/**
 * \file Geocentric.cpp
 * \brief Implementation for GeographicLib::Geocentric class
 *
 * Copyright (c) Charles Karney (2008-2017) <charles@karney.com> and licensed
 * under the MIT/X11 License. For more information, see
 * https://geographiclib.sourceforge.io/
 **********************************************************************/

#include <GeographicLib/Geocentric.hpp>

namespace GeographicLib {

 using namespace std;

 Geocentric::Geocentric(real a, real f)
 : _a(a)
 , _f(f)
 , _e2(_f * (2 - _f))
 , _e2m(Math::sq(1 - _f)) // 1 - _e2
 , _e2a(abs(_e2))
 , _e4a(Math::sq(_e2))
 , _maxrad(2 * _a / numeric_limits<real>::epsilon())
 {
 if (!(Math::isfinite(_a) && _a > 0))
 throw GeographicErr("Equatorial radius is not positive");
 if (!(Math::isfinite(_f) && _f < 1))
 throw GeographicErr("Polar semi-axis is not positive");
 }

 const Geocentric& Geocentric::WGS84() {
 static const Geocentric wgs84(Constants::WGS84_a(), Constants::WGS84_f());
 return wgs84;
 }

 void Geocentric::IntForward(real lat, real lon, real h,
 real& X, real& Y, real& Z,
 real M[dim2_]) const {
 real sphi, cphi, slam, clam;
 Math::sincosd(Math::LatFix(lat), sphi, cphi);
 Math::sincosd(lon, slam, clam);
 real n = _a/sqrt(1 - _e2 * Math::sq(sphi));
 Z = (_e2m * n + h) * sphi;
 X = (n + h) * cphi;
 Y = X * slam;
 X *= clam;
 if (M)
 Rotation(sphi, cphi, slam, clam, M);
 }

 void Geocentric::IntReverse(real X, real Y, real Z,
 real& lat, real& lon, real& h,
 real M[dim2_]) const {
 real
 R = Math::hypot(X, Y),
 slam = R != 0 ? Y / R : 0,
 clam = R != 0 ? X / R : 1;
 h = Math::hypot(R, Z); // Distance to center of earth
 real sphi, cphi;
 if (h > _maxrad) {
 // We really far away (> 12 million light years); treat the earth as a
 // point and h, above, is an acceptable approximation to the height.
 // This avoids overflow, e.g., in the computation of disc below. It's
 // possible that h has overflowed to inf; but that's OK.
 //
 // Treat the case X, Y finite, but R overflows to +inf by scaling by 2.
 R = Math::hypot(X/2, Y/2);
 slam = R != 0 ? (Y/2) / R : 0;
 clam = R != 0 ? (X/2) / R : 1;
 real H = Math::hypot(Z/2, R);
 sphi = (Z/2) / H;
 cphi = R / H;
 } else if (_e4a == 0) {
 // Treat the spherical case. Dealing with underflow in the general case
 // with _e2 = 0 is difficult. Origin maps to N pole same as with
 // ellipsoid.
 real H = Math::hypot(h == 0 ? 1 : Z, R);
 sphi = (h == 0 ? 1 : Z) / H;
 cphi = R / H;
 h -= _a;
 } else {
 // Treat prolate spheroids by swapping R and Z here and by switching
 // the arguments to phi = atan2(...) at the end.
 real
 p = Math::sq(R / _a),
 q = _e2m * Math::sq(Z / _a),
 r = (p + q - _e4a) / 6;
 if (_f < 0) swap(p, q);
 if ( !(_e4a * q == 0 && r <= 0) ) {
 real
 // Avoid possible division by zero when r = 0 by multiplying
 // equations for s and t by r^3 and r, resp.
 S = _e4a * p * q / 4, // S = r^3 * s
 r2 = Math::sq(r),
 r3 = r * r2,
 disc = S * (2 * r3 + S);
 real u = r;
 if (disc >= 0) {
 real T3 = S + r3;
 // Pick the sign on the sqrt to maximize abs(T3). This minimizes
 // loss of precision due to cancellation. The result is unchanged
 // because of the way the T is used in definition of u.
 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
 // N.B. cbrt always returns the real root. cbrt(-8) = -2.
 real T = Math::cbrt(T3); // T = r * t
 // T can be zero; but then r2 / T -> 0.
 u += T + (T != 0 ? r2 / T : 0);
 } else {
 // T is complex, but the way u is defined the result is real.
 real ang = atan2(sqrt(-disc), -(S + r3));
 // There are three possible cube roots. We choose the root which
 // avoids cancellation. Note that disc < 0 implies that r < 0.
 u += 2 * r * cos(ang / 3);
 }
 real
 v = sqrt(Math::sq(u) + _e4a * q), // guaranteed positive
 // Avoid loss of accuracy when u < 0. Underflow doesn't occur in
 // e4 * q / (v - u) because u ~ e^4 when q is small and u < 0.
 uv = u < 0 ? _e4a * q / (v - u) : u + v, // u+v, guaranteed positive
 // Need to guard against w going negative due to roundoff in uv - q.
 w = max(real(0), _e2a * (uv - q) / (2 * v)),
 // Rearrange expression for k to avoid loss of accuracy due to
 // subtraction. Division by 0 not possible because uv > 0, w >= 0.
 k = uv / (sqrt(uv + Math::sq(w)) + w),
 k1 = _f >= 0 ? k : k - _e2,
 k2 = _f >= 0 ? k + _e2 : k,
 d = k1 * R / k2,
 H = Math::hypot(Z/k1, R/k2);
 sphi = (Z/k1) / H;
 cphi = (R/k2) / H;
 h = (1 - _e2m/k1) * Math::hypot(d, Z);
 } else { // e4 * q == 0 && r <= 0
 // This leads to k = 0 (oblate, equatorial plane) and k + e^2 = 0
 // (prolate, rotation axis) and the generation of 0/0 in the general
 // formulas for phi and h. using the general formula and division by 0
 // in formula for h. So handle this case by taking the limits:
 // f > 0: z -> 0, k -> e2 * sqrt(q)/sqrt(e4 - p)
 // f < 0: R -> 0, k + e2 -> - e2 * sqrt(q)/sqrt(e4 - p)
 real
 zz = sqrt((_f >= 0 ? _e4a - p : p) / _e2m),
 xx = sqrt( _f < 0 ? _e4a - p : p ),
 H = Math::hypot(zz, xx);
 sphi = zz / H;
 cphi = xx / H;
 if (Z < 0) sphi = -sphi; // for tiny negative Z (not for prolate)
 h = - _a * (_f >= 0 ? _e2m : 1) * H / _e2a;
 }
 }
 lat = Math::atan2d(sphi, cphi);
 lon = Math::atan2d(slam, clam);
 if (M)
 Rotation(sphi, cphi, slam, clam, M);
 }

 void Geocentric::Rotation(real sphi, real cphi, real slam, real clam,
 real M[dim2_]) {
 // This rotation matrix is given by the following quaternion operations
 // qrot(lam, [0,0,1]) * qrot(phi, [0,-1,0]) * [1,1,1,1]/2
 // or
 // qrot(pi/2 + lam, [0,0,1]) * qrot(-pi/2 + phi , [-1,0,0])
 // where
 // qrot(t,v) = [cos(t/2), sin(t/2)*v[1], sin(t/2)*v[2], sin(t/2)*v[3]]

 // Local X axis (east) in geocentric coords
 M[0] = -slam; M[3] = clam; M[6] = 0;
 // Local Y axis (north) in geocentric coords
 M[1] = -clam * sphi; M[4] = -slam * sphi; M[7] = cphi;
 // Local Z axis (up) in geocentric coords
 M[2] = clam * cphi; M[5] = slam * cphi; M[8] = sphi;
 }

} // namespace GeographicLib