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EllipticFunction.cpp
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/**
* \file EllipticFunction.cpp
* \brief Implementation for GeographicLib::EllipticFunction class
*
* Copyright (c) Charles Karney (2008-2024) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/EllipticFunction.hpp>
namespace GeographicLib {
using namespace std;
/*
* Implementation of methods given in
*
* B. C. Carlson
* Computation of elliptic integrals
* Numerical Algorithms 10, 13-26 (1995)
*/
Math::real EllipticFunction::RF(real x, real y, real z) {
// Carlson, eqs 2.2 - 2.7
static const real tolRF =
pow(3 * numeric_limits<real>::epsilon() * real(0.01), 1/real(8));
real
A0 = (x + y + z)/3,
An = A0,
Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRF,
x0 = x,
y0 = y,
z0 = z,
mul = 1;
while (Q >= mul * fabs(An)) {
// Max 6 trips
real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
An = (An + lam)/4;
x0 = (x0 + lam)/4;
y0 = (y0 + lam)/4;
z0 = (z0 + lam)/4;
mul *= 4;
}
real
X = (A0 - x) / (mul * An),
Y = (A0 - y) / (mul * An),
Z = - (X + Y),
E2 = X*Y - Z*Z,
E3 = X*Y*Z;
// https://dlmf.nist.gov/19.36.E1
// Polynomial is
// (1 - E2/10 + E3/14 + E2^2/24 - 3*E2*E3/44
// - 5*E2^3/208 + 3*E3^2/104 + E2^2*E3/16)
// convert to Horner form...
return (E3 * (6930 * E3 + E2 * (15015 * E2 - 16380) + 17160) +
E2 * ((10010 - 5775 * E2) * E2 - 24024) + 240240) /
(240240 * sqrt(An));
}
Math::real EllipticFunction::RF(real x, real y) {
// Carlson, eqs 2.36 - 2.38
static const real tolRG0 =
real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
real xn = sqrt(x), yn = sqrt(y);
if (xn < yn) swap(xn, yn);
while (fabs(xn-yn) > tolRG0 * xn) {
// Max 4 trips
real t = (xn + yn) /2;
yn = sqrt(xn * yn);
xn = t;
}
return Math::pi() / (xn + yn);
}
Math::real EllipticFunction::RC(real x, real y) {
// Defined only for y != 0 and x >= 0.
return ( !(x >= y) ? // x < y and catch nans
// https://dlmf.nist.gov/19.2.E18
atan(sqrt((y - x) / x)) / sqrt(y - x) :
( x == y ? 1 / sqrt(y) :
asinh( y > 0 ?
// https://dlmf.nist.gov/19.2.E19
// atanh(sqrt((x - y) / x))
sqrt((x - y) / y) :
// https://dlmf.nist.gov/19.2.E20
// atanh(sqrt(x / (x - y)))
sqrt(-x / y) ) / sqrt(x - y) ) );
}
Math::real EllipticFunction::RG(real x, real y, real z) {
return (x == 0 ? RG(y, z) :
(y == 0 ? RG(z, x) :
(z == 0 ? RG(x, y) :
// Carlson, eq 1.7
(z * RF(x, y, z) - (x-z) * (y-z) * RD(x, y, z) / 3
+ sqrt(x * y / z)) / 2 )));
}
Math::real EllipticFunction::RG(real x, real y) {
// Carlson, eqs 2.36 - 2.39
static const real tolRG0 =
real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
real
x0 = sqrt(fmax(x, y)),
y0 = sqrt(fmin(x, y)),
xn = x0,
yn = y0,
s = 0,
mul = real(0.25);
while (fabs(xn-yn) > tolRG0 * xn) {
// Max 4 trips
real t = (xn + yn) /2;
yn = sqrt(xn * yn);
xn = t;
mul *= 2;
t = xn - yn;
s += mul * t * t;
}
return (Math::sq( (x0 + y0)/2 ) - s) * Math::pi() / (2 * (xn + yn));
}
Math::real EllipticFunction::RJ(real x, real y, real z, real p) {
// Carlson, eqs 2.17 - 2.25
static const real
tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
1/real(8));
real
A0 = (x + y + z + 2*p)/5,
An = A0,
delta = (p-x) * (p-y) * (p-z),
Q = fmax(fmax(fabs(A0-x), fabs(A0-y)),
fmax(fabs(A0-z), fabs(A0-p))) / tolRD,
x0 = x,
y0 = y,
z0 = z,
p0 = p,
mul = 1,
mul3 = 1,
s = 0;
while (Q >= mul * fabs(An)) {
// Max 7 trips
real
lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
e0 = delta/(mul3 * Math::sq(d0));
s += RC(1, 1 + e0)/(mul * d0);
An = (An + lam)/4;
x0 = (x0 + lam)/4;
y0 = (y0 + lam)/4;
z0 = (z0 + lam)/4;
p0 = (p0 + lam)/4;
mul *= 4;
mul3 *= 64;
}
real
X = (A0 - x) / (mul * An),
Y = (A0 - y) / (mul * An),
Z = (A0 - z) / (mul * An),
P = -(X + Y + Z) / 2,
E2 = X*Y + X*Z + Y*Z - 3*P*P,
E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
E5 = X*Y*Z*P*P;
// https://dlmf.nist.gov/19.36.E2
// Polynomial is
// (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
// - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
// - 9*(E3*E4+E2*E5)/68)
return ((471240 - 540540 * E2) * E5 +
(612612 * E2 - 540540 * E3 - 556920) * E4 +
E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
(4084080 * mul * An * sqrt(An)) + 6 * s;
}
Math::real EllipticFunction::RD(real x, real y, real z) {
// Carlson, eqs 2.28 - 2.34
static const real
tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
1/real(8));
real
A0 = (x + y + 3*z)/5,
An = A0,
Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRD,
x0 = x,
y0 = y,
z0 = z,
mul = 1,
s = 0;
while (Q >= mul * fabs(An)) {
// Max 7 trips
real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
s += 1/(mul * sqrt(z0) * (z0 + lam));
An = (An + lam)/4;
x0 = (x0 + lam)/4;
y0 = (y0 + lam)/4;
z0 = (z0 + lam)/4;
mul *= 4;
}
real
X = (A0 - x) / (mul * An),
Y = (A0 - y) / (mul * An),
Z = -(X + Y) / 3,
E2 = X*Y - 6*Z*Z,
E3 = (3*X*Y - 8*Z*Z)*Z,
E4 = 3 * (X*Y - Z*Z) * Z*Z,
E5 = X*Y*Z*Z*Z;
// https://dlmf.nist.gov/19.36.E2
// Polynomial is
// (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
// - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
// - 9*(E3*E4+E2*E5)/68)
return ((471240 - 540540 * E2) * E5 +
(612612 * E2 - 540540 * E3 - 556920) * E4 +
E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
(4084080 * mul * An * sqrt(An)) + 3 * s;
}
void EllipticFunction::Reset(real k2, real alpha2,
real kp2, real alphap2) {
// Accept nans here (needed for GeodesicExact)
if (k2 > 1)
throw GeographicErr("Parameter k2 is not in (-inf, 1]");
if (alpha2 > 1)
throw GeographicErr("Parameter alpha2 is not in (-inf, 1]");
if (kp2 < 0)
throw GeographicErr("Parameter kp2 is not in [0, inf)");
if (alphap2 < 0)
throw GeographicErr("Parameter alphap2 is not in [0, inf)");
_k2 = k2;
_kp2 = kp2;
_alpha2 = alpha2;
_alphap2 = alphap2;
_eps = _k2/Math::sq(sqrt(_kp2) + 1);
// Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
// K E D
// k = 0: pi/2 pi/2 pi/4
// k = 1: inf 1 inf
// Pi G H
// k = 0, alpha = 0: pi/2 pi/2 pi/4
// k = 1, alpha = 0: inf 1 1
// k = 0, alpha = 1: inf inf pi/2
// k = 1, alpha = 1: inf inf inf
//
// Pi(0, k) = K(k)
// G(0, k) = E(k)
// H(0, k) = K(k) - D(k)
// Pi(0, k) = K(k)
// G(0, k) = E(k)
// H(0, k) = K(k) - D(k)
// Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2))
// G(alpha2, 0) = pi/(2*sqrt(1-alpha2))
// H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2)))
// Pi(alpha2, 1) = inf
// H(1, k) = K(k)
// G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
if (_k2 != 0) {
// Complete elliptic integral K(k), Carlson eq. 4.1
// https://dlmf.nist.gov/19.25.E1
_kKc = _kp2 != 0 ? RF(_kp2, 1) : Math::infinity();
// Complete elliptic integral E(k), Carlson eq. 4.2
// https://dlmf.nist.gov/19.25.E1
_eEc = _kp2 != 0 ? 2 * RG(_kp2, 1) : 1;
// D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3
// https://dlmf.nist.gov/19.25.E1
_dDc = _kp2 != 0 ? RD(0, _kp2, 1) / 3 : Math::infinity();
} else {
_kKc = _eEc = Math::pi()/2; _dDc = _kKc/2;
}
if (_alpha2 != 0) {
// https://dlmf.nist.gov/19.25.E2
real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) :
Math::infinity(),
// Only use rc if _kp2 = 0.
rc = _kp2 != 0 ? 0 :
(_alphap2 != 0 ? RC(1, _alphap2) : Math::infinity());
// Pi(alpha^2, k)
_pPic = _kp2 != 0 ? _kKc + _alpha2 * rj / 3 : Math::infinity();
// G(alpha^2, k)
_gGc = _kp2 != 0 ? _kKc + (_alpha2 - _k2) * rj / 3 : rc;
// H(alpha^2, k)
_hHc = _kp2 != 0 ? _kKc - (_alphap2 != 0 ? _alphap2 * rj : 0) / 3 : rc;
} else {
_pPic = _kKc; _gGc = _eEc;
// Hc = Kc - Dc but this involves large cancellations if k2 is close to
// 1. So write (for alpha2 = 0)
// Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2)
// = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...)
// = 1/kp * D(i*k/kp)
// and use D(k) = RD(0, kp2, 1) / 3
// so Hc = 1/kp * RD(0, 1/kp2, 1) / 3
// = kp2 * RD(0, 1, kp2) / 3
// using https://dlmf.nist.gov/19.20.E18
// Equivalently
// RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
// For k2 = 0 and alpha2 = 0, we have
// Hc = int(cos(phi)^2,...) = pi/4
// For k2 = 1 and alpha2 = 0, we have
// Hc = int(cos(phi),...) = 1
_hHc = _kp2 == 1 ? Math::pi()/4 :
(_kp2 == 0 ? 1 : _kp2 * RD(0, 1, _kp2) / 3);
}
}
/*
* Implementation of methods given in
*
* R. Bulirsch
* Numerical Calculation of Elliptic Integrals and Elliptic Functions
* Numericshe Mathematik 7, 78-90 (1965)
*/
void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn) const {
// Bulirsch's sncndn routine, p 89.
static const real tolJAC =
sqrt(numeric_limits<real>::epsilon() * real(0.01));
if (_kp2 != 0) {
real mc = _kp2, d = 0;
if (signbit(_kp2)) {
// This implements DLMF Eqs 22.17.2 - 22.17.4. But this only
// accomodates kp2 < 0 or k2 > 1 and these are outside the advertized
// ranges for the contructor for this class.
d = 1 - mc;
mc /= -d;
d = sqrt(d);
x *= d;
}
real c = 0; // To suppress warning about uninitialized variable
real m[num_], n[num_];
unsigned l = 0;
for (real a = 1;
l < num_ ||
GEOGRAPHICLIB_PANIC
("Convergence failure in EllipticFunction::sncndn");
++l) {
// This converges quadratically. Max 5 trips
m[l] = a;
n[l] = mc = sqrt(mc);
c = (a + mc) / 2;
if (!(fabs(a - mc) > tolJAC * a)) {
++l;
break;
}
mc *= a;
a = c;
}
x *= c;
sn = sin(x);
cn = cos(x);
dn = 1;
if (sn != 0) {
real a = cn / sn;
c *= a;
while (l--) {
real b = m[l];
a *= c;
c *= dn;
dn = (n[l] + a) / (b + a);
a = c / b;
}
a = 1 / sqrt(c*c + 1);
sn = signbit(sn) ? -a : a;
cn = c * sn;
if (signbit(_kp2)) {
// See DLMF Eqs 22.17.2 - 22.17.4
swap(cn, dn);
sn /= d;
}
}
} else {
sn = tanh(x);
dn = cn = 1 / cosh(x);
}
}
Math::real EllipticFunction::am(real x) const {
// This implements DLMF Sec 22.20(ii).
// See also Sala (1989), https://doi.org/10.1137/0520100, Sec 5.
static const real tolJAC =
pow(numeric_limits<real>::epsilon(), real(0.75));
real k2 = _k2, kp2 = _kp2;
if (_k2 == 0)
return x;
else if (_kp2 == 0) {
return atan(sinh(x)); // gd(x)
} else if (_k2 < 0) {
// Sala Eq. 5.8
k2 = -_k2 / _kp2; kp2 = 1 / _kp2;
x *= sqrt(_kp2);
}
real a[num_], b, c[num_];
a[0] = 1; b = sqrt(kp2); c[0] = sqrt(k2);
int l = 1;
for (; l < num_ ||
GEOGRAPHICLIB_PANIC
("Convergence failure in EllipticFunction::am");) {
a[l] = (a[l-1] + b) / 2;
c[l] = (a[l-1] - b) / 2;
b = sqrt(a[l-1] * b);
if (!(c[l] > tolJAC * a[l])) break;
++l;
}
// Now a[l] = pi/(2*K)
// Need to initialize phi1 to stop Visual Studio complaining
real phi = a[l] * x * real(1 << l), phi1 = 0;
for (; l > 0; --l) {
phi1 = phi;
phi = (phi + asin(c[l] * sin(phi) / a[l])) / 2;
}
// For k2 < 0, see Sala Eq. 5.8
return _k2 < 0 ? phi1 - phi : phi;
}
Math::real EllipticFunction::am(real x, real& sn, real& cn, real& dn) const {
real phi = am(x);
if (_kp2 == 0) {
// Could rely on sin(gd(x)) = tanh(x) and cos(gd(x)) = 1 / cosh(x). But
// this is more accurate for large |x|.
sn = tanh(x); cn = dn = 1 / cosh(x);
} else {
sn = sin(phi); cn = cos(phi);
// See comment following DLMF Eq. 22.20.5
// dn = cn / cos(phi1 - phi)
dn = Delta(sn, cn);
}
return phi;
}
Math::real EllipticFunction::F(real sn, real cn, real dn) const {
// Carlson, eq. 4.5 and
// https://dlmf.nist.gov/19.25.E5
real cn2 = cn*cn, dn2 = dn*dn,
fi = cn2 != 0 ? fabs(sn) * RF(cn2, dn2, 1) : K();
// Enforce usual trig-like symmetries
if (signbit(cn))
fi = 2 * K() - fi;
return copysign(fi, sn);
}
Math::real EllipticFunction::E(real sn, real cn, real dn) const {
real
cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
ei = cn2 != 0 ?
fabs(sn) * ( _k2 <= 0 ?
// Carlson, eq. 4.6 and
// https://dlmf.nist.gov/19.25.E9
RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
( _kp2 >= 0 ?
// https://dlmf.nist.gov/19.25.E10
_kp2 * RF(cn2, dn2, 1) +
_k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
_k2 * fabs(cn) / dn :
// https://dlmf.nist.gov/19.25.E11
- _kp2 * sn2 * RD(dn2, 1, cn2) / 3 +
dn / fabs(cn) ) ) :
E();
// Enforce usual trig-like symmetries
if (signbit(cn))
ei = 2 * E() - ei;
return copysign(ei, sn);
}
Math::real EllipticFunction::D(real sn, real cn, real dn) const {
// Carlson, eq. 4.8 and
// https://dlmf.nist.gov/19.25.E13
real
cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
di = cn2 != 0 ? fabs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D();
// Enforce usual trig-like symmetries
if (signbit(cn))
di = 2 * D() - di;
return copysign(di, sn);
}
Math::real EllipticFunction::Pi(real sn, real cn, real dn) const {
// Carlson, eq. 4.7 and
// https://dlmf.nist.gov/19.25.E14
real
cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
pii = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) +
_alpha2 * sn2 *
RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
Pi();
// Enforce usual trig-like symmetries
if (signbit(cn))
pii = 2 * Pi() - pii;
return copysign(pii, sn);
}
Math::real EllipticFunction::G(real sn, real cn, real dn) const {
real
cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
gi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) +
(_alpha2 - _k2) * sn2 *
RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
G();
// Enforce usual trig-like symmetries
if (signbit(cn))
gi = 2 * G() - gi;
return copysign(gi, sn);
}
Math::real EllipticFunction::H(real sn, real cn, real dn) const {
real
cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
// WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2
hi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) -
_alphap2 * sn2 *
RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
H();
// Enforce usual trig-like symmetries
if (signbit(cn))
hi = 2 * H() - hi;
return copysign(hi, sn);
}
Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn);
}
Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn);
}
Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn);
}
Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn);
}
Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn);
}
Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const {
// Function is periodic with period pi
if (signbit(cn)) { cn = -cn; sn = -sn; }
return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn);
}
Math::real EllipticFunction::F(real phi) const {
if (_k2 == 0)
return phi;
else if (_kp2 == 0)
return asinh(tan(phi));
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? F(sn, cn, dn) :
(deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2);
}
Math::real EllipticFunction::E(real phi) const {
if (_k2 == 0)
return phi;
// else if (_kp2 == 0)
// Despite DLMF Eq 19.6.9 this is probably wrong, since
// sqrt(1 - k^2*sin(phi)^2) -> abs(cos(phi)) in the limit k -> 1.
// return sin(phi);
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? E(sn, cn, dn) :
(deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2);
}
Math::real EllipticFunction::Ed(real ang) const {
// ang - Math::AngNormalize(ang) is (nearly) an exact multiple of 360
real n = round((ang - Math::AngNormalize(ang))/Math::td);
real sn, cn;
Math::sincosd(ang, sn, cn);
return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
}
Math::real EllipticFunction::Pi(real phi) const {
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? Pi(sn, cn, dn) :
(deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2);
}
Math::real EllipticFunction::D(real phi) const {
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? D(sn, cn, dn) :
(deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2);
}
Math::real EllipticFunction::G(real phi) const {
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? G(sn, cn, dn) :
(deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2);
}
Math::real EllipticFunction::H(real phi) const {
real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
return fabs(phi) < Math::pi() ? H(sn, cn, dn) :
(deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2);
}
Math::real EllipticFunction::Einv(real x) const {
static const real tolJAC =
sqrt(numeric_limits<real>::epsilon() / real(100));
real n = floor(x / (2 * _eEc) + real(0.5));
x -= 2 * _eEc * n; // x now in [-ec, ec)
// Linear approximation
real phi = Math::pi() * x / (2 * _eEc); // phi in [-pi/2, pi/2)
// First order correction
phi -= _eps * sin(2 * phi) / 2;
// For kp2 close to zero use asin(x/_eEc) or
// J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012)
// https://doi.org/10.1016/j.amc.2011.12.021
for (int i = 0;
i < num_ ||
GEOGRAPHICLIB_PANIC("Convergence failure in EllipticFunction::Einv");
++i) {
real
sn = sin(phi),
cn = cos(phi),
dn = Delta(sn, cn),
err = (E(sn, cn, dn) - x)/dn;
phi -= err;
if (!(fabs(err) > tolJAC))
break;
}
return n * Math::pi() + phi;
}
Math::real EllipticFunction::deltaEinv(real stau, real ctau) const {
// Function is periodic with period pi
if (signbit(ctau)) { ctau = -ctau; stau = -stau; }
real tau = atan2(stau, ctau);
return Einv( tau * E() / (Math::pi()/2) ) - tau;
}
} // namespace GeographicLib
EllipticFunction.hpp
Header for GeographicLib::EllipticFunction class.
ang
GeographicLib::Angle ang
Definition Geod3Solve.cpp:26
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GEOGRAPHICLIB_PANIC
#define GEOGRAPHICLIB_PANIC(msg)
Definition Math.hpp:67
GeographicLib::EllipticFunction::G
Math::real G() const
Definition EllipticFunction.hpp:251
GeographicLib::EllipticFunction::sncndn
void sncndn(real x, real &sn, real &cn, real &dn) const
Definition EllipticFunction.cpp:315
GeographicLib::EllipticFunction::RJ
static real RJ(real x, real y, real z, real p)
Definition EllipticFunction.cpp:123
GeographicLib::EllipticFunction::H
Math::real H() const
Definition EllipticFunction.hpp:265
GeographicLib::EllipticFunction::deltaG
Math::real deltaG(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:543
GeographicLib::EllipticFunction::RG
static real RG(real x, real y, real z)
Definition EllipticFunction.cpp:91
GeographicLib::EllipticFunction::deltaE
Math::real deltaE(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:525
GeographicLib::EllipticFunction::F
Math::real F(real phi) const
Definition EllipticFunction.cpp:555
GeographicLib::EllipticFunction::RC
static real RC(real x, real y)
Definition EllipticFunction.cpp:76
GeographicLib::EllipticFunction::Einv
Math::real Einv(real x) const
Definition EllipticFunction.cpp:609
GeographicLib::EllipticFunction::RD
static real RD(real x, real y, real z)
Definition EllipticFunction.cpp:177
GeographicLib::EllipticFunction::Pi
Math::real Pi() const
Definition EllipticFunction.hpp:238
GeographicLib::EllipticFunction::E
Math::real E() const
Definition EllipticFunction.hpp:203
GeographicLib::EllipticFunction::k2
Math::real k2() const
Definition EllipticFunction.hpp:158
GeographicLib::EllipticFunction::alphap2
Math::real alphap2() const
Definition EllipticFunction.hpp:175
GeographicLib::EllipticFunction::Reset
void Reset(real k2=0, real alpha2=0)
Definition EllipticFunction.hpp:126
GeographicLib::EllipticFunction::am
Math::real am(real x) const
Definition EllipticFunction.cpp:378
GeographicLib::EllipticFunction::Delta
Math::real Delta(real sn, real cn) const
Definition EllipticFunction.hpp:607
GeographicLib::EllipticFunction::deltaD
Math::real deltaD(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:537
GeographicLib::EllipticFunction::Ed
Math::real Ed(real ang) const
Definition EllipticFunction.cpp:577
GeographicLib::EllipticFunction::K
Math::real K() const
Definition EllipticFunction.hpp:191
GeographicLib::EllipticFunction::kp2
Math::real kp2() const
Definition EllipticFunction.hpp:164
GeographicLib::EllipticFunction::deltaH
Math::real deltaH(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:549
GeographicLib::EllipticFunction::deltaF
Math::real deltaF(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:519
GeographicLib::EllipticFunction::RF
static real RF(real x, real y, real z)
Definition EllipticFunction.cpp:24
GeographicLib::EllipticFunction::deltaPi
Math::real deltaPi(real sn, real cn, real dn) const
Definition EllipticFunction.cpp:531
GeographicLib::EllipticFunction::deltaEinv
Math::real deltaEinv(real stau, real ctau) const
Definition EllipticFunction.cpp:637
GeographicLib::EllipticFunction::alpha2
Math::real alpha2() const
Definition EllipticFunction.hpp:169
GeographicLib::EllipticFunction::D
Math::real D() const
Definition EllipticFunction.hpp:216
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:344
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition Math.cpp:104
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::infinity
static T infinity()
Definition Math.cpp:308
GeographicLib::Math::td
static constexpr int td
degrees per turn
Definition Math.hpp:146
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:187
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition NearestNeighbor.hpp:819