14 # pragma warning (disable: 4127)
32 pow(3 * numeric_limits<real>::epsilon() *
real(0.01), 1/
real(8));
36 Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRF,
41 while (Q >= mul * abs(An)) {
43 real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
51 X = (A0 - x) / (mul * An),
52 Y = (A0 - y) / (mul * An),
61 return (E3 * (6930 * E3 + E2 * (15015 * E2 - 16380) + 17160) +
62 E2 * ((10010 - 5775 * E2) * E2 - 24024) + 240240) /
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) {
74 real t = (xn + yn) /2;
85 atan(sqrt((y - x) / x)) / sqrt(y - x) :
86 ( x == y ? 1 / sqrt(y) :
93 sqrt(-x / y) ) / sqrt(x - y) ) );
100 return (z * RF(x, y, z) - (x-z) * (y-z) * RD(x, y, z) / 3
101 + sqrt(x * y / z)) / 2;
107 real(2.7) * sqrt((numeric_limits<real>::epsilon() *
real(0.01)));
109 x0 = sqrt(max(x, y)),
110 y0 = sqrt(min(x, y)),
115 while (abs(xn-yn) > tolRG0 * xn) {
117 real t = (xn + yn) /2;
129 real tolRD = pow(
real(0.2) * (numeric_limits<real>::epsilon() *
real(0.01)),
132 A0 = (x + y + z + 2*p)/5,
134 delta = (p-x) * (p-y) * (p-z),
135 Q = max(max(abs(A0-x), abs(A0-y)), max(abs(A0-z), abs(A0-p))) / tolRD,
143 while (Q >= mul * abs(An)) {
146 lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
147 d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
149 s += RC(1, 1 + e0)/(mul * d0);
159 X = (A0 - x) / (mul * An),
160 Y = (A0 - y) / (mul * An),
161 Z = (A0 - z) / (mul * An),
162 P = -(X + Y + Z) / 2,
163 E2 = X*Y + X*Z + Y*Z - 3*P*P,
164 E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
165 E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
172 return ((471240 - 540540 * E2) * E5 +
173 (612612 * E2 - 540540 * E3 - 556920) * E4 +
174 E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
175 E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
176 (4084080 * mul * An * sqrt(An)) + 6 * s;
181 real tolRD = pow(
real(0.2) * (numeric_limits<real>::epsilon() *
real(0.01)),
184 A0 = (x + y + 3*z)/5,
186 Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRD,
192 while (Q >= mul * abs(An)) {
194 real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
195 s += 1/(mul * sqrt(z0) * (z0 + lam));
203 X = (A0 - x) / (mul * An),
204 Y = (A0 - y) / (mul * An),
207 E3 = (3*X*Y - 8*Z*Z)*Z,
208 E4 = 3 * (X*Y - Z*Z) * Z*Z,
215 return ((471240 - 540540 * E2) * E5 +
216 (612612 * E2 - 540540 * E3 - 556920) * E4 +
217 E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
218 E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
219 (4084080 * mul * An * sqrt(An)) + 3 * s;
223 real kp2, real alphap2) {
228 _eps = _k2/
Math::sq(sqrt(_kp2) + 1);
235 _Ec = _kp2 ? 2 * RG(_kp2, 1) : 1;
240 _Kc = _Ec =
Math::pi()/2; _Dc = _Kc/2;
246 _Pic = _Kc + _alpha2 * rj / 3;
248 _Gc = _kp2 ? _Kc + (_alpha2 - _k2) * rj / 3 : RC(1, _alphap2);
250 _Hc = _kp2 ? _Kc - _alphap2 * rj / 3 : RC(1, _alphap2);
252 _Pic = _Kc; _Gc = _Ec; _Hc = _Kc - _Dc;
267 real tolJAC = sqrt(numeric_limits<real>::epsilon() *
real(0.01));
269 real mc = _kp2, d = 0;
277 real m[num_], n[num_];
282 n[l] = mc = sqrt(mc);
284 if (!(abs(a - mc) > tolJAC * a)) {
302 dn = (n[l] + a) / (b + a);
305 a = 1 / sqrt(c*c + 1);
306 sn = sn < 0 ? -a : a;
315 dn = cn = 1 / cosh(x);
322 real fi = abs(sn) * RF(cn*cn, dn*dn, 1);
333 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
337 RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
340 _kp2 * RF(cn2, dn2, 1) +
341 _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
344 - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 + dn / abs(cn) ) );
357 real di = abs(sn) * sn*sn * RD(cn*cn, dn*dn, 1) / 3;
370 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
371 pii = abs(sn) * (RF(cn2, dn2, 1) +
372 _alpha2 * sn2 * RJ(cn2, dn2, 1, 1 - _alpha2 * sn2) / 3);
375 pii = 2 * Pi() - pii;
383 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
384 gi = abs(sn) * (RF(cn2, dn2, 1) +
385 (_alpha2 - _k2) * sn2 *
386 RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3);
397 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
398 hi = abs(sn) * (RF(cn2, dn2, 1) -
400 RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3);
411 if (cn < 0) { cn = -cn; sn = -sn; }
412 return F(sn, cn, dn) * (
Math::pi()/2) / K() - atan2(sn, cn);
417 if (cn < 0) { cn = -cn; sn = -sn; }
418 return E(sn, cn, dn) * (
Math::pi()/2) / E() - atan2(sn, cn);
424 if (cn < 0) { cn = -cn; sn = -sn; }
425 return Pi(sn, cn, dn) * (
Math::pi()/2) / Pi() - atan2(sn, cn);
430 if (cn < 0) { cn = -cn; sn = -sn; }
431 return D(sn, cn, dn) * (
Math::pi()/2) / D() - atan2(sn, cn);
436 if (cn < 0) { cn = -cn; sn = -sn; }
437 return G(sn, cn, dn) * (
Math::pi()/2) / G() - atan2(sn, cn);
442 if (cn < 0) { cn = -cn; sn = -sn; }
443 return H(sn, cn, dn) * (
Math::pi()/2) / H() - atan2(sn, cn);
447 real sn = sin(phi), cn = cos(phi);
448 return (deltaF(sn, cn, Delta(sn, cn)) + phi) * K() / (
Math::pi()/2);
452 real sn = sin(phi), cn = cos(phi);
453 return (deltaE(sn, cn, Delta(sn, cn)) + phi) * E() / (
Math::pi()/2);
457 real n = ceil(ang/360 -
real(0.5));
461 return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
465 real sn = sin(phi), cn = cos(phi);
466 return (deltaPi(sn, cn, Delta(sn, cn)) + phi) * Pi() / (
Math::pi()/2);
470 real sn = sin(phi), cn = cos(phi);
471 return (deltaD(sn, cn, Delta(sn, cn)) + phi) * D() / (
Math::pi()/2);
475 real sn = sin(phi), cn = cos(phi);
476 return (deltaG(sn, cn, Delta(sn, cn)) + phi) * G() / (
Math::pi()/2);
480 real sn = sin(phi), cn = cos(phi);
481 return (deltaH(sn, cn, Delta(sn, cn)) + phi) * H() / (
Math::pi()/2);
485 real tolJAC = sqrt(numeric_limits<real>::epsilon() *
real(0.01));
486 real n = floor(x / (2 * _Ec) + 0.5);
489 real phi =
Math::pi() * x / (2 * _Ec);
491 phi -= _eps * sin(2 * phi) / 2;
497 err = (E(sn, cn, dn) - x)/dn;
499 if (abs(err) < tolJAC)
507 if (ctau < 0) { ctau = -ctau; stau = -stau; }
508 real tau = atan2(stau, ctau);
509 return Einv( tau * E() / (
Math::pi()/2) ) - tau;
Math::real deltaEinv(real stau, real ctau) const
void Reset(real k2=0, real alpha2=0)
Math::real deltaF(real sn, real cn, real dn) const
GeographicLib::Math::real real
static void sincosd(T x, T &sinx, T &cosx)
Math::real deltaPi(real sn, real cn, real dn) const
static real RG(real x, real y, real z)
Math::real Ed(real ang) const
Math::real Einv(real x) const
static real RF(real x, real y, real z)
static real RC(real x, real y)
Math::real F(real phi) const
Namespace for GeographicLib.
void sncndn(real x, real &sn, real &cn, real &dn) const
Header for GeographicLib::EllipticFunction class.
Math::real deltaE(real sn, real cn, real dn) const
Math::real deltaH(real sn, real cn, real dn) const
static real RD(real x, real y, real z)
static real RJ(real x, real y, real z, real p)
Math::real deltaD(real sn, real cn, real dn) const
Math::real deltaG(real sn, real cn, real dn) const
#define GEOGRAPHICLIB_PANIC