GeographicLib  2.0
EllipticFunction.cpp
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1 /**
2  * \file EllipticFunction.cpp
3  * \brief Implementation for GeographicLib::EllipticFunction class
4  *
5  * Copyright (c) Charles Karney (2008-2022) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  **********************************************************************/
9 
11 
12 #if defined(_MSC_VER)
13 // Squelch warnings about constant conditional expressions
14 # pragma warning (disable: 4127)
15 #endif
16 
17 namespace GeographicLib {
18 
19  using namespace std;
20 
21  /*
22  * Implementation of methods given in
23  *
24  * B. C. Carlson
25  * Computation of elliptic integrals
26  * Numerical Algorithms 10, 13-26 (1995)
27  */
28 
29  Math::real EllipticFunction::RF(real x, real y, real z) {
30  // Carlson, eqs 2.2 - 2.7
31  static const real tolRF =
32  pow(3 * numeric_limits<real>::epsilon() * real(0.01), 1/real(8));
33  real
34  A0 = (x + y + z)/3,
35  An = A0,
36  Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRF,
37  x0 = x,
38  y0 = y,
39  z0 = z,
40  mul = 1;
41  while (Q >= mul * fabs(An)) {
42  // Max 6 trips
43  real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
44  An = (An + lam)/4;
45  x0 = (x0 + lam)/4;
46  y0 = (y0 + lam)/4;
47  z0 = (z0 + lam)/4;
48  mul *= 4;
49  }
50  real
51  X = (A0 - x) / (mul * An),
52  Y = (A0 - y) / (mul * An),
53  Z = - (X + Y),
54  E2 = X*Y - Z*Z,
55  E3 = X*Y*Z;
56  // https://dlmf.nist.gov/19.36.E1
57  // Polynomial is
58  // (1 - E2/10 + E3/14 + E2^2/24 - 3*E2*E3/44
59  // - 5*E2^3/208 + 3*E3^2/104 + E2^2*E3/16)
60  // convert to Horner form...
61  return (E3 * (6930 * E3 + E2 * (15015 * E2 - 16380) + 17160) +
62  E2 * ((10010 - 5775 * E2) * E2 - 24024) + 240240) /
63  (240240 * sqrt(An));
64  }
65 
66  Math::real EllipticFunction::RF(real x, real y) {
67  // Carlson, eqs 2.36 - 2.38
68  static const real tolRG0 =
69  real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
70  real xn = sqrt(x), yn = sqrt(y);
71  if (xn < yn) swap(xn, yn);
72  while (fabs(xn-yn) > tolRG0 * xn) {
73  // Max 4 trips
74  real t = (xn + yn) /2;
75  yn = sqrt(xn * yn);
76  xn = t;
77  }
78  return Math::pi() / (xn + yn);
79  }
80 
81  Math::real EllipticFunction::RC(real x, real y) {
82  // Defined only for y != 0 and x >= 0.
83  return ( !(x >= y) ? // x < y and catch nans
84  // https://dlmf.nist.gov/19.2.E18
85  atan(sqrt((y - x) / x)) / sqrt(y - x) :
86  ( x == y ? 1 / sqrt(y) :
87  asinh( y > 0 ?
88  // https://dlmf.nist.gov/19.2.E19
89  // atanh(sqrt((x - y) / x))
90  sqrt((x - y) / y) :
91  // https://dlmf.nist.gov/19.2.E20
92  // atanh(sqrt(x / (x - y)))
93  sqrt(-x / y) ) / sqrt(x - y) ) );
94  }
95 
96  Math::real EllipticFunction::RG(real x, real y, real z) {
97  if (z == 0)
98  swap(y, z);
99  // Carlson, eq 1.7
100  return (z * RF(x, y, z) - (x-z) * (y-z) * RD(x, y, z) / 3
101  + sqrt(x * y / z)) / 2;
102  }
103 
105  // Carlson, eqs 2.36 - 2.39
106  static const real tolRG0 =
107  real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
108  real
109  x0 = sqrt(fmax(x, y)),
110  y0 = sqrt(fmin(x, y)),
111  xn = x0,
112  yn = y0,
113  s = 0,
114  mul = real(0.25);
115  while (fabs(xn-yn) > tolRG0 * xn) {
116  // Max 4 trips
117  real t = (xn + yn) /2;
118  yn = sqrt(xn * yn);
119  xn = t;
120  mul *= 2;
121  t = xn - yn;
122  s += mul * t * t;
123  }
124  return (Math::sq( (x0 + y0)/2 ) - s) * Math::pi() / (2 * (xn + yn));
125  }
126 
127  Math::real EllipticFunction::RJ(real x, real y, real z, real p) {
128  // Carlson, eqs 2.17 - 2.25
129  static const real
130  tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
131  1/real(8));
132  real
133  A0 = (x + y + z + 2*p)/5,
134  An = A0,
135  delta = (p-x) * (p-y) * (p-z),
136  Q = fmax(fmax(fabs(A0-x), fabs(A0-y)),
137  fmax(fabs(A0-z), fabs(A0-p))) / tolRD,
138  x0 = x,
139  y0 = y,
140  z0 = z,
141  p0 = p,
142  mul = 1,
143  mul3 = 1,
144  s = 0;
145  while (Q >= mul * fabs(An)) {
146  // Max 7 trips
147  real
148  lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
149  d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
150  e0 = delta/(mul3 * Math::sq(d0));
151  s += RC(1, 1 + e0)/(mul * d0);
152  An = (An + lam)/4;
153  x0 = (x0 + lam)/4;
154  y0 = (y0 + lam)/4;
155  z0 = (z0 + lam)/4;
156  p0 = (p0 + lam)/4;
157  mul *= 4;
158  mul3 *= 64;
159  }
160  real
161  X = (A0 - x) / (mul * An),
162  Y = (A0 - y) / (mul * An),
163  Z = (A0 - z) / (mul * An),
164  P = -(X + Y + Z) / 2,
165  E2 = X*Y + X*Z + Y*Z - 3*P*P,
166  E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
167  E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
168  E5 = X*Y*Z*P*P;
169  // https://dlmf.nist.gov/19.36.E2
170  // Polynomial is
171  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
172  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
173  // - 9*(E3*E4+E2*E5)/68)
174  return ((471240 - 540540 * E2) * E5 +
175  (612612 * E2 - 540540 * E3 - 556920) * E4 +
176  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
177  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
178  (4084080 * mul * An * sqrt(An)) + 6 * s;
179  }
180 
181  Math::real EllipticFunction::RD(real x, real y, real z) {
182  // Carlson, eqs 2.28 - 2.34
183  static const real
184  tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
185  1/real(8));
186  real
187  A0 = (x + y + 3*z)/5,
188  An = A0,
189  Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRD,
190  x0 = x,
191  y0 = y,
192  z0 = z,
193  mul = 1,
194  s = 0;
195  while (Q >= mul * fabs(An)) {
196  // Max 7 trips
197  real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
198  s += 1/(mul * sqrt(z0) * (z0 + lam));
199  An = (An + lam)/4;
200  x0 = (x0 + lam)/4;
201  y0 = (y0 + lam)/4;
202  z0 = (z0 + lam)/4;
203  mul *= 4;
204  }
205  real
206  X = (A0 - x) / (mul * An),
207  Y = (A0 - y) / (mul * An),
208  Z = -(X + Y) / 3,
209  E2 = X*Y - 6*Z*Z,
210  E3 = (3*X*Y - 8*Z*Z)*Z,
211  E4 = 3 * (X*Y - Z*Z) * Z*Z,
212  E5 = X*Y*Z*Z*Z;
213  // https://dlmf.nist.gov/19.36.E2
214  // Polynomial is
215  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
216  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
217  // - 9*(E3*E4+E2*E5)/68)
218  return ((471240 - 540540 * E2) * E5 +
219  (612612 * E2 - 540540 * E3 - 556920) * E4 +
220  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
221  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
222  (4084080 * mul * An * sqrt(An)) + 3 * s;
223  }
224 
225  void EllipticFunction::Reset(real k2, real alpha2,
226  real kp2, real alphap2) {
227  // Accept nans here (needed for GeodesicExact)
228  if (k2 > 1)
229  throw GeographicErr("Parameter k2 is not in (-inf, 1]");
230  if (alpha2 > 1)
231  throw GeographicErr("Parameter alpha2 is not in (-inf, 1]");
232  if (kp2 < 0)
233  throw GeographicErr("Parameter kp2 is not in [0, inf)");
234  if (alphap2 < 0)
235  throw GeographicErr("Parameter alphap2 is not in [0, inf)");
236  _k2 = k2;
237  _kp2 = kp2;
238  _alpha2 = alpha2;
239  _alphap2 = alphap2;
240  _eps = _k2/Math::sq(sqrt(_kp2) + 1);
241  // Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
242  // K E D
243  // k = 0: pi/2 pi/2 pi/4
244  // k = 1: inf 1 inf
245  // Pi G H
246  // k = 0, alpha = 0: pi/2 pi/2 pi/4
247  // k = 1, alpha = 0: inf 1 1
248  // k = 0, alpha = 1: inf inf pi/2
249  // k = 1, alpha = 1: inf inf inf
250  //
251  // Pi(0, k) = K(k)
252  // G(0, k) = E(k)
253  // H(0, k) = K(k) - D(k)
254  // Pi(0, k) = K(k)
255  // G(0, k) = E(k)
256  // H(0, k) = K(k) - D(k)
257  // Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2))
258  // G(alpha2, 0) = pi/(2*sqrt(1-alpha2))
259  // H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2)))
260  // Pi(alpha2, 1) = inf
261  // H(1, k) = K(k)
262  // G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
263  if (_k2 != 0) {
264  // Complete elliptic integral K(k), Carlson eq. 4.1
265  // https://dlmf.nist.gov/19.25.E1
266  _kKc = _kp2 != 0 ? RF(_kp2, 1) : Math::infinity();
267  // Complete elliptic integral E(k), Carlson eq. 4.2
268  // https://dlmf.nist.gov/19.25.E1
269  _eEc = _kp2 != 0 ? 2 * RG(_kp2, 1) : 1;
270  // D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3
271  // https://dlmf.nist.gov/19.25.E1
272  _dDc = _kp2 != 0 ? RD(0, _kp2, 1) / 3 : Math::infinity();
273  } else {
274  _kKc = _eEc = Math::pi()/2; _dDc = _kKc/2;
275  }
276  if (_alpha2 != 0) {
277  // https://dlmf.nist.gov/19.25.E2
278  real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) :
279  Math::infinity(),
280  // Only use rc if _kp2 = 0.
281  rc = _kp2 != 0 ? 0 :
282  (_alphap2 != 0 ? RC(1, _alphap2) : Math::infinity());
283  // Pi(alpha^2, k)
284  _pPic = _kp2 != 0 ? _kKc + _alpha2 * rj / 3 : Math::infinity();
285  // G(alpha^2, k)
286  _gGc = _kp2 != 0 ? _kKc + (_alpha2 - _k2) * rj / 3 : rc;
287  // H(alpha^2, k)
288  _hHc = _kp2 != 0 ? _kKc - (_alphap2 != 0 ? _alphap2 * rj : 0) / 3 : rc;
289  } else {
290  _pPic = _kKc; _gGc = _eEc;
291  // Hc = Kc - Dc but this involves large cancellations if k2 is close to
292  // 1. So write (for alpha2 = 0)
293  // Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2)
294  // = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...)
295  // = 1/kp * D(i*k/kp)
296  // and use D(k) = RD(0, kp2, 1) / 3
297  // so Hc = 1/kp * RD(0, 1/kp2, 1) / 3
298  // = kp2 * RD(0, 1, kp2) / 3
299  // using https://dlmf.nist.gov/19.20.E18
300  // Equivalently
301  // RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
302  // For k2 = 1 and alpha2 = 0, we have
303  // Hc = int(cos(phi),...) = 1
304  _hHc = _kp2 != 0 ? _kp2 * RD(0, 1, _kp2) / 3 : 1;
305  }
306  }
307 
308  /*
309  * Implementation of methods given in
310  *
311  * R. Bulirsch
312  * Numerical Calculation of Elliptic Integrals and Elliptic Functions
313  * Numericshe Mathematik 7, 78-90 (1965)
314  */
315 
316  void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn) const {
317  // Bulirsch's sncndn routine, p 89.
318  static const real tolJAC =
319  sqrt(numeric_limits<real>::epsilon() * real(0.01));
320  if (_kp2 != 0) {
321  real mc = _kp2, d = 0;
322  if (signbit(_kp2)) {
323  d = 1 - mc;
324  mc /= -d;
325  d = sqrt(d);
326  x *= d;
327  }
328  real c = 0; // To suppress warning about uninitialized variable
329  real m[num_], n[num_];
330  unsigned l = 0;
331  for (real a = 1; l < num_ || GEOGRAPHICLIB_PANIC; ++l) {
332  // This converges quadratically. Max 5 trips
333  m[l] = a;
334  n[l] = mc = sqrt(mc);
335  c = (a + mc) / 2;
336  if (!(fabs(a - mc) > tolJAC * a)) {
337  ++l;
338  break;
339  }
340  mc *= a;
341  a = c;
342  }
343  x *= c;
344  sn = sin(x);
345  cn = cos(x);
346  dn = 1;
347  if (sn != 0) {
348  real a = cn / sn;
349  c *= a;
350  while (l--) {
351  real b = m[l];
352  a *= c;
353  c *= dn;
354  dn = (n[l] + a) / (b + a);
355  a = c / b;
356  }
357  a = 1 / sqrt(c*c + 1);
358  sn = signbit(sn) ? -a : a;
359  cn = c * sn;
360  if (signbit(_kp2)) {
361  swap(cn, dn);
362  sn /= d;
363  }
364  }
365  } else {
366  sn = tanh(x);
367  dn = cn = 1 / cosh(x);
368  }
369  }
370 
371  Math::real EllipticFunction::F(real sn, real cn, real dn) const {
372  // Carlson, eq. 4.5 and
373  // https://dlmf.nist.gov/19.25.E5
374  real cn2 = cn*cn, dn2 = dn*dn,
375  fi = cn2 != 0 ? fabs(sn) * RF(cn2, dn2, 1) : K();
376  // Enforce usual trig-like symmetries
377  if (signbit(cn))
378  fi = 2 * K() - fi;
379  return copysign(fi, sn);
380  }
381 
382  Math::real EllipticFunction::E(real sn, real cn, real dn) const {
383  real
384  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
385  ei = cn2 != 0 ?
386  fabs(sn) * ( _k2 <= 0 ?
387  // Carlson, eq. 4.6 and
388  // https://dlmf.nist.gov/19.25.E9
389  RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
390  ( _kp2 >= 0 ?
391  // https://dlmf.nist.gov/19.25.E10
392  _kp2 * RF(cn2, dn2, 1) +
393  _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
394  _k2 * fabs(cn) / dn :
395  // https://dlmf.nist.gov/19.25.E11
396  - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 +
397  dn / fabs(cn) ) ) :
398  E();
399  // Enforce usual trig-like symmetries
400  if (signbit(cn))
401  ei = 2 * E() - ei;
402  return copysign(ei, sn);
403  }
404 
405  Math::real EllipticFunction::D(real sn, real cn, real dn) const {
406  // Carlson, eq. 4.8 and
407  // https://dlmf.nist.gov/19.25.E13
408  real
409  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
410  di = cn2 != 0 ? fabs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D();
411  // Enforce usual trig-like symmetries
412  if (signbit(cn))
413  di = 2 * D() - di;
414  return copysign(di, sn);
415  }
416 
417  Math::real EllipticFunction::Pi(real sn, real cn, real dn) const {
418  // Carlson, eq. 4.7 and
419  // https://dlmf.nist.gov/19.25.E14
420  real
421  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
422  pii = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) +
423  _alpha2 * sn2 *
424  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
425  Pi();
426  // Enforce usual trig-like symmetries
427  if (signbit(cn))
428  pii = 2 * Pi() - pii;
429  return copysign(pii, sn);
430  }
431 
432  Math::real EllipticFunction::G(real sn, real cn, real dn) const {
433  real
434  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
435  gi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) +
436  (_alpha2 - _k2) * sn2 *
437  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
438  G();
439  // Enforce usual trig-like symmetries
440  if (signbit(cn))
441  gi = 2 * G() - gi;
442  return copysign(gi, sn);
443  }
444 
445  Math::real EllipticFunction::H(real sn, real cn, real dn) const {
446  real
447  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
448  // WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2
449  hi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) -
450  _alphap2 * sn2 *
451  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
452  H();
453  // Enforce usual trig-like symmetries
454  if (signbit(cn))
455  hi = 2 * H() - hi;
456  return copysign(hi, sn);
457  }
458 
459  Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const {
460  // Function is periodic with period pi
461  if (signbit(cn)) { cn = -cn; sn = -sn; }
462  return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn);
463  }
464 
465  Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const {
466  // Function is periodic with period pi
467  if (signbit(cn)) { cn = -cn; sn = -sn; }
468  return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn);
469  }
470 
471  Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const {
472  // Function is periodic with period pi
473  if (signbit(cn)) { cn = -cn; sn = -sn; }
474  return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn);
475  }
476 
477  Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const {
478  // Function is periodic with period pi
479  if (signbit(cn)) { cn = -cn; sn = -sn; }
480  return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn);
481  }
482 
483  Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const {
484  // Function is periodic with period pi
485  if (signbit(cn)) { cn = -cn; sn = -sn; }
486  return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn);
487  }
488 
489  Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const {
490  // Function is periodic with period pi
491  if (signbit(cn)) { cn = -cn; sn = -sn; }
492  return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn);
493  }
494 
495  Math::real EllipticFunction::F(real phi) const {
496  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
497  return fabs(phi) < Math::pi() ? F(sn, cn, dn) :
498  (deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2);
499  }
500 
501  Math::real EllipticFunction::E(real phi) const {
502  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
503  return fabs(phi) < Math::pi() ? E(sn, cn, dn) :
504  (deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2);
505  }
506 
508  // ang - Math::AngNormalize(ang) is (nearly) an exact multiple of 360
509  real n = round((ang - Math::AngNormalize(ang))/Math::td);
510  real sn, cn;
511  Math::sincosd(ang, sn, cn);
512  return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
513  }
514 
516  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
517  return fabs(phi) < Math::pi() ? Pi(sn, cn, dn) :
518  (deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2);
519  }
520 
521  Math::real EllipticFunction::D(real phi) const {
522  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
523  return fabs(phi) < Math::pi() ? D(sn, cn, dn) :
524  (deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2);
525  }
526 
527  Math::real EllipticFunction::G(real phi) const {
528  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
529  return fabs(phi) < Math::pi() ? G(sn, cn, dn) :
530  (deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2);
531  }
532 
533  Math::real EllipticFunction::H(real phi) const {
534  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
535  return fabs(phi) < Math::pi() ? H(sn, cn, dn) :
536  (deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2);
537  }
538 
540  static const real tolJAC =
541  sqrt(numeric_limits<real>::epsilon() * real(0.01));
542  real n = floor(x / (2 * _eEc) + real(0.5));
543  x -= 2 * _eEc * n; // x now in [-ec, ec)
544  // Linear approximation
545  real phi = Math::pi() * x / (2 * _eEc); // phi in [-pi/2, pi/2)
546  // First order correction
547  phi -= _eps * sin(2 * phi) / 2;
548  // For kp2 close to zero use asin(x/_eEc) or
549  // J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012)
550  // https://doi.org/10.1016/j.amc.2011.12.021
551  for (int i = 0; i < num_ || GEOGRAPHICLIB_PANIC; ++i) {
552  real
553  sn = sin(phi),
554  cn = cos(phi),
555  dn = Delta(sn, cn),
556  err = (E(sn, cn, dn) - x)/dn;
557  phi -= err;
558  if (!(fabs(err) > tolJAC))
559  break;
560  }
561  return n * Math::pi() + phi;
562  }
563 
564  Math::real EllipticFunction::deltaEinv(real stau, real ctau) const {
565  // Function is periodic with period pi
566  if (signbit(ctau)) { ctau = -ctau; stau = -stau; }
567  real tau = atan2(stau, ctau);
568  return Einv( tau * E() / (Math::pi()/2) ) - tau;
569  }
570 
571 } // namespace GeographicLib
Header for GeographicLib::EllipticFunction class.
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:61
void sncndn(real x, real &sn, real &cn, real &dn) const
static real RJ(real x, real y, real z, real p)
Math::real deltaG(real sn, real cn, real dn) const
static real RG(real x, real y, real z)
Math::real deltaE(real sn, real cn, real dn) const
Math::real F(real phi) const
static real RC(real x, real y)
Math::real Einv(real x) const
static real RD(real x, real y, real z)
void Reset(real k2=0, real alpha2=0)
Math::real deltaD(real sn, real cn, real dn) const
Math::real Ed(real ang) const
Math::real deltaH(real sn, real cn, real dn) const
Math::real deltaF(real sn, real cn, real dn) const
static real RF(real x, real y, real z)
Math::real deltaPi(real sn, real cn, real dn) const
Math::real deltaEinv(real stau, real ctau) const
Exception handling for GeographicLib.
Definition: Constants.hpp:316
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:106
static T sq(T x)
Definition: Math.hpp:212
static T AngNormalize(T x)
Definition: Math.cpp:71
static T infinity()
Definition: Math.cpp:262
static T pi()
Definition: Math.hpp:190
@ td
degrees per turn
Definition: Math.hpp:145
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)