GeographicLib 2.1.1
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 and enum-float expressions
14# pragma warning (disable: 4127 5055)
15#endif
16
17namespace 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
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
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) :
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
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
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
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
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
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.
#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 Delta(real sn, real cn) const
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)