GeographicLib  1.51
EllipticFunction.cpp
Go to the documentation of this file.
1 /**
2  * \file EllipticFunction.cpp
3  * \brief Implementation for GeographicLib::EllipticFunction class
4  *
5  * Copyright (c) Charles Karney (2008-2020) <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 = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRF,
37  x0 = x,
38  y0 = y,
39  z0 = z,
40  mul = 1;
41  while (Q >= mul * abs(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 (abs(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(max(x, y)),
110  y0 = sqrt(min(x, y)),
111  xn = x0,
112  yn = y0,
113  s = 0,
114  mul = real(0.25);
115  while (abs(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 = max(max(abs(A0-x), abs(A0-y)), max(abs(A0-z), abs(A0-p))) / tolRD,
137  x0 = x,
138  y0 = y,
139  z0 = z,
140  p0 = p,
141  mul = 1,
142  mul3 = 1,
143  s = 0;
144  while (Q >= mul * abs(An)) {
145  // Max 7 trips
146  real
147  lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
148  d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
149  e0 = delta/(mul3 * Math::sq(d0));
150  s += RC(1, 1 + e0)/(mul * d0);
151  An = (An + lam)/4;
152  x0 = (x0 + lam)/4;
153  y0 = (y0 + lam)/4;
154  z0 = (z0 + lam)/4;
155  p0 = (p0 + lam)/4;
156  mul *= 4;
157  mul3 *= 64;
158  }
159  real
160  X = (A0 - x) / (mul * An),
161  Y = (A0 - y) / (mul * An),
162  Z = (A0 - z) / (mul * An),
163  P = -(X + Y + Z) / 2,
164  E2 = X*Y + X*Z + Y*Z - 3*P*P,
165  E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
166  E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
167  E5 = X*Y*Z*P*P;
168  // https://dlmf.nist.gov/19.36.E2
169  // Polynomial is
170  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
171  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
172  // - 9*(E3*E4+E2*E5)/68)
173  return ((471240 - 540540 * E2) * E5 +
174  (612612 * E2 - 540540 * E3 - 556920) * E4 +
175  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
176  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
177  (4084080 * mul * An * sqrt(An)) + 6 * s;
178  }
179 
180  Math::real EllipticFunction::RD(real x, real y, real z) {
181  // Carlson, eqs 2.28 - 2.34
182  static const real
183  tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
184  1/real(8));
185  real
186  A0 = (x + y + 3*z)/5,
187  An = A0,
188  Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRD,
189  x0 = x,
190  y0 = y,
191  z0 = z,
192  mul = 1,
193  s = 0;
194  while (Q >= mul * abs(An)) {
195  // Max 7 trips
196  real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
197  s += 1/(mul * sqrt(z0) * (z0 + lam));
198  An = (An + lam)/4;
199  x0 = (x0 + lam)/4;
200  y0 = (y0 + lam)/4;
201  z0 = (z0 + lam)/4;
202  mul *= 4;
203  }
204  real
205  X = (A0 - x) / (mul * An),
206  Y = (A0 - y) / (mul * An),
207  Z = -(X + Y) / 3,
208  E2 = X*Y - 6*Z*Z,
209  E3 = (3*X*Y - 8*Z*Z)*Z,
210  E4 = 3 * (X*Y - Z*Z) * Z*Z,
211  E5 = X*Y*Z*Z*Z;
212  // https://dlmf.nist.gov/19.36.E2
213  // Polynomial is
214  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
215  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
216  // - 9*(E3*E4+E2*E5)/68)
217  return ((471240 - 540540 * E2) * E5 +
218  (612612 * E2 - 540540 * E3 - 556920) * E4 +
219  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
220  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
221  (4084080 * mul * An * sqrt(An)) + 3 * s;
222  }
223 
224  void EllipticFunction::Reset(real k2, real alpha2,
225  real kp2, real alphap2) {
226  // Accept nans here (needed for GeodesicExact)
227  if (k2 > 1)
228  throw GeographicErr("Parameter k2 is not in (-inf, 1]");
229  if (alpha2 > 1)
230  throw GeographicErr("Parameter alpha2 is not in (-inf, 1]");
231  if (kp2 < 0)
232  throw GeographicErr("Parameter kp2 is not in [0, inf)");
233  if (alphap2 < 0)
234  throw GeographicErr("Parameter alphap2 is not in [0, inf)");
235  _k2 = k2;
236  _kp2 = kp2;
237  _alpha2 = alpha2;
238  _alphap2 = alphap2;
239  _eps = _k2/Math::sq(sqrt(_kp2) + 1);
240  // Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
241  // K E D
242  // k = 0: pi/2 pi/2 pi/4
243  // k = 1: inf 1 inf
244  // Pi G H
245  // k = 0, alpha = 0: pi/2 pi/2 pi/4
246  // k = 1, alpha = 0: inf 1 1
247  // k = 0, alpha = 1: inf inf pi/2
248  // k = 1, alpha = 1: inf inf inf
249  //
250  // Pi(0, k) = K(k)
251  // G(0, k) = E(k)
252  // H(0, k) = K(k) - D(k)
253  // Pi(0, k) = K(k)
254  // G(0, k) = E(k)
255  // H(0, k) = K(k) - D(k)
256  // Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2))
257  // G(alpha2, 0) = pi/(2*sqrt(1-alpha2))
258  // H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2)))
259  // Pi(alpha2, 1) = inf
260  // H(1, k) = K(k)
261  // G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
262  if (_k2 != 0) {
263  // Complete elliptic integral K(k), Carlson eq. 4.1
264  // https://dlmf.nist.gov/19.25.E1
265  _Kc = _kp2 != 0 ? RF(_kp2, 1) : Math::infinity();
266  // Complete elliptic integral E(k), Carlson eq. 4.2
267  // https://dlmf.nist.gov/19.25.E1
268  _Ec = _kp2 != 0 ? 2 * RG(_kp2, 1) : 1;
269  // D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3
270  // https://dlmf.nist.gov/19.25.E1
271  _Dc = _kp2 != 0 ? RD(0, _kp2, 1) / 3 : Math::infinity();
272  } else {
273  _Kc = _Ec = Math::pi()/2; _Dc = _Kc/2;
274  }
275  if (_alpha2 != 0) {
276  // https://dlmf.nist.gov/19.25.E2
277  real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) :
278  Math::infinity(),
279  // Only use rc if _kp2 = 0.
280  rc = _kp2 != 0 ? 0 :
281  (_alphap2 != 0 ? RC(1, _alphap2) : Math::infinity());
282  // Pi(alpha^2, k)
283  _Pic = _kp2 != 0 ? _Kc + _alpha2 * rj / 3 : Math::infinity();
284  // G(alpha^2, k)
285  _Gc = _kp2 != 0 ? _Kc + (_alpha2 - _k2) * rj / 3 : rc;
286  // H(alpha^2, k)
287  _Hc = _kp2 != 0 ? _Kc - (_alphap2 != 0 ? _alphap2 * rj : 0) / 3 : rc;
288  } else {
289  _Pic = _Kc; _Gc = _Ec;
290  // Hc = Kc - Dc but this involves large cancellations if k2 is close to
291  // 1. So write (for alpha2 = 0)
292  // Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2)
293  // = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...)
294  // = 1/kp * D(i*k/kp)
295  // and use D(k) = RD(0, kp2, 1) / 3
296  // so Hc = 1/kp * RD(0, 1/kp2, 1) / 3
297  // = kp2 * RD(0, 1, kp2) / 3
298  // using https://dlmf.nist.gov/19.20.E18
299  // Equivalently
300  // RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
301  // For k2 = 1 and alpha2 = 0, we have
302  // Hc = int(cos(phi),...) = 1
303  _Hc = _kp2 != 0 ? _kp2 * RD(0, 1, _kp2) / 3 : 1;
304  }
305  }
306 
307  /*
308  * Implementation of methods given in
309  *
310  * R. Bulirsch
311  * Numerical Calculation of Elliptic Integrals and Elliptic Functions
312  * Numericshe Mathematik 7, 78-90 (1965)
313  */
314 
315  void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn) const {
316  // Bulirsch's sncndn routine, p 89.
317  static const real tolJAC =
318  sqrt(numeric_limits<real>::epsilon() * real(0.01));
319  if (_kp2 != 0) {
320  real mc = _kp2, d = 0;
321  if (_kp2 < 0) {
322  d = 1 - mc;
323  mc /= -d;
324  d = sqrt(d);
325  x *= d;
326  }
327  real c = 0; // To suppress warning about uninitialized variable
328  real m[num_], n[num_];
329  unsigned l = 0;
330  for (real a = 1; l < num_ || GEOGRAPHICLIB_PANIC; ++l) {
331  // This converges quadratically. Max 5 trips
332  m[l] = a;
333  n[l] = mc = sqrt(mc);
334  c = (a + mc) / 2;
335  if (!(abs(a - mc) > tolJAC * a)) {
336  ++l;
337  break;
338  }
339  mc *= a;
340  a = c;
341  }
342  x *= c;
343  sn = sin(x);
344  cn = cos(x);
345  dn = 1;
346  if (sn != 0) {
347  real a = cn / sn;
348  c *= a;
349  while (l--) {
350  real b = m[l];
351  a *= c;
352  c *= dn;
353  dn = (n[l] + a) / (b + a);
354  a = c / b;
355  }
356  a = 1 / sqrt(c*c + 1);
357  sn = sn < 0 ? -a : a;
358  cn = c * sn;
359  if (_kp2 < 0) {
360  swap(cn, dn);
361  sn /= d;
362  }
363  }
364  } else {
365  sn = tanh(x);
366  dn = cn = 1 / cosh(x);
367  }
368  }
369 
370  Math::real EllipticFunction::F(real sn, real cn, real dn) const {
371  // Carlson, eq. 4.5 and
372  // https://dlmf.nist.gov/19.25.E5
373  real cn2 = cn*cn, dn2 = dn*dn,
374  fi = cn2 != 0 ? abs(sn) * RF(cn2, dn2, 1) : K();
375  // Enforce usual trig-like symmetries
376  if (cn < 0)
377  fi = 2 * K() - fi;
378  return copysign(fi, sn);
379  }
380 
381  Math::real EllipticFunction::E(real sn, real cn, real dn) const {
382  real
383  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
384  ei = cn2 != 0 ?
385  abs(sn) * ( _k2 <= 0 ?
386  // Carlson, eq. 4.6 and
387  // https://dlmf.nist.gov/19.25.E9
388  RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
389  ( _kp2 >= 0 ?
390  // https://dlmf.nist.gov/19.25.E10
391  _kp2 * RF(cn2, dn2, 1) +
392  _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
393  _k2 * abs(cn) / dn :
394  // https://dlmf.nist.gov/19.25.E11
395  - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 +
396  dn / abs(cn) ) ) :
397  E();
398  // Enforce usual trig-like symmetries
399  if (cn < 0)
400  ei = 2 * E() - ei;
401  return copysign(ei, sn);
402  }
403 
404  Math::real EllipticFunction::D(real sn, real cn, real dn) const {
405  // Carlson, eq. 4.8 and
406  // https://dlmf.nist.gov/19.25.E13
407  real
408  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
409  di = cn2 != 0 ? abs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D();
410  // Enforce usual trig-like symmetries
411  if (cn < 0)
412  di = 2 * D() - di;
413  return copysign(di, sn);
414  }
415 
416  Math::real EllipticFunction::Pi(real sn, real cn, real dn) const {
417  // Carlson, eq. 4.7 and
418  // https://dlmf.nist.gov/19.25.E14
419  real
420  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
421  pii = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) +
422  _alpha2 * sn2 *
423  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
424  Pi();
425  // Enforce usual trig-like symmetries
426  if (cn < 0)
427  pii = 2 * Pi() - pii;
428  return copysign(pii, sn);
429  }
430 
431  Math::real EllipticFunction::G(real sn, real cn, real dn) const {
432  real
433  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
434  gi = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) +
435  (_alpha2 - _k2) * sn2 *
436  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
437  G();
438  // Enforce usual trig-like symmetries
439  if (cn < 0)
440  gi = 2 * G() - gi;
441  return copysign(gi, sn);
442  }
443 
444  Math::real EllipticFunction::H(real sn, real cn, real dn) const {
445  real
446  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
447  // WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2
448  hi = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) -
449  _alphap2 * sn2 *
450  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
451  H();
452  // Enforce usual trig-like symmetries
453  if (cn < 0)
454  hi = 2 * H() - hi;
455  return copysign(hi, sn);
456  }
457 
458  Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const {
459  // Function is periodic with period pi
460  if (cn < 0) { cn = -cn; sn = -sn; }
461  return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn);
462  }
463 
464  Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const {
465  // Function is periodic with period pi
466  if (cn < 0) { cn = -cn; sn = -sn; }
467  return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn);
468  }
469 
470  Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const {
471  // Function is periodic with period pi
472  if (cn < 0) { cn = -cn; sn = -sn; }
473  return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn);
474  }
475 
476  Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const {
477  // Function is periodic with period pi
478  if (cn < 0) { cn = -cn; sn = -sn; }
479  return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn);
480  }
481 
482  Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const {
483  // Function is periodic with period pi
484  if (cn < 0) { cn = -cn; sn = -sn; }
485  return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn);
486  }
487 
488  Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const {
489  // Function is periodic with period pi
490  if (cn < 0) { cn = -cn; sn = -sn; }
491  return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn);
492  }
493 
494  Math::real EllipticFunction::F(real phi) const {
495  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
496  return abs(phi) < Math::pi() ? F(sn, cn, dn) :
497  (deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2);
498  }
499 
500  Math::real EllipticFunction::E(real phi) const {
501  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
502  return abs(phi) < Math::pi() ? E(sn, cn, dn) :
503  (deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2);
504  }
505 
507  real n = ceil(ang/360 - real(0.5));
508  ang -= 360 * n;
509  real sn, cn;
510  Math::sincosd(ang, sn, cn);
511  return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
512  }
513 
515  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
516  return abs(phi) < Math::pi() ? Pi(sn, cn, dn) :
517  (deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2);
518  }
519 
520  Math::real EllipticFunction::D(real phi) const {
521  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
522  return abs(phi) < Math::pi() ? D(sn, cn, dn) :
523  (deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2);
524  }
525 
526  Math::real EllipticFunction::G(real phi) const {
527  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
528  return abs(phi) < Math::pi() ? G(sn, cn, dn) :
529  (deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2);
530  }
531 
532  Math::real EllipticFunction::H(real phi) const {
533  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
534  return abs(phi) < Math::pi() ? H(sn, cn, dn) :
535  (deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2);
536  }
537 
539  static const real tolJAC =
540  sqrt(numeric_limits<real>::epsilon() * real(0.01));
541  real n = floor(x / (2 * _Ec) + real(0.5));
542  x -= 2 * _Ec * n; // x now in [-ec, ec)
543  // Linear approximation
544  real phi = Math::pi() * x / (2 * _Ec); // phi in [-pi/2, pi/2)
545  // First order correction
546  phi -= _eps * sin(2 * phi) / 2;
547  // For kp2 close to zero use asin(x/_Ec) or
548  // J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012)
549  // https://doi.org/10.1016/j.amc.2011.12.021
550  for (int i = 0; i < num_ || GEOGRAPHICLIB_PANIC; ++i) {
551  real
552  sn = sin(phi),
553  cn = cos(phi),
554  dn = Delta(sn, cn),
555  err = (E(sn, cn, dn) - x)/dn;
556  phi -= err;
557  if (!(abs(err) > tolJAC))
558  break;
559  }
560  return n * Math::pi() + phi;
561  }
562 
563  Math::real EllipticFunction::deltaEinv(real stau, real ctau) const {
564  // Function is periodic with period pi
565  if (ctau < 0) { ctau = -ctau; stau = -stau; }
566  real tau = atan2(stau, ctau);
567  return Einv( tau * E() / (Math::pi()/2) ) - tau;
568  }
569 
570 } // 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:315
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T sq(T x)
Definition: Math.hpp:171
static T infinity()
Definition: Math.cpp:284
static T pi()
Definition: Math.hpp:149
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)