cbet2) / degree;
473 
474  if (outmask & GEOD_AZIMUTH)
475  /* minus signs give range [-180, 180). 0- converts -0 to +0. */
476  azi2 = 0 - atan2(-salp2, calp2) / degree;
477 
478  if (outmask & (GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) {
479  real
480  B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2),
481  AB2 = (1 + l->A2m1) * (B22 - l->B21),
482  J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2);
483  if (outmask & GEOD_REDUCEDLENGTH)
484  /* Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
485  * accurate cancellation in the case of coincident points. */
486  m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2))
487  - l->csig1 * csig2 * J12);
488  if (outmask & GEOD_GEODESICSCALE) {
489  real t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) / (l->dn1 + dn2);
490  M12 = csig12 + (t * ssig2 - csig2 * J12) * l->ssig1 / l->dn1;
491  M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) * ssig2 / dn2;
492  }
493  }
494 
495  if (outmask & GEOD_AREA) {
496  real
497  B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4);
498  real salp12, calp12;
499  if (l->calp0 == 0 || l->salp0 == 0) {
500  /* alp12 = alp2 - alp1, used in atan2 so no need to normalized */
501  salp12 = salp2 * l->calp1 - calp2 * l->salp1;
502  calp12 = calp2 * l->calp1 + salp2 * l->salp1;
503  /* The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
504  * salp12 = -0 and alp12 = -180. However this depends on the sign being
505  * attached to 0 correctly. The following ensures the correct
506  * behavior. */
507  if (salp12 == 0 && calp12 < 0) {
508  salp12 = tiny * l->calp1;
509  calp12 = -1;
510  }
511  } else {
512  /* tan(alp) = tan(alp0) * sec(sig)
513  * tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
514  * = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
515  * If csig12 > 0, write
516  * csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
517  * else
518  * csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
519  * No need to normalize */
520  salp12 = l->calp0 * l->salp0 *
521  (csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 :
522  ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1));
523  calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2;
524  }
525  S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41);
526  }
527 
528  if (outmask & GEOD_LATITUDE)
529  *plat2 = lat2;
530  if (outmask & GEOD_LONGITUDE)
531  *plon2 = lon2;
532  if (outmask & GEOD_AZIMUTH)
533  *pazi2 = azi2;
534  if (outmask & GEOD_DISTANCE)
535  *ps12 = s12;
536  if (outmask & GEOD_REDUCEDLENGTH)
537  *pm12 = m12;
538  if (outmask & GEOD_GEODESICSCALE) {
539  if (pM12) *pM12 = M12;
540  if (pM21) *pM21 = M21;
541  }
542  if (outmask & GEOD_AREA)
543  *pS12 = S12;
544 
545  return arcmode ? s12_a12 : sig12 / degree;
546 }
547 
548 void geod_position(const struct geod_geodesicline* l, real s12,
549  real* plat2, real* plon2, real* pazi2) {
550  geod_genposition(l, FALSE, s12, plat2, plon2, pazi2, 0, 0, 0, 0, 0);
551 }
552 
553 real geod_gendirect(const struct geod_geodesic* g,
554  real lat1, real lon1, real azi1,
555  boolx arcmode, real s12_a12,
556  real* plat2, real* plon2, real* pazi2,
557  real* ps12, real* pm12, real* pM12, real* pM21,
558  real* pS12) {
559  struct geod_geodesicline l;
560  unsigned outmask =
561  (plat2 ? GEOD_LATITUDE : 0U) |
562  (plon2 ? GEOD_LONGITUDE : 0U) |
563  (pazi2 ? GEOD_AZIMUTH : 0U) |
564  (ps12 ? GEOD_DISTANCE : 0U) |
565  (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
566  (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
567  (pS12 ? GEOD_AREA : 0U);
568 
569  geod_lineinit(&l, g, lat1, lon1, azi1,
570  /* Automatically supply GEOD_DISTANCE_IN if necessary */
571  outmask | (arcmode ? GEOD_NONE : GEOD_DISTANCE_IN));
572  return geod_genposition(&l, arcmode, s12_a12,
573  plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12);
574 }
575 
576 void geod_direct(const struct geod_geodesic* g,
577  real lat1, real lon1, real azi1,
578  real s12,
579  real* plat2, real* plon2, real* pazi2) {
580  geod_gendirect(g, lat1, lon1, azi1, FALSE, s12, plat2, plon2, pazi2,
581  0, 0, 0, 0, 0);
582 }
583 
584 real geod_geninverse(const struct geod_geodesic* g,
585  real lat1, real lon1, real lat2, real lon2,
586  real* ps12, real* pazi1, real* pazi2,
587  real* pm12, real* pM12, real* pM21, real* pS12) {
588  real s12 = 0, azi1 = 0, azi2 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0;
589  real lon12;
590  int latsign, lonsign, swapp;
591  real phi, sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0;
592  real dn1, dn2, lam12, slam12, clam12;
593  real a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0;
594  /* index zero elements of these arrays are unused */
595  real C1a[nC1 + 1], C2a[nC2 + 1], C3a[nC3];
596  boolx meridian;
597  real omg12 = 0;
598 
599  unsigned outmask =
600  (ps12 ? GEOD_DISTANCE : 0U) |
601  (pazi1 || pazi2 ? GEOD_AZIMUTH : 0U) |
602  (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
603  (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
604  (pS12 ? GEOD_AREA : 0U);
605 
606  outmask &= OUT_ALL;
607  /* Compute longitude difference (AngDiff does this carefully). Result is
608  * in [-180, 180] but -180 is only for west-going geodesics. 180 is for
609  * east-going and meridional geodesics. */
610  lon12 = AngDiff(AngNormalize(lon1), AngNormalize(lon2));
611  /* If very close to being on the same half-meridian, then make it so. */
612  lon12 = AngRound(lon12);
613  /* Make longitude difference positive. */
614  lonsign = lon12 >= 0 ? 1 : -1;
615  lon12 *= lonsign;
616  /* If really close to the equator, treat as on equator. */
617  lat1 = AngRound(lat1);
618  lat2 = AngRound(lat2);
619  /* Swap points so that point with higher (abs) latitude is point 1 */
620  swapp = fabs(lat1) >= fabs(lat2) ? 1 : -1;
621  if (swapp < 0) {
622  lonsign *= -1;
623  swapx(&lat1, &lat2);
624  }
625  /* Make lat1 <= 0 */
626  latsign = lat1 < 0 ? 1 : -1;
627  lat1 *= latsign;
628  lat2 *= latsign;
629  /* Now we have
630  *
631  * 0 <= lon12 <= 180
632  * -90 <= lat1 <= 0
633  * lat1 <= lat2 <= -lat1
634  *
635  * longsign, swapp, latsign register the transformation to bring the
636  * coordinates to this canonical form. In all cases, 1 means no change was
637  * made. We make these transformations so that there are few cases to
638  * check, e.g., on verifying quadrants in atan2. In addition, this
639  * enforces some symmetries in the results returned. */
640 
641  phi = lat1 * degree;
642  /* Ensure cbet1 = +epsilon at poles */
643  sbet1 = g->f1 * sin(phi);
644  cbet1 = lat1 == -90 ? tiny : cos(phi);
645  SinCosNorm(&sbet1, &cbet1);
646 
647  phi = lat2 * degree;
648  /* Ensure cbet2 = +epsilon at poles */
649  sbet2 = g->f1 * sin(phi);
650  cbet2 = fabs(lat2) == 90 ? tiny : cos(phi);
651  SinCosNorm(&sbet2, &cbet2);
652 
653  /* If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
654  * |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
655  * a better measure. This logic is used in assigning calp2 in Lambda12.
656  * Sometimes these quantities vanish and in that case we force bet2 = +/-
657  * bet1 exactly. An example where is is necessary is the inverse problem
658  * 48.522876735459 0 -48.52287673545898293 179.599720456223079643
659  * which failed with Visual Studio 10 (Release and Debug) */
660 
661  if (cbet1 < -sbet1) {
662  if (cbet2 == cbet1)
663  sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
664  } else {
665  if (fabs(sbet2) == -sbet1)
666  cbet2 = cbet1;
667  }
668 
669  dn1 = sqrt(1 + g->ep2 * sq(sbet1));
670  dn2 = sqrt(1 + g->ep2 * sq(sbet2));
671 
672  lam12 = lon12 * degree;
673  slam12 = lon12 == 180 ? 0 : sin(lam12);
674  clam12 = cos(lam12); /* lon12 == 90 isn't interesting */
675 
676  meridian = lat1 == -90 || slam12 == 0;
677 
678  if (meridian) {
679 
680  /* Endpoints are on a single full meridian, so the geodesic might lie on
681  * a meridian. */
682 
683  real ssig1, csig1, ssig2, csig2;
684  calp1 = clam12; salp1 = slam12; /* Head to the target longitude */
685  calp2 = 1; salp2 = 0; /* At the target we're heading north */
686 
687  /* tan(bet) = tan(sig) * cos(alp) */
688  ssig1 = sbet1; csig1 = calp1 * cbet1;
689  ssig2 = sbet2; csig2 = calp2 * cbet2;
690 
691  /* sig12 = sig2 - sig1 */
692  sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (real)(0)),
693  csig1 * csig2 + ssig1 * ssig2);
694  {
695  real dummy;
696  Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
697  cbet1, cbet2, &s12x, &m12x, &dummy,
698  (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
699  }
700  /* Add the check for sig12 since zero length geodesics might yield m12 <
701  * 0. Test case was
702  *
703  * echo 20.001 0 20.001 0 | GeodSolve -i
704  *
705  * In fact, we will have sig12 > pi/2 for meridional geodesic which is
706  * not a shortest path. */
707  if (sig12 < 1 || m12x >= 0) {
708  m12x *= g->b;
709  s12x *= g->b;
710  a12 = sig12 / degree;
711  } else
712  /* m12 < 0, i.e., prolate and too close to anti-podal */
713  meridian = FALSE;
714  }
715 
716  if (!meridian &&
717  sbet1 == 0 && /* and sbet2 == 0 */
718  /* Mimic the way Lambda12 works with calp1 = 0 */
719  (g->f <= 0 || lam12 <= pi - g->f * pi)) {
720 
721  /* Geodesic runs along equator */
722  calp1 = calp2 = 0; salp1 = salp2 = 1;
723  s12x = g->a * lam12;
724  sig12 = omg12 = lam12 / g->f1;
725  m12x = g->b * sin(sig12);
726  if (outmask & GEOD_GEODESICSCALE)
727  M12 = M21 = cos(sig12);
728  a12 = lon12 / g->f1;
729 
730  } else if (!meridian) {
731 
732  /* Now point1 and point2 belong within a hemisphere bounded by a
733  * meridian and geodesic is neither meridional or equatorial. */
734 
735  /* Figure a starting point for Newton's method */
736  real dnm = 0;
737  sig12 = InverseStart(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
738  lam12,
739  &salp1, &calp1, &salp2, &calp2, &dnm,
740  C1a, C2a);
741 
742  if (sig12 >= 0) {
743  /* Short lines (InverseStart sets salp2, calp2, dnm) */
744  s12x = sig12 * g->b * dnm;
745  m12x = sq(dnm) * g->b * sin(sig12 / dnm);
746  if (outmask & GEOD_GEODESICSCALE)
747  M12 = M21 = cos(sig12 / dnm);
748  a12 = sig12 / degree;
749  omg12 = lam12 / (g->f1 * dnm);
750  } else {
751 
752  /* Newton's method. This is a straightforward solution of f(alp1) =
753  * lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
754  * root in the interval (0, pi) and its derivative is positive at the
755  * root. Thus f(alp) is positive for alp > alp1 and negative for alp <
756  * alp1. During the course of the iteration, a range (alp1a, alp1b) is
757  * maintained which brackets the root and with each evaluation of
758  * f(alp) the range is shrunk, if possible. Newton's method is
759  * restarted whenever the derivative of f is negative (because the new
760  * value of alp1 is then further from the solution) or if the new
761  * estimate of alp1 lies outside (0,pi); in this case, the new starting
762  * guess is taken to be (alp1a + alp1b) / 2. */
763  real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0;
764  unsigned numit = 0;
765  /* Bracketing range */
766  real salp1a = tiny, calp1a = 1, salp1b = tiny, calp1b = -1;
767  boolx tripn, tripb;
768  for (tripn = FALSE, tripb = FALSE; numit < maxit2; ++numit) {
769  /* the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
770  * WGS84 and random input: mean = 2.85, sd = 0.60 */
771  real dv,
772  v = (Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
773  &salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2,
774  &eps, &omg12, numit < maxit1, &dv, C1a, C2a, C3a)
775  - lam12);
776  /* 2 * tol0 is approximately 1 ulp for a number in [0, pi]. */
777  /* Reversed test to allow escape with NaNs */
778  if (tripb || !(fabs(v) >= (tripn ? 8 : 2) * tol0)) break;
779  /* Update bracketing values */
780  if (v > 0 && (numit > maxit1 || calp1/salp1 > calp1b/salp1b))
781  { salp1b = salp1; calp1b = calp1; }
782  else if (v < 0 && (numit > maxit1 || calp1/salp1 < calp1a/salp1a))
783  { salp1a = salp1; calp1a = calp1; }
784  if (numit < maxit1 && dv > 0) {
785  real
786  dalp1 = -v/dv;
787  real
788  sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
789  nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
790  if (nsalp1 > 0 && fabs(dalp1) < pi) {
791  calp1 = calp1 * cdalp1 - salp1 * sdalp1;
792  salp1 = nsalp1;
793  SinCosNorm(&salp1, &calp1);
794  /* In some regimes we don't get quadratic convergence because
795  * slope -> 0. So use convergence conditions based on epsilon
796  * instead of sqrt(epsilon). */
797  tripn = fabs(v) <= 16 * tol0;
798  continue;
799  }
800  }
801  /* Either dv was not postive or updated value was outside legal
802  * range. Use the midpoint of the bracket as the next estimate.
803  * This mechanism is not needed for the WGS84 ellipsoid, but it does
804  * catch problems with more eccentric ellipsoids. Its efficacy is
805  * such for the WGS84 test set with the starting guess set to alp1 =
806  * 90deg:
807  * the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
808  * WGS84 and random input: mean = 4.74, sd = 0.99 */
809  salp1 = (salp1a + salp1b)/2;
810  calp1 = (calp1a + calp1b)/2;
811  SinCosNorm(&salp1, &calp1);
812  tripn = FALSE;
813  tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb ||
814  fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb);
815  }
816  {
817  real dummy;
818  Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
819  cbet1, cbet2, &s12x, &m12x, &dummy,
820  (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
821  }
822  m12x *= g->b;
823  s12x *= g->b;
824  a12 = sig12 / degree;
825  omg12 = lam12 - omg12;
826  }
827  }
828 
829  if (outmask & GEOD_DISTANCE)
830  s12 = 0 + s12x; /* Convert -0 to 0 */
831 
832  if (outmask & GEOD_REDUCEDLENGTH)
833  m12 = 0 + m12x; /* Convert -0 to 0 */
834 
835  if (outmask & GEOD_AREA) {
836  real
837  /* From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) */
838  salp0 = salp1 * cbet1,
839  calp0 = hypotx(calp1, salp1 * sbet1); /* calp0 > 0 */
840  real alp12;
841  if (calp0 != 0 && salp0 != 0) {
842  real
843  /* From Lambda12: tan(bet) = tan(sig) * cos(alp) */
844  ssig1 = sbet1, csig1 = calp1 * cbet1,
845  ssig2 = sbet2, csig2 = calp2 * cbet2,
846  k2 = sq(calp0) * g->ep2,
847  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
848  /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). */
849  A4 = sq(g->a) * calp0 * salp0 * g->e2;
850  real C4a[nC4];
851  real B41, B42;
852  SinCosNorm(&ssig1, &csig1);
853  SinCosNorm(&ssig2, &csig2);
854  C4f(g, eps, C4a);
855  B41 = SinCosSeries(FALSE, ssig1, csig1, C4a, nC4);
856  B42 = SinCosSeries(FALSE, ssig2, csig2, C4a, nC4);
857  S12 = A4 * (B42 - B41);
858  } else
859  /* Avoid problems with indeterminate sig1, sig2 on equator */
860  S12 = 0;
861 
862  if (!meridian &&
863  omg12 < (real)(0.75) * pi && /* Long difference too big */
864  sbet2 - sbet1 < (real)(1.75)) { /* Lat difference too big */
865  /* Use tan(Gamma/2) = tan(omg12/2)
866  * * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
867  * with tan(x/2) = sin(x)/(1+cos(x)) */
868  real
869  somg12 = sin(omg12), domg12 = 1 + cos(omg12),
870  dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
871  alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
872  domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
873  } else {
874  /* alp12 = alp2 - alp1, used in atan2 so no need to normalize */
875  real
876  salp12 = salp2 * calp1 - calp2 * salp1,
877  calp12 = calp2 * calp1 + salp2 * salp1;
878  /* The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
879  * salp12 = -0 and alp12 = -180. However this depends on the sign
880  * being attached to 0 correctly. The following ensures the correct
881  * behavior. */
882  if (salp12 == 0 && calp12 < 0) {
883  salp12 = tiny * calp1;
884  calp12 = -1;
885  }
886  alp12 = atan2(salp12, calp12);
887  }
888  S12 += g->c2 * alp12;
889  S12 *= swapp * lonsign * latsign;
890  /* Convert -0 to 0 */
891  S12 += 0;
892  }
893 
894  /* Convert calp, salp to azimuth accounting for lonsign, swapp, latsign. */
895  if (swapp < 0) {
896  swapx(&salp1, &salp2);
897  swapx(&calp1, &calp2);
898  if (outmask & GEOD_GEODESICSCALE)
899  swapx(&M12, &M21);
900  }
901 
902  salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
903  salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
904 
905  if (outmask & GEOD_AZIMUTH) {
906  /* minus signs give range [-180, 180). 0- converts -0 to +0. */
907  azi1 = 0 - atan2(-salp1, calp1) / degree;
908  azi2 = 0 - atan2(-salp2, calp2) / degree;
909  }
910 
911  if (outmask & GEOD_DISTANCE)
912  *ps12 = s12;
913  if (outmask & GEOD_AZIMUTH) {
914  if (pazi1) *pazi1 = azi1;
915  if (pazi2) *pazi2 = azi2;
916  }
917  if (outmask & GEOD_REDUCEDLENGTH)
918  *pm12 = m12;
919  if (outmask & GEOD_GEODESICSCALE) {
920  if (pM12) *pM12 = M12;
921  if (pM21) *pM21 = M21;
922  }
923  if (outmask & GEOD_AREA)
924  *pS12 = S12;
925 
926  /* Returned value in [0, 180] */
927  return a12;
928 }
929 
930 void geod_inverse(const struct geod_geodesic* g,
931  real lat1, real lon1, real lat2, real lon2,
932  real* ps12, real* pazi1, real* pazi2) {
933  geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, 0, 0, 0, 0);
934 }
935 
936 real SinCosSeries(boolx sinp, real sinx, real cosx, const real c[], int n) {
937  /* Evaluate
938  * y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
939  * sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
940  * using Clenshaw summation. N.B. c[0] is unused for sin series
941  * Approx operation count = (n + 5) mult and (2 * n + 2) add */
942  real ar, y0, y1;
943  c += (n + sinp); /* Point to one beyond last element */
944  ar = 2 * (cosx - sinx) * (cosx + sinx); /* 2 * cos(2 * x) */
945  y0 = n & 1 ? *--c : 0; y1 = 0; /* accumulators for sum */
946  /* Now n is even */
947  n /= 2;
948  while (n--) {
949  /* Unroll loop x 2, so accumulators return to their original role */
950  y1 = ar * y0 - y1 + *--c;
951  y0 = ar * y1 - y0 + *--c;
952  }
953  return sinp
954  ? 2 * sinx * cosx * y0 /* sin(2 * x) * y0 */
955  : cosx * (y0 - y1); /* cos(x) * (y0 - y1) */
956 }
957 
958 void Lengths(const struct geod_geodesic* g,
959  real eps, real sig12,
960  real ssig1, real csig1, real dn1,
961  real ssig2, real csig2, real dn2,
962  real cbet1, real cbet2,
963  real* ps12b, real* pm12b, real* pm0,
964  boolx scalep, real* pM12, real* pM21,
965  /* Scratch areas of the right size */
966  real C1a[], real C2a[]) {
967  real s12b = 0, m12b = 0, m0 = 0, M12 = 0, M21 = 0;
968  real A1m1, AB1, A2m1, AB2, J12;
969 
970  /* Return m12b = (reduced length)/b; also calculate s12b = distance/b,
971  * and m0 = coefficient of secular term in expression for reduced length. */
972  C1f(eps, C1a);
973  C2f(eps, C2a);
974  A1m1 = A1m1f(eps);
975  AB1 = (1 + A1m1) * (SinCosSeries(TRUE, ssig2, csig2, C1a, nC1) -
976  SinCosSeries(TRUE, ssig1, csig1, C1a, nC1));
977  A2m1 = A2m1f(eps);
978  AB2 = (1 + A2m1) * (SinCosSeries(TRUE, ssig2, csig2, C2a, nC2) -
979  SinCosSeries(TRUE, ssig1, csig1, C2a, nC2));
980  m0 = A1m1 - A2m1;
981  J12 = m0 * sig12 + (AB1 - AB2);
982  /* Missing a factor of b.
983  * Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure accurate
984  * cancellation in the case of coincident points. */
985  m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
986  /* Missing a factor of b */
987  s12b = (1 + A1m1) * sig12 + AB1;
988  if (scalep) {
989  real csig12 = csig1 * csig2 + ssig1 * ssig2;
990  real t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
991  M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
992  M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
993  }
994  *ps12b = s12b;
995  *pm12b = m12b;
996  *pm0 = m0;
997  if (scalep) {
998  *pM12 = M12;
999  *pM21 = M21;
1000  }
1001 }
1002 
1003 real Astroid(real x, real y) {
1004  /* Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
1005  * This solution is adapted from Geocentric::Reverse. */
1006  real k;
1007  real
1008  p = sq(x),
1009  q = sq(y),
1010  r = (p + q - 1) / 6;
1011  if ( !(q == 0 && r <= 0) ) {
1012  real
1013  /* Avoid possible division by zero when r = 0 by multiplying equations
1014  * for s and t by r^3 and r, resp. */
1015  S = p * q / 4, /* S = r^3 * s */
1016  r2 = sq(r),
1017  r3 = r * r2,
1018  /* The discrimant of the quadratic equation for T3. This is zero on
1019  * the evolute curve p^(1/3)+q^(1/3) = 1 */
1020  disc = S * (S + 2 * r3);
1021  real u = r;
1022  real v, uv, w;
1023  if (disc >= 0) {
1024  real T3 = S + r3, T;
1025  /* Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
1026  * of precision due to cancellation. The result is unchanged because
1027  * of the way the T is used in definition of u. */
1028  T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); /* T3 = (r * t)^3 */
1029  /* N.B. cbrtx always returns the real root. cbrtx(-8) = -2. */
1030  T = cbrtx(T3); /* T = r * t */
1031  /* T can be zero; but then r2 / T -> 0. */
1032  u += T + (T != 0 ? r2 / T : 0);
1033  } else {
1034  /* T is complex, but the way u is defined the result is real. */
1035  real ang = atan2(sqrt(-disc), -(S + r3));
1036  /* There are three possible cube roots. We choose the root which
1037  * avoids cancellation. Note that disc < 0 implies that r < 0. */
1038  u += 2 * r * cos(ang / 3);
1039  }
1040  v = sqrt(sq(u) + q); /* guaranteed positive */
1041  /* Avoid loss of accuracy when u < 0. */
1042  uv = u < 0 ? q / (v - u) : u + v; /* u+v, guaranteed positive */
1043  w = (uv - q) / (2 * v); /* positive? */
1044  /* Rearrange expression for k to avoid loss of accuracy due to
1045  * subtraction. Division by 0 not possible because uv > 0, w >= 0. */
1046  k = uv / (sqrt(uv + sq(w)) + w); /* guaranteed positive */
1047  } else { /* q == 0 && r <= 0 */
1048  /* y = 0 with |x| <= 1. Handle this case directly.
1049  * for y small, positive root is k = abs(y)/sqrt(1-x^2) */
1050  k = 0;
1051  }
1052  return k;
1053 }
1054 
1055 real InverseStart(const struct geod_geodesic* g,
1056  real sbet1, real cbet1, real dn1,
1057  real sbet2, real cbet2, real dn2,
1058  real lam12,
1059  real* psalp1, real* pcalp1,
1060  /* Only updated if return val >= 0 */
1061  real* psalp2, real* pcalp2,
1062  /* Only updated for short lines */
1063  real* pdnm,
1064  /* Scratch areas of the right size */
1065  real C1a[], real C2a[]) {
1066  real salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0;
1067 
1068  /* Return a starting point for Newton's method in salp1 and calp1 (function
1069  * value is -1). If Newton's method doesn't need to be used, return also
1070  * salp2 and calp2 and function value is sig12. */
1071  real
1072  sig12 = -1, /* Return value */
1073  /* bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0] */
1074  sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
1075  cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
1076 #if defined(__GNUC__) && __GNUC__ == 4 && \
1077  (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
1078  /* Volatile declaration needed to fix inverse cases
1079  * 88.202499451857 0 -88.202499451857 179.981022032992859592
1080  * 89.262080389218 0 -89.262080389218 179.992207982775375662
1081  * 89.333123580033 0 -89.333123580032997687 179.99295812360148422
1082  * which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
1083  * and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw). */
1084  real sbet12a;
1085  {
1086  volatile real xx1 = sbet2 * cbet1;
1087  volatile real xx2 = cbet2 * sbet1;
1088  sbet12a = xx1 + xx2;
1089  }
1090 #else
1091  real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
1092 #endif
1093  boolx shortline = cbet12 >= 0 && sbet12 < (real)(0.5) &&
1094  cbet2 * lam12 < (real)(0.5);
1095  real omg12 = lam12, somg12, comg12, ssig12, csig12;
1096  if (shortline) {
1097  real sbetm2 = sq(sbet1 + sbet2);
1098  /* sin((bet1+bet2)/2)^2
1099  * = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) */
1100  sbetm2 /= sbetm2 + sq(cbet1 + cbet2);
1101  dnm = sqrt(1 + g->ep2 * sbetm2);
1102  omg12 /= g->f1 * dnm;
1103  }
1104  somg12 = sin(omg12); comg12 = cos(omg12);
1105 
1106  salp1 = cbet2 * somg12;
1107  calp1 = comg12 >= 0 ?
1108  sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) :
1109  sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1110 
1111  ssig12 = hypotx(salp1, calp1);
1112  csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
1113 
1114  if (shortline && ssig12 < g->etol2) {
1115  /* really short lines */
1116  salp2 = cbet1 * somg12;
1117  calp2 = sbet12 - cbet1 * sbet2 *
1118  (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12);
1119  SinCosNorm(&salp2, &calp2);
1120  /* Set return value */
1121  sig12 = atan2(ssig12, csig12);
1122  } else if (fabs(g->n) > (real)(0.1) || /* No astroid calc if too eccentric */
1123  csig12 >= 0 ||
1124  ssig12 >= 6 * fabs(g->n) * pi * sq(cbet1)) {
1125  /* Nothing to do, zeroth order spherical approximation is OK */
1126  } else {
1127  /* Scale lam12 and bet2 to x, y coordinate system where antipodal point
1128  * is at origin and singular point is at y = 0, x = -1. */
1129  real y, lamscale, betscale;
1130  /* Volatile declaration needed to fix inverse case
1131  * 56.320923501171 0 -56.320923501171 179.664747671772880215
1132  * which otherwise fails with g++ 4.4.4 x86 -O3 */
1133  volatile real x;
1134  if (g->f >= 0) { /* In fact f == 0 does not get here */
1135  /* x = dlong, y = dlat */
1136  {
1137  real
1138  k2 = sq(sbet1) * g->ep2,
1139  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1140  lamscale = g->f * cbet1 * A3f(g, eps) * pi;
1141  }
1142  betscale = lamscale * cbet1;
1143 
1144  x = (lam12 - pi) / lamscale;
1145  y = sbet12a / betscale;
1146  } else { /* f < 0 */
1147  /* x = dlat, y = dlong */
1148  real
1149  cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
1150  bet12a = atan2(sbet12a, cbet12a);
1151  real m12b, m0, dummy;
1152  /* In the case of lon12 = 180, this repeats a calculation made in
1153  * Inverse. */
1154  Lengths(g, g->n, pi + bet12a,
1155  sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
1156  cbet1, cbet2, &dummy, &m12b, &m0, FALSE,
1157  &dummy, &dummy, C1a, C2a);
1158  x = -1 + m12b / (cbet1 * cbet2 * m0 * pi);
1159  betscale = x < -(real)(0.01) ? sbet12a / x :
1160  -g->f * sq(cbet1) * pi;
1161  lamscale = betscale / cbet1;
1162  y = (lam12 - pi) / lamscale;
1163  }
1164 
1165  if (y > -tol1 && x > -1 - xthresh) {
1166  /* strip near cut */
1167  if (g->f >= 0) {
1168  salp1 = minx((real)(1), -(real)(x)); calp1 = - sqrt(1 - sq(salp1));
1169  } else {
1170  calp1 = maxx((real)(x > -tol1 ? 0 : -1), (real)(x));
1171  salp1 = sqrt(1 - sq(calp1));
1172  }
1173  } else {
1174  /* Estimate alp1, by solving the astroid problem.
1175  *
1176  * Could estimate alpha1 = theta + pi/2, directly, i.e.,
1177  * calp1 = y/k; salp1 = -x/(1+k); for f >= 0
1178  * calp1 = x/(1+k); salp1 = -y/k; for f < 0 (need to check)
1179  *
1180  * However, it's better to estimate omg12 from astroid and use
1181  * spherical formula to compute alp1. This reduces the mean number of
1182  * Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
1183  * (min 0 max 5). The changes in the number of iterations are as
1184  * follows:
1185  *
1186  * change percent
1187  * 1 5
1188  * 0 78
1189  * -1 16
1190  * -2 0.6
1191  * -3 0.04
1192  * -4 0.002
1193  *
1194  * The histogram of iterations is (m = number of iterations estimating
1195  * alp1 directly, n = number of iterations estimating via omg12, total
1196  * number of trials = 148605):
1197  *
1198  * iter m n
1199  * 0 148 186
1200  * 1 13046 13845
1201  * 2 93315 102225
1202  * 3 36189 32341
1203  * 4 5396 7
1204  * 5 455 1
1205  * 6 56 0
1206  *
1207  * Because omg12 is near pi, estimate work with omg12a = pi - omg12 */
1208  real k = Astroid(x, y);
1209  real
1210  omg12a = lamscale * ( g->f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
1211  somg12 = sin(omg12a); comg12 = -cos(omg12a);
1212  /* Update spherical estimate of alp1 using omg12 instead of lam12 */
1213  salp1 = cbet2 * somg12;
1214  calp1 = sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1215  }
1216  }
1217  if (salp1 > 0) /* Sanity check on starting guess */
1218  SinCosNorm(&salp1, &calp1);
1219  else {
1220  salp1 = 1; calp1 = 0;
1221  }
1222 
1223  *psalp1 = salp1;
1224  *pcalp1 = calp1;
1225  if (shortline)
1226  *pdnm = dnm;
1227  if (sig12 >= 0) {
1228  *psalp2 = salp2;
1229  *pcalp2 = calp2;
1230  }
1231  return sig12;
1232 }
1233 
1234 real Lambda12(const struct geod_geodesic* g,
1235  real sbet1, real cbet1, real dn1,
1236  real sbet2, real cbet2, real dn2,
1237  real salp1, real calp1,
1238  real* psalp2, real* pcalp2,
1239  real* psig12,
1240  real* pssig1, real* pcsig1,
1241  real* pssig2, real* pcsig2,
1242  real* peps, real* pdomg12,
1243  boolx diffp, real* pdlam12,
1244  /* Scratch areas of the right size */
1245  real C1a[], real C2a[], real C3a[]) {
1246  real salp2 = 0, calp2 = 0, sig12 = 0,
1247  ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0, dlam12 = 0;
1248  real salp0, calp0;
1249  real somg1, comg1, somg2, comg2, omg12, lam12;
1250  real B312, h0, k2;
1251 
1252  if (sbet1 == 0 && calp1 == 0)
1253  /* Break degeneracy of equatorial line. This case has already been
1254  * handled. */
1255  calp1 = -tiny;
1256 
1257  /* sin(alp1) * cos(bet1) = sin(alp0) */
1258  salp0 = salp1 * cbet1;
1259  calp0 = hypotx(calp1, salp1 * sbet1); /* calp0 > 0 */
1260 
1261  /* tan(bet1) = tan(sig1) * cos(alp1)
1262  * tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1) */
1263  ssig1 = sbet1; somg1 = salp0 * sbet1;
1264  csig1 = comg1 = calp1 * cbet1;
1265  SinCosNorm(&ssig1, &csig1);
1266  /* SinCosNorm(&somg1, &comg1); -- don't need to normalize! */
1267 
1268  /* Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
1269  * about this case, since this can yield singularities in the Newton
1270  * iteration.
1271  * sin(alp2) * cos(bet2) = sin(alp0) */
1272  salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
1273  /* calp2 = sqrt(1 - sq(salp2))
1274  * = sqrt(sq(calp0) - sq(sbet2)) / cbet2
1275  * and subst for calp0 and rearrange to give (choose positive sqrt
1276  * to give alp2 in [0, pi/2]). */
1277  calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
1278  sqrt(sq(calp1 * cbet1) +
1279  (cbet1 < -sbet1 ?
1280  (cbet2 - cbet1) * (cbet1 + cbet2) :
1281  (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
1282  fabs(calp1);
1283  /* tan(bet2) = tan(sig2) * cos(alp2)
1284  * tan(omg2) = sin(alp0) * tan(sig2). */
1285  ssig2 = sbet2; somg2 = salp0 * sbet2;
1286  csig2 = comg2 = calp2 * cbet2;
1287  SinCosNorm(&ssig2, &csig2);
1288  /* SinCosNorm(&somg2, &comg2); -- don't need to normalize! */
1289 
1290  /* sig12 = sig2 - sig1, limit to [0, pi] */
1291  sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (real)(0)),
1292  csig1 * csig2 + ssig1 * ssig2);
1293 
1294  /* omg12 = omg2 - omg1, limit to [0, pi] */
1295  omg12 = atan2(maxx(comg1 * somg2 - somg1 * comg2, (real)(0)),
1296  comg1 * comg2 + somg1 * somg2);
1297  k2 = sq(calp0) * g->ep2;
1298  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1299  C3f(g, eps, C3a);
1300  B312 = (SinCosSeries(TRUE, ssig2, csig2, C3a, nC3-1) -
1301  SinCosSeries(TRUE, ssig1, csig1, C3a, nC3-1));
1302  h0 = -g->f * A3f(g, eps);
1303  domg12 = salp0 * h0 * (sig12 + B312);
1304  lam12 = omg12 + domg12;
1305 
1306  if (diffp) {
1307  if (calp2 == 0)
1308  dlam12 = - 2 * g->f1 * dn1 / sbet1;
1309  else {
1310  real dummy;
1311  Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
1312  cbet1, cbet2, &dummy, &dlam12, &dummy,
1313  FALSE, &dummy, &dummy, C1a, C2a);
1314  dlam12 *= g->f1 / (calp2 * cbet2);
1315  }
1316  }
1317 
1318  *psalp2 = salp2;
1319  *pcalp2 = calp2;
1320  *psig12 = sig12;
1321  *pssig1 = ssig1;
1322  *pcsig1 = csig1;
1323  *pssig2 = ssig2;
1324  *pcsig2 = csig2;
1325  *peps = eps;
1326  *pdomg12 = domg12;
1327  if (diffp)
1328  *pdlam12 = dlam12;
1329 
1330  return lam12;
1331 }
1332 
1333 real A3f(const struct geod_geodesic* g, real eps) {
1334  /* Evaluate sum(A3x[k] * eps^k, k, 0, nA3x-1) by Horner's method */
1335  real v = 0;
1336  int i;
1337  for (i = nA3x; i; )
1338  v = eps * v + g->A3x[--i];
1339  return v;
1340 }
1341 
1342 void C3f(const struct geod_geodesic* g, real eps, real c[]) {
1343  /* Evaluate C3 coeffs by Horner's method
1344  * Elements c[1] thru c[nC3 - 1] are set */
1345  int i, j, k;
1346  real mult = 1;
1347  for (j = nC3x, k = nC3 - 1; k; ) {
1348  real t = 0;
1349  for (i = nC3 - k; i; --i)
1350  t = eps * t + g->C3x[--j];
1351  c[k--] = t;
1352  }
1353 
1354  for (k = 1; k < nC3; ) {
1355  mult *= eps;
1356  c[k++] *= mult;
1357  }
1358 }
1359 
1360 void C4f(const struct geod_geodesic* g, real eps, real c[]) {
1361  /* Evaluate C4 coeffs by Horner's method
1362  * Elements c[0] thru c[nC4 - 1] are set */
1363  int i, j, k;
1364  real mult = 1;
1365  for (j = nC4x, k = nC4; k; ) {
1366  real t = 0;
1367  for (i = nC4 - k + 1; i; --i)
1368  t = eps * t + g->C4x[--j];
1369  c[--k] = t;
1370  }
1371 
1372  for (k = 1; k < nC4; ) {
1373  mult *= eps;
1374  c[k++] *= mult;
1375  }
1376 }
1377 
1378 /* Generated by Maxima on 2010-09-04 10:26:17-04:00 */
1379 
1380 /* The scale factor A1-1 = mean value of (d/dsigma)I1 - 1 */
1381 real A1m1f(real eps) {
1382  real
1383  eps2 = sq(eps),
1384  t = eps2*(eps2*(eps2+4)+64)/256;
1385  return (t + eps) / (1 - eps);
1386 }
1387 
1388 /* The coefficients C1[l] in the Fourier expansion of B1 */
1389 void C1f(real eps, real c[]) {
1390  real
1391  eps2 = sq(eps),
1392  d = eps;
1393  c[1] = d*((6-eps2)*eps2-16)/32;
1394  d *= eps;
1395  c[2] = d*((64-9*eps2)*eps2-128)/2048;
1396  d *= eps;
1397  c[3] = d*(9*eps2-16)/768;
1398  d *= eps;
1399  c[4] = d*(3*eps2-5)/512;
1400  d *= eps;
1401  c[5] = -7*d/1280;
1402  d *= eps;
1403  c[6] = -7*d/2048;
1404 }
1405 
1406 /* The coefficients C1p[l] in the Fourier expansion of B1p */
1407 void C1pf(real eps, real c[]) {
1408  real
1409  eps2 = sq(eps),
1410  d = eps;
1411  c[1] = d*(eps2*(205*eps2-432)+768)/1536;
1412  d *= eps;
1413  c[2] = d*(eps2*(4005*eps2-4736)+3840)/12288;
1414  d *= eps;
1415  c[3] = d*(116-225*eps2)/384;
1416  d *= eps;
1417  c[4] = d*(2695-7173*eps2)/7680;
1418  d *= eps;
1419  c[5] = 3467*d/7680;
1420  d *= eps;
1421  c[6] = 38081*d/61440;
1422 }
1423 
1424 /* The scale factor A2-1 = mean value of (d/dsigma)I2 - 1 */
1425 real A2m1f(real eps) {
1426  real
1427  eps2 = sq(eps),
1428  t = eps2*(eps2*(25*eps2+36)+64)/256;
1429  return t * (1 - eps) - eps;
1430 }
1431 
1432 /* The coefficients C2[l] in the Fourier expansion of B2 */
1433 void C2f(real eps, real c[]) {
1434  real
1435  eps2 = sq(eps),
1436  d = eps;
1437  c[1] = d*(eps2*(eps2+2)+16)/32;
1438  d *= eps;
1439  c[2] = d*(eps2*(35*eps2+64)+384)/2048;
1440  d *= eps;
1441  c[3] = d*(15*eps2+80)/768;
1442  d *= eps;
1443  c[4] = d*(7*eps2+35)/512;
1444  d *= eps;
1445  c[5] = 63*d/1280;
1446  d *= eps;
1447  c[6] = 77*d/2048;
1448 }
1449 
1450 /* The scale factor A3 = mean value of (d/dsigma)I3 */
1451 void A3coeff(struct geod_geodesic* g) {
1452  g->A3x[0] = 1;
1453  g->A3x[1] = (g->n-1)/2;
1454  g->A3x[2] = (g->n*(3*g->n-1)-2)/8;
1455  g->A3x[3] = ((-g->n-3)*g->n-1)/16;
1456  g->A3x[4] = (-2*g->n-3)/64;
1457  g->A3x[5] = -3/(real)(128);
1458 }
1459 
1460 /* The coefficients C3[l] in the Fourier expansion of B3 */
1461 void C3coeff(struct geod_geodesic* g) {
1462  g->C3x[0] = (1-g->n)/4;
1463  g->C3x[1] = (1-g->n*g->n)/8;
1464  g->C3x[2] = ((3-g->n)*g->n+3)/64;
1465  g->C3x[3] = (2*g->n+5)/128;
1466  g->C3x[4] = 3/(real)(128);
1467  g->C3x[5] = ((g->n-3)*g->n+2)/32;
1468  g->C3x[6] = ((-3*g->n-2)*g->n+3)/64;
1469  g->C3x[7] = (g->n+3)/128;
1470  g->C3x[8] = 5/(real)(256);
1471  g->C3x[9] = (g->n*(5*g->n-9)+5)/192;
1472  g->C3x[10] = (9-10*g->n)/384;
1473  g->C3x[11] = 7/(real)(512);
1474  g->C3x[12] = (7-14*g->n)/512;
1475  g->C3x[13] = 7/(real)(512);
1476  g->C3x[14] = 21/(real)(2560);
1477 }
1478 
1479 /* Generated by Maxima on 2012-10-19 08:02:34-04:00 */
1480 
1481 /* The coefficients C4[l] in the Fourier expansion of I4 */
1482 void C4coeff(struct geod_geodesic* g) {
1483  g->C4x[0] = (g->n*(g->n*(g->n*(g->n*(100*g->n+208)+572)+3432)-12012)+30030)/
1484  45045;
1485  g->C4x[1] = (g->n*(g->n*(g->n*(64*g->n+624)-4576)+6864)-3003)/15015;
1486  g->C4x[2] = (g->n*((14144-10656*g->n)*g->n-4576)-858)/45045;
1487  g->C4x[3] = ((-224*g->n-4784)*g->n+1573)/45045;
1488  g->C4x[4] = (1088*g->n+156)/45045;
1489  g->C4x[5] = 97/(real)(15015);
1490  g->C4x[6] = (g->n*(g->n*((-64*g->n-624)*g->n+4576)-6864)+3003)/135135;
1491  g->C4x[7] = (g->n*(g->n*(5952*g->n-11648)+9152)-2574)/135135;
1492  g->C4x[8] = (g->n*(5792*g->n+1040)-1287)/135135;
1493  g->C4x[9] = (468-2944*g->n)/135135;
1494  g->C4x[10] = 1/(real)(9009);
1495  g->C4x[11] = (g->n*((4160-1440*g->n)*g->n-4576)+1716)/225225;
1496  g->C4x[12] = ((4992-8448*g->n)*g->n-1144)/225225;
1497  g->C4x[13] = (1856*g->n-936)/225225;
1498  g->C4x[14] = 8/(real)(10725);
1499  g->C4x[15] = (g->n*(3584*g->n-3328)+1144)/315315;
1500  g->C4x[16] = (1024*g->n-208)/105105;
1501  g->C4x[17] = -136/(real)(63063);
1502  g->C4x[18] = (832-2560*g->n)/405405;
1503  g->C4x[19] = -128/(real)(135135);
1504  g->C4x[20] = 128/(real)(99099);
1505 }
1506 
1507 int transit(real lon1, real lon2) {
1508  real lon12;
1509  /* Return 1 or -1 if crossing prime meridian in east or west direction.
1510  * Otherwise return zero. */
1511  /* Compute lon12 the same way as Geodesic::Inverse. */
1512  lon1 = AngNormalize(lon1);
1513  lon2 = AngNormalize(lon2);
1514  lon12 = AngDiff(lon1, lon2);
1515  return lon1 < 0 && lon2 >= 0 && lon12 > 0 ? 1 :
1516  (lon2 < 0 && lon1 >= 0 && lon12 < 0 ? -1 : 0);
1517 }
1518 
1519 void accini(real s[]) {
1520  /* Initialize an accumulator; this is an array with two elements. */
1521  s[0] = s[1] = 0;
1522 }
1523 
1524 void acccopy(const real s[], real t[]) {
1525  /* Copy an accumulator; t = s. */
1526  t[0] = s[0]; t[1] = s[1];
1527 }
1528 
1529 void accadd(real s[], real y) {
1530  /* Add y to an accumulator. */
1531  real u, z = sumx(y, s[1], &u);
1532  s[0] = sumx(z, s[0], &s[1]);
1533  if (s[0] == 0)
1534  s[0] = u;
1535  else
1536  s[1] = s[1] + u;
1537 }
1538 
1539 real accsum(const real s[], real y) {
1540  /* Return accumulator + y (but don't add to accumulator). */
1541  real t[2];
1542  acccopy(s, t);
1543  accadd(t, y);
1544  return t[0];
1545 }
1546 
1547 void accneg(real s[]) {
1548  /* Negate an accumulator. */
1549  s[0] = -s[0]; s[1] = -s[1];
1550 }
1551 
1552 void geod_polygon_init(struct geod_polygon* p, boolx polylinep) {
1553  p->lat0 = p->lon0 = p->lat = p->lon = NaN;
1554  p->polyline = (polylinep != 0);
1555  accini(p->P);
1556  accini(p->A);
1557  p->num = p->crossings = 0;
1558 }
1559 
1560 void geod_polygon_addpoint(const struct geod_geodesic* g,
1561  struct geod_polygon* p,
1562  real lat, real lon) {
1563  lon = AngNormalize(lon);
1564  if (p->num == 0) {
1565  p->lat0 = p->lat = lat;
1566  p->lon0 = p->lon = lon;
1567  } else {
1568  real s12, S12;
1569  geod_geninverse(g, p->lat, p->lon, lat, lon,
1570  &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1571  accadd(p->P, s12);
1572  if (!p->polyline) {
1573  accadd(p->A, S12);
1574  p->crossings += transit(p->lon, lon);
1575  }
1576  p->lat = lat; p->lon = lon;
1577  }
1578  ++p->num;
1579 }
1580 
1581 void geod_polygon_addedge(const struct geod_geodesic* g,
1582  struct geod_polygon* p,
1583  real azi, real s) {
1584  if (p->num) { /* Do nothing is num is zero */
1585  real lat, lon, S12;
1586  geod_gendirect(g, p->lat, p->lon, azi, FALSE, s,
1587  &lat, &lon, 0,
1588  0, 0, 0, 0, p->polyline ? 0 : &S12);
1589  accadd(p->P, s);
1590  if (!p->polyline) {
1591  accadd(p->A, S12);
1592  p->crossings += transit(p->lon, lon);
1593  }
1594  p->lat = lat; p->lon = lon;
1595  ++p->num;
1596  }
1597 }
1598 
1599 unsigned geod_polygon_compute(const struct geod_geodesic* g,
1600  const struct geod_polygon* p,
1601  boolx reverse, boolx sign,
1602  real* pA, real* pP) {
1603  real s12, S12, t[2], area0;
1604  int crossings;
1605  if (p->num < 2) {
1606  if (pP) *pP = 0;
1607  if (!p->polyline && pA) *pA = 0;
1608  return p->num;
1609  }
1610  if (p->polyline) {
1611  if (pP) *pP = p->P[0];
1612  return p->num;
1613  }
1614  geod_geninverse(g, p->lat, p->lon, p->lat0, p->lon0,
1615  &s12, 0, 0, 0, 0, 0, &S12);
1616  if (pP) *pP = accsum(p->P, s12);
1617  acccopy(p->A, t);
1618  accadd(t, S12);
1619  crossings = p->crossings + transit(p->lon, p->lon0);
1620  area0 = 4 * pi * g->c2;
1621  if (crossings & 1)
1622  accadd(t, (t[0] < 0 ? 1 : -1) * area0/2);
1623  /* area is with the clockwise sense. If !reverse convert to
1624  * counter-clockwise convention. */
1625  if (!reverse)
1626  accneg(t);
1627  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
1628  if (sign) {
1629  if (t[0] > area0/2)
1630  accadd(t, -area0);
1631  else if (t[0] <= -area0/2)
1632  accadd(t, +area0);
1633  } else {
1634  if (t[0] >= area0)
1635  accadd(t, -area0);
1636  else if (t[0] < 0)
1637  accadd(t, +area0);
1638  }
1639  if (pA) *pA = 0 + t[0];
1640  return p->num;
1641 }
1642 
1643 unsigned geod_polygon_testpoint(const struct geod_geodesic* g,
1644  const struct geod_polygon* p,
1645  real lat, real lon,
1646  boolx reverse, boolx sign,
1647  real* pA, real* pP) {
1648  real perimeter, tempsum, area0;
1649  int crossings, i;
1650  unsigned num = p->num + 1;
1651  if (num == 1) {
1652  if (pP) *pP = 0;
1653  if (!p->polyline && pA) *pA = 0;
1654  return num;
1655  }
1656  perimeter = p->P[0];
1657  tempsum = p->polyline ? 0 : p->A[0];
1658  crossings = p->crossings;
1659  for (i = 0; i < (p->polyline ? 1 : 2); ++i) {
1660  real s12, S12;
1661  geod_geninverse(g,
1662  i == 0 ? p->lat : lat, i == 0 ? p->lon : lon,
1663  i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon,
1664  &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1665  perimeter += s12;
1666  if (!p->polyline) {
1667  tempsum += S12;
1668  crossings += transit(i == 0 ? p->lon : lon,
1669  i != 0 ? p->lon0 : lon);
1670  }
1671  }
1672 
1673  if (pP) *pP = perimeter;
1674  if (p->polyline)
1675  return num;
1676 
1677  area0 = 4 * pi * g->c2;
1678  if (crossings & 1)
1679  tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1680  /* area is with the clockwise sense. If !reverse convert to
1681  * counter-clockwise convention. */
1682  if (!reverse)
1683  tempsum *= -1;
1684  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
1685  if (sign) {
1686  if (tempsum > area0/2)
1687  tempsum -= area0;
1688  else if (tempsum <= -area0/2)
1689  tempsum += area0;
1690  } else {
1691  if (tempsum >= area0)
1692  tempsum -= area0;
1693  else if (tempsum < 0)
1694  tempsum += area0;
1695  }
1696  if (pA) *pA = 0 + tempsum;
1697  return num;
1698 }
1699 
1700 unsigned geod_polygon_testedge(const struct geod_geodesic* g,
1701  const struct geod_polygon* p,
1702  real azi, real s,
1703  boolx reverse, boolx sign,
1704  real* pA, real* pP) {
1705  real perimeter, tempsum, area0;
1706  int crossings;
1707  unsigned num = p->num + 1;
1708  if (num == 1) { /* we don't have a starting point! */
1709  if (pP) *pP = NaN;
1710  if (!p->polyline && pA) *pA = NaN;
1711  return 0;
1712  }
1713  perimeter = p->P[0] + s;
1714  if (p->polyline) {
1715  if (pP) *pP = perimeter;
1716  return num;
1717  }
1718 
1719  tempsum = p->A[0];
1720  crossings = p->crossings;
1721  {
1722  real lat, lon, s12, S12;
1723  geod_gendirect(g, p->lat, p->lon, azi, FALSE, s,
1724  &lat, &lon, 0,
1725  0, 0, 0, 0, &S12);
1726  tempsum += S12;
1727  crossings += transit(p->lon, lon);
1728  geod_geninverse(g, lat, lon, p->lat0, p->lon0,
1729  &s12, 0, 0, 0, 0, 0, &S12);
1730  perimeter += s12;
1731  tempsum += S12;
1732  crossings += transit(lon, p->lon0);
1733  }
1734 
1735  area0 = 4 * pi * g->c2;
1736  if (crossings & 1)
1737  tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1738  /* area is with the clockwise sense. If !reverse convert to
1739  * counter-clockwise convention. */
1740  if (!reverse)
1741  tempsum *= -1;
1742  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
1743  if (sign) {
1744  if (tempsum > area0/2)
1745  tempsum -= area0;
1746  else if (tempsum <= -area0/2)
1747  tempsum += area0;
1748  } else {
1749  if (tempsum >= area0)
1750  tempsum -= area0;
1751  else if (tempsum < 0)
1752  tempsum += area0;
1753  }
1754  if (pP) *pP = perimeter;
1755  if (pA) *pA = 0 + tempsum;
1756  return num;
1757 }
1758 
1759 void geod_polygonarea(const struct geod_geodesic* g,
1760  real lats[], real lons[], int n,
1761  real* pA, real* pP) {
1762  int i;
1763  struct geod_polygon p;
1764  geod_polygon_init(&p, FALSE);
1765  for (i = 0; i < n; ++i)
1766  geod_polygon_addpoint(g, &p, lats[i], lons[i]);
1767  geod_polygon_compute(g, &p, FALSE, TRUE, pA, pP);
1768 }
1769 
1770 /** @endcond */
double geod_genposition(const struct geod_geodesicline *l, int arcmode, double s12_a12, double *plat2, double *plon2, double *pazi2, double *ps12, double *pm12, double *pM12, double *pM21, double *pS12)
unsigned geod_polygon_testedge(const struct geod_geodesic *g, const struct geod_polygon *p, double azi, double s, int reverse, int sign, double *pA, double *pP)
GeographicLib::Math::real real
double lon
Definition: geodesic.h:181
void geod_polygon_addedge(const struct geod_geodesic *g, struct geod_polygon *p, double azi, double s)
unsigned num
Definition: geodesic.h:190
void geod_position(const struct geod_geodesicline *l, double s12, double *plat2, double *plon2, double *pazi2)
double f
Definition: geodesic.h:148
void geod_lineinit(struct geod_geodesicline *l, const struct geod_geodesic *g, double lat1, double lon1, double azi1, unsigned caps)
unsigned caps
Definition: geodesic.h:171
double geod_geninverse(const struct geod_geodesic *g, double lat1, double lon1, double lat2, double lon2, double *ps12, double *pazi1, double *pazi2, double *pm12, double *pM12, double *pM21, double *pS12)
void geod_polygon_addpoint(const struct geod_geodesic *g, struct geod_polygon *p, double lat, double lon)
void geod_polygon_init(struct geod_polygon *p, int polylinep)
void geod_direct(const struct geod_geodesic *g, double lat1, double lon1, double azi1, double s12, double *plat2, double *plon2, double *pazi2)
unsigned geod_polygon_compute(const struct geod_geodesic *g, const struct geod_polygon *p, int reverse, int sign, double *pA, double *pP)
void geod_polygonarea(const struct geod_geodesic *g, double lats[], double lons[], int n, double *pA, double *pP)
double a
Definition: geodesic.h:147
double geod_gendirect(const struct geod_geodesic *g, double lat1, double lon1, double azi1, int arcmode, double s12_a12, double *plat2, double *plon2, double *pazi2, double *ps12, double *pm12, double *pM12, double *pM21, double *pS12)
unsigned geod_polygon_testpoint(const struct geod_geodesic *g, const struct geod_polygon *p, double lat, double lon, int reverse, int sign, double *pA, double *pP)
void geod_inverse(const struct geod_geodesic *g, double lat1, double lon1, double lat2, double lon2, double *ps12, double *pazi1, double *pazi2)
void geod_init(struct geod_geodesic *g, double a, double f)
double lat
Definition: geodesic.h:180
Header for the geodesic routines in C.