toys/gps/source/geodesic.cpp

/**
 * \file Geodesic.cpp
 * \brief Implementation for GeographicLib::Geodesic class
 *
 * Copyright (c) Charles Karney (2009-2017) <charles@karney.com> and licensed
 * under the MIT/X11 License.  For more information, see
 * https://geographiclib.sourceforge.io/
 *
 * This is a reformulation of the geodesic problem.  The notation is as
 * follows:
 * - at a general point (no suffix or 1 or 2 as suffix)
 *   - phi = latitude
 *   - beta = latitude on auxiliary sphere
 *   - omega = longitude on auxiliary sphere
 *   - lambda = longitude
 *   - alpha = azimuth of great circle
 *   - sigma = arc length along great circle
 *   - s = distance
 *   - tau = scaled distance (= sigma at multiples of pi/2)
 * - at northwards equator crossing
 *   - beta = phi = 0
 *   - omega = lambda = 0
 *   - alpha = alpha0
 *   - sigma = s = 0
 * - a 12 suffix means a difference, e.g., s12 = s2 - s1.
 * - s and c prefixes mean sin and cos
 **********************************************************************/
#include "geodesic.h"

#include <iostream>

void static_assertC(bool status, const char *putLog)
{
    if (!status)
    {
        std::cout << putLog << std::endl;
        exit(-1);
    }
}

namespace GeographicLib {
using namespace std;

Geodesic::Geodesic(real a, real f)
    : maxit2_(maxit1_ + Math::digits() + 10)
      // Underflow guard.  We require
      //   tiny_ * epsilon() > 0
      //   tiny_ + epsilon() == epsilon()
      ,
      tiny_(sqrt(numeric_limits<real>::min())),
      tol0_(numeric_limits<real>::epsilon())
      // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
      // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
      // which otherwise failed for Visual Studio 10 (Release and Debug)
      ,
      tol1_(200 * tol0_),
      tol2_(sqrt(tol0_)),
      tolb_(tol0_ * tol2_) // Check on bisection interval
      ,
      xthresh_(1000 * tol2_),
      _a(a),
      _f(f),
      _f1(1 - _f),
      _e2(_f * (2 - _f)),
      _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
      ,
      _n(_f / (2 - _f)),
      _b(_a * _f1),
      _c2((Math::sq(_a) +
           Math::sq(_b) *
               (_e2 == 0 ? 1
                         : Math::eatanhe(real(1),
                                         (_f < 0 ? -1 : 1) * sqrt(abs(_e2))) /
                               _e2)) /
          2) // authalic radius squared
      // The sig12 threshold for "really short".  Using the auxiliary sphere
      // solution with dnm computed at (bet1 + bet2) / 2, the relative error in
      // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
      // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.  For a
      // given f and sig12, the max error occurs for lines near the pole.  If
      // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
      // increases by a factor of 2.)  Setting this equal to epsilon gives
      // sig12 = etol2.  Here 0.1 is a safety factor (error decreased by 100)
      // and max(0.001, abs(f)) stops etol2 getting too large in the nearly
      // spherical case.
      ,
      _etol2(0.1 * tol2_ /
             sqrt(max(real(0.001), abs(_f)) * min(real(1), 1 - _f / 2) / 2))
{
    if (!(Math::isfinite(_a) && _a > 0))
    {
        std::cerr << "Equatorial radius is not positive" << std::endl;
        return;
    }

    if (!(Math::isfinite(_b) && _b > 0))
    {
        std::cerr << "Polar semi-axis is not positive" << std::endl;
        return;
    }

    A3coeff();
    C3coeff();
    C4coeff();
}

const Geodesic &Geodesic::WGS84()
{
    static const Geodesic wgs84(6378137,
                                1 / ((double)(298257223563LL) / 1000000000));
    //    static const Geodesic wgs84(Constants::WGS84_a(),
    //    Constants::WGS84_f());
    return wgs84;
}

Math::real Geodesic::SinCosSeries(bool sinp, real sinx, real cosx,
                                  const real c[], int n)
{
    // Evaluate
    // y = sinp ? sum(c[i] * sin( 2*i    * x), i, 1, n) :
    //            sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
    // using Clenshaw summation.  N.B. c[0] is unused for sin series
    // Approx operation count = (n + 5) mult and (2 * n + 2) add
    c += (n + sinp); // Point to one beyond last element
    real ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
        y0 = n & 1 ? *--c : 0, y1 = 0;           // accumulators for sum
    // Now n is even
    n /= 2;

    while (n--)
    {
        // Unroll loop x 2, so accumulators return to their original role
        y1 = ar * y0 - y1 + *--c;
        y0 = ar * y1 - y0 + *--c;
    }

    return sinp ? 2 * sinx * cosx * y0 // sin(2 * x) * y0
                : cosx * (y0 - y1);    // cos(x) * (y0 - y1)
}

Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
                                unsigned outmask, real &s12, real &salp1,
                                real &calp1, real &salp2, real &calp2,
                                real &m12, real &M12, real &M21,
                                real &S12) const
{
    // Compute longitude difference (AngDiff does this carefully).  Result is
    // in [-180, 180] but -180 is only for west-going geodesics.  180 is for
    // east-going and meridional geodesics.
    real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
    // Make longitude difference positive.
    int lonsign = lon12 >= 0 ? 1 : -1;
    // If very close to being on the same half-meridian, then make it so.
    lon12 = lonsign * Math::AngRound(lon12);
    lon12s = Math::AngRound((180 - lon12) - lonsign * lon12s);
    real lam12 = lon12 * Math::degree(), slam12, clam12;

    if (lon12 > 90)
    {
        Math::sincosd(lon12s, slam12, clam12);
        clam12 = -clam12;
    }
    else
    {
        Math::sincosd(lon12, slam12, clam12);
    }

    // If really close to the equator, treat as on equator.
    lat1 = Math::AngRound(Math::LatFix(lat1));
    lat2 = Math::AngRound(Math::LatFix(lat2));
    // Swap points so that point with higher (abs) latitude is point 1
    // If one latitude is a nan, then it becomes lat1.
    int swapp = abs(lat1) < abs(lat2) ? -1 : 1;

    if (swapp < 0)
    {
        lonsign *= -1;
        swap(lat1, lat2);
    }

    // Make lat1 <= 0
    int latsign = lat1 < 0 ? 1 : -1;
    lat1 *= latsign;
    lat2 *= latsign;
    // Now we have
    //
    //     0 <= lon12 <= 180
    //     -90 <= lat1 <= 0
    //     lat1 <= lat2 <= -lat1
    //
    // longsign, swapp, latsign register the transformation to bring the
    // coordinates to this canonical form.  In all cases, 1 means no change was
    // made.  We make these transformations so that there are few cases to
    // check, e.g., on verifying quadrants in atan2.  In addition, this
    // enforces some symmetries in the results returned.
    real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
    Math::sincosd(lat1, sbet1, cbet1);
    sbet1 *= _f1;
    // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
    // will be <= 2*tiny for two points at the same pole.
    Math::norm(sbet1, cbet1);
    cbet1 = max(tiny_, cbet1);
    Math::sincosd(lat2, sbet2, cbet2);
    sbet2 *= _f1;
    // Ensure cbet2 = +epsilon at poles
    Math::norm(sbet2, cbet2);
    cbet2 = max(tiny_, cbet2);

    // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
    // |bet1| - |bet2|.  Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
    // a better measure.  This logic is used in assigning calp2 in Lambda12.
    // Sometimes these quantities vanish and in that case we force bet2 = +/-
    // bet1 exactly.  An example where is is necessary is the inverse problem
    // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
    // which failed with Visual Studio 10 (Release and Debug)

    if (cbet1 < -sbet1)
    {
        if (cbet2 == cbet1)
        {
            sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
        }
    }
    else
    {
        if (abs(sbet2) == -sbet1)
        {
            cbet2 = cbet1;
        }
    }

    real dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
         dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
    real a12, sig12;
    // index zero element of this array is unused
    real Ca[nC_];
    bool meridian = lat1 == -90 || slam12 == 0;

    if (meridian)
    {
        // Endpoints are on a single full meridian, so the geodesic might lie on
        // a meridian.
        calp1 = clam12;
        salp1 = slam12; // Head to the target longitude
        calp2 = 1;
        salp2 = 0; // At the target we're heading north
        real
            // tan(bet) = tan(sig) * cos(alp)
            ssig1 = sbet1,
            csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2;
        // sig12 = sig2 - sig1
        sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
                      csig1 * csig2 + ssig1 * ssig2);
        {
            real dummy;
            Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1,
                    cbet2, outmask | DISTANCE | REDUCEDLENGTH, s12x, m12x,
                    dummy, M12, M21, Ca);
        }

        // Add the check for sig12 since zero length geodesics might yield m12 <
        // 0.  Test case was
        //
        //    echo 20.001 0 20.001 0 | GeodSolve -i
        //
        // In fact, we will have sig12 > pi/2 for meridional geodesic which is
        // not a shortest path.
        if (sig12 < 1 || m12x >= 0)
        {
            // Need at least 2, to handle 90 0 90 180
            if (sig12 < 3 * tiny_)
            {
                sig12 = m12x = s12x = 0;
            }

            m12x *= _b;
            s12x *= _b;
            a12 = sig12 / Math::degree();
        }
        else
        // m12 < 0, i.e., prolate and too close to anti-podal
        {
            meridian = false;
        }
    }

    // somg12 > 1 marks that it needs to be calculated
    real omg12 = 0, somg12 = 2, comg12 = 0;

    if (!meridian && sbet1 == 0 && // and sbet2 == 0
        (_f <= 0 || lon12s >= _f * 180))
    {
        // Geodesic runs along equator
        calp1 = calp2 = 0;
        salp1 = salp2 = 1;
        s12x = _a * lam12;
        sig12 = omg12 = lam12 / _f1;
        m12x = _b * sin(sig12);

        if (outmask & GEODESICSCALE)
        {
            M12 = M21 = cos(sig12);
        }

        a12 = lon12 / _f1;
    }
    else if (!meridian)
    {
        // Now point1 and point2 belong within a hemisphere bounded by a
        // meridian and geodesic is neither meridional or equatorial.
        // Figure a starting point for Newton's method
        real dnm;
        sig12 =
            InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12, slam12,
                         clam12, salp1, calp1, salp2, calp2, dnm, Ca);

        if (sig12 >= 0)
        {
            // Short lines (InverseStart sets salp2, calp2, dnm)
            s12x = sig12 * _b * dnm;
            m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);

            if (outmask & GEODESICSCALE)
            {
                M12 = M21 = cos(sig12 / dnm);
            }

            a12 = sig12 / Math::degree();
            omg12 = lam12 / (_f1 * dnm);
        }
        else
        {
            // Newton's method.  This is a straightforward solution of f(alp1) =
            // lambda12(alp1) - lam12 = 0 with one wrinkle.  f(alp) has exactly
            // one root in the interval (0, pi) and its derivative is positive
            // at the root.  Thus f(alp) is positive for alp > alp1 and negative
            // for alp < alp1.  During the course of the iteration, a range
            // (alp1a, alp1b) is maintained which brackets the root and with
            // each evaluation of f(alp) the range is shrunk, if possible.
            // Newton's method is restarted whenever the derivative of f is
            // negative (because the new value of alp1 is then further from the
            // solution) or if the new estimate of alp1 lies outside (0,pi); in
            // this case, the new starting guess is taken to be (alp1a + alp1b)
            // / 2.
            //
            // initial values to suppress warnings (if loop is executed 0 times)
            real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0,
                 domg12 = 0;
            unsigned numit = 0;
            // Bracketing range
            real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;

            for (bool tripn = false, tripb = false;
                 numit < maxit2_ || GEOGRAPHICLIB_PANIC; ++numit)
            {
                // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
                // WGS84 and random input: mean = 2.85, sd = 0.60
                real dv;
                real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1,
                                  calp1, slam12, clam12, salp2, calp2, sig12,
                                  ssig1, csig1, ssig2, csig2, eps, domg12,
                                  numit < maxit1_, dv, Ca);

                // Reversed test to allow escape with NaNs
                if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_))
                {
                    break;
                }

                // Update bracketing values
                if (v > 0 &&
                    (numit > maxit1_ || calp1 / salp1 > calp1b / salp1b))
                {
                    salp1b = salp1;
                    calp1b = calp1;
                }
                else if (v < 0 &&
                         (numit > maxit1_ || calp1 / salp1 < calp1a / salp1a))
                {
                    salp1a = salp1;
                    calp1a = calp1;
                }

                if (numit < maxit1_ && dv > 0)
                {
                    real dalp1 = -v / dv;
                    real sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
                         nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;

                    if (nsalp1 > 0 && abs(dalp1) < Math::pi())
                    {
                        calp1 = calp1 * cdalp1 - salp1 * sdalp1;
                        salp1 = nsalp1;
                        Math::norm(salp1, calp1);
                        // In some regimes we don't get quadratic convergence
                        // because slope -> 0.  So use convergence conditions
                        // based on epsilon instead of sqrt(epsilon).
                        tripn = abs(v) <= 16 * tol0_;
                        continue;
                    }
                }

                // Either dv was not positive or updated value was outside legal
                // range.  Use the midpoint of the bracket as the next estimate.
                // This mechanism is not needed for the WGS84 ellipsoid, but it
                // does catch problems with more eccentric ellipsoids.  Its
                // efficacy is such for the WGS84 test set with the starting
                // guess set to alp1 = 90deg: the WGS84 test set: mean = 5.21,
                // sd = 3.93, max = 24 WGS84 and random input: mean = 4.74, sd =
                // 0.99
                salp1 = (salp1a + salp1b) / 2;
                calp1 = (calp1a + calp1b) / 2;
                Math::norm(salp1, calp1);
                tripn = false;
                tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
                         abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
            }

            {
                real dummy;
                // Ensure that the reduced length and geodesic scale are
                // computed in a "canonical" way, with the I2 integral.
                unsigned lengthmask =
                    outmask |
                    (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE
                                                               : NONE);
                Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1,
                        cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
            }

            m12x *= _b;
            s12x *= _b;
            a12 = sig12 / Math::degree();

            if (outmask & AREA)
            {
                // omg12 = lam12 - domg12
                real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
                somg12 = slam12 * cdomg12 - clam12 * sdomg12;
                comg12 = clam12 * cdomg12 + slam12 * sdomg12;
            }
        }
    }

    if (outmask & DISTANCE)
    {
        s12 = 0 + s12x; // Convert -0 to 0
    }

    if (outmask & REDUCEDLENGTH)
    {
        m12 = 0 + m12x; // Convert -0 to 0
    }

    if (outmask & AREA)
    {
        real
            // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
            salp0 = salp1 * cbet1,
            calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
        real alp12;

        if (calp0 != 0 && salp0 != 0)
        {
            real
                // From Lambda12: tan(bet) = tan(sig) * cos(alp)
                ssig1 = sbet1,
                csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2,
                k2 = Math::sq(calp0) * _ep2,
                eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
                // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
                A4 = Math::sq(_a) * calp0 * salp0 * _e2;
            Math::norm(ssig1, csig1);
            Math::norm(ssig2, csig2);
            C4f(eps, Ca);
            real B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
                 B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
            S12 = A4 * (B42 - B41);
        }
        else
        // Avoid problems with indeterminate sig1, sig2 on equator
        {
            S12 = 0;
        }

        if (!meridian && somg12 > 1)
        {
            somg12 = sin(omg12);
            comg12 = cos(omg12);
        }

        if (!meridian &&
            // omg12 < 3/4 * pi
            comg12 > -real(0.7071) &&   // Long difference not too big
            sbet2 - sbet1 < real(1.75)) // Lat difference not too big
        {
            // Use tan(Gamma/2) = tan(omg12/2)
            // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
            // with tan(x/2) = sin(x)/(1+cos(x))
            real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
            alp12 = 2 * atan2(somg12 * (sbet1 * dbet2 + sbet2 * dbet1),
                              domg12 * (sbet1 * sbet2 + dbet1 * dbet2));
        }
        else
        {
            // alp12 = alp2 - alp1, used in atan2 so no need to normalize
            real salp12 = salp2 * calp1 - calp2 * salp1,
                 calp12 = calp2 * calp1 + salp2 * salp1;

            // The right thing appears to happen if alp1 = +/-180 and alp2 = 0,
            // viz salp12 = -0 and alp12 = -180.  However this depends on the
            // sign being attached to 0 correctly.  The following ensures the
            // correct behavior.
            if (salp12 == 0 && calp12 < 0)
            {
                salp12 = tiny_ * calp1;
                calp12 = -1;
            }

            alp12 = atan2(salp12, calp12);
        }

        S12 += _c2 * alp12;
        S12 *= swapp * lonsign * latsign;
        // Convert -0 to 0
        S12 += 0;
    }

    // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
    if (swapp < 0)
    {
        swap(salp1, salp2);
        swap(calp1, calp2);

        if (outmask & GEODESICSCALE)
        {
            swap(M12, M21);
        }
    }

    salp1 *= swapp * lonsign;
    calp1 *= swapp * latsign;
    salp2 *= swapp * lonsign;
    calp2 *= swapp * latsign;
    // Returned value in [0, 180]
    return a12;
}

Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
                                unsigned outmask, real &s12, real &azi1,
                                real &azi2, real &m12, real &M12, real &M21,
                                real &S12) const
{
    // Ignore the warning C4100
    azi1 = azi1;
    azi2 = azi2;
    outmask &= OUT_MASK;
    real salp1, calp1, salp2, calp2,
        a12 = GenInverse(lat1, lon1, lat2, lon2, outmask, s12, salp1, calp1,
                         salp2, calp2, m12, M12, M21, S12);
    return a12;
}

void Geodesic::Lengths(real eps, real sig12, real ssig1, real csig1, real dn1,
                       real ssig2, real csig2, real dn2, real cbet1, real cbet2,
                       unsigned outmask, real &s12b, real &m12b, real &m0,
                       real &M12, real &M21,
                       // Scratch area of the right size
                       real Ca[]) const
{
    // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
    // and m0 = coefficient of secular term in expression for reduced length.
    outmask &= OUT_MASK;
    // outmask & DISTANCE: set s12b
    // outmask & REDUCEDLENGTH: set m12b & m0
    // outmask & GEODESICSCALE: set M12 & M21
    real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
    real Cb[nC2_ + 1];

    if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE))
    {
        A1 = A1m1f(eps);
        C1f(eps, Ca);

        if (outmask & (REDUCEDLENGTH | GEODESICSCALE))
        {
            A2 = A2m1f(eps);
            C2f(eps, Cb);
            m0x = A1 - A2;
            A2 = 1 + A2;
        }

        A1 = 1 + A1;
    }

    if (outmask & DISTANCE)
    {
        real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
                  SinCosSeries(true, ssig1, csig1, Ca, nC1_);
        // Missing a factor of _b
        s12b = A1 * (sig12 + B1);

        if (outmask & (REDUCEDLENGTH | GEODESICSCALE))
        {
            real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
                      SinCosSeries(true, ssig1, csig1, Cb, nC2_);
            J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
        }
    }
    else if (outmask & (REDUCEDLENGTH | GEODESICSCALE))
    {
        // Assume here that nC1_ >= nC2_
        for (int l = 1; l <= nC2_; ++l)
        {
            Cb[l] = A1 * Ca[l] - A2 * Cb[l];
        }

        J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
                             SinCosSeries(true, ssig1, csig1, Cb, nC2_));
    }

    if (outmask & REDUCEDLENGTH)
    {
        m0 = m0x;
        // Missing a factor of _b.
        // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
        // accurate cancellation in the case of coincident points.
        m12b =
            dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
    }

    if (outmask & GEODESICSCALE)
    {
        real csig12 = csig1 * csig2 + ssig1 * ssig2;
        real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
        M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
        M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
    }
}

Math::real Geodesic::Astroid(real x, real y)
{
    // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
    // This solution is adapted from Geocentric::Reverse.
    real k;
    real p = Math::sq(x), q = Math::sq(y), r = (p + q - 1) / 6;

    if (!(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 = p * q / 4, // S = r^3 * s
            r2 = Math::sq(r), r3 = r * r2,
                // The discriminant of the quadratic equation for T3.  This is
                // zero on the evolute curve p^(1/3)+q^(1/3) = 1
            disc = S * (S + 2 * r3);
        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 ? 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) + q), // guaranteed positive
            // Avoid loss of accuracy when u < 0.
            uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
            w = (uv - q) / (2 * v);           // positive?
        // 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); // guaranteed positive
    }
    else // q == 0 && r <= 0
    {
        // y = 0 with |x| <= 1.  Handle this case directly.
        // for y small, positive root is k = abs(y)/sqrt(1-x^2)
        k = 0;
    }

    return k;
}

Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1, real sbet2,
                                  real cbet2, real dn2, real lam12, real slam12,
                                  real clam12, real &salp1, real &calp1,
                                  // Only updated if return val >= 0
                                  real &salp2, real &calp2,
                                  // Only updated for short lines
                                  real &dnm,
                                  // Scratch area of the right size
                                  real Ca[]) const
{
    // Return a starting point for Newton's method in salp1 and calp1 (function
    // value is -1).  If Newton's method doesn't need to be used, return also
    // salp2 and calp2 and function value is sig12.
    real sig12 = -1, // Return value
        // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
        sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
         cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
#if defined(__GNUC__) && __GNUC__ == 4 && \
    (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
    // Volatile declaration needed to fix inverse cases
    // 88.202499451857 0 -88.202499451857 179.981022032992859592
    // 89.262080389218 0 -89.262080389218 179.992207982775375662
    // 89.333123580033 0 -89.333123580032997687 179.99295812360148422
    // which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
    // and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw).
    real sbet12a;
    {
        GEOGRAPHICLIB_VOLATILE real xx1 = sbet2 * cbet1;
        GEOGRAPHICLIB_VOLATILE real xx2 = cbet2 * sbet1;
        sbet12a = xx1 + xx2;
    }
#else
    real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
#endif
    bool shortline =
        cbet12 >= 0 && sbet12 < real(0.5) && cbet2 * lam12 < real(0.5);
    real somg12, comg12;

    if (shortline)
    {
        real sbetm2 = Math::sq(sbet1 + sbet2);
        // sin((bet1+bet2)/2)^2
        // =  (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
        sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
        dnm = sqrt(1 + _ep2 * sbetm2);
        real omg12 = lam12 / (_f1 * dnm);
        somg12 = sin(omg12);
        comg12 = cos(omg12);
    }
    else
    {
        somg12 = slam12;
        comg12 = clam12;
    }

    salp1 = cbet2 * somg12;
    calp1 = comg12 >= 0
                ? sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12)
                : sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
    real ssig12 = Math::hypot(salp1, calp1),
         csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;

    if (shortline && ssig12 < _etol2)
    {
        // really short lines
        salp2 = cbet1 * somg12;
        calp2 = sbet12 - cbet1 * sbet2 *
                             (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12)
                                          : 1 - comg12);
        Math::norm(salp2, calp2);
        // Set return value
        sig12 = atan2(ssig12, csig12);
    }
    else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
             csig12 >= 0 ||
             ssig12 >= 6 * abs(_n) * Math::pi() * Math::sq(cbet1))
    {
        // Nothing to do, zeroth order spherical approximation is OK
    }
    else
    {
        // Scale lam12 and bet2 to x, y coordinate system where antipodal point
        // is at origin and singular point is at y = 0, x = -1.
        real y, lamscale, betscale;
        // Volatile declaration needed to fix inverse case
        // 56.320923501171 0 -56.320923501171 179.664747671772880215
        // which otherwise fails with g++ 4.4.4 x86 -O3
        GEOGRAPHICLIB_VOLATILE real x;
        real lam12x = atan2(-slam12, -clam12); // lam12 - pi

        if (_f >= 0) // In fact f == 0 does not get here
        {
            // x = dlong, y = dlat
            {
                real k2 = Math::sq(sbet1) * _ep2,
                     eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
                lamscale = _f * cbet1 * A3f(eps) * Math::pi();
            }
            betscale = lamscale * cbet1;
            x = lam12x / lamscale;
            y = sbet12a / betscale;
        }
        else // _f < 0
        {
            // x = dlat, y = dlong
            real cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
                 bet12a = atan2(sbet12a, cbet12a);
            real m12b, m0, dummy;
            // In the case of lon12 = 180, this repeats a calculation made in
            // Inverse.
            Lengths(_n, Math::pi() + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2,
                    dn2, cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy,
                    dummy, Ca);
            x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
            betscale = x < -real(0.01) ? sbet12a / x
                                       : -_f * Math::sq(cbet1) * Math::pi();
            lamscale = betscale / cbet1;
            y = lam12x / lamscale;
        }

        if (y > -tol1_ && x > -1 - xthresh_)
        {
            // strip near cut
            // Need real(x) here to cast away the volatility of x for min/max
            if (_f >= 0)
            {
                salp1 = min(real(1), -real(x));
                calp1 = -sqrt(1 - Math::sq(salp1));
            }
            else
            {
                calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
                salp1 = sqrt(1 - Math::sq(calp1));
            }
        }
        else
        {
            // Estimate alp1, by solving the astroid problem.
            //
            // Could estimate alpha1 = theta + pi/2, directly, i.e.,
            //   calp1 = y/k; salp1 = -x/(1+k);  for _f >= 0
            //   calp1 = x/(1+k); salp1 = -y/k;  for _f < 0 (need to check)
            //
            // However, it's better to estimate omg12 from astroid and use
            // spherical formula to compute alp1.  This reduces the mean number
            // of Newton iterations for astroid cases from 2.24 (min 0, max 6)
            // to 2.12 (min 0 max 5).  The changes in the number of iterations
            // are as follows:
            //
            // change percent
            //    1       5
            //    0      78
            //   -1      16
            //   -2       0.6
            //   -3       0.04
            //   -4       0.002
            //
            // The histogram of iterations is (m = number of iterations
            // estimating alp1 directly, n = number of iterations estimating via
            // omg12, total number of trials = 148605):
            //
            //  iter    m      n
            //    0   148    186
            //    1 13046  13845
            //    2 93315 102225
            //    3 36189  32341
            //    4  5396      7
            //    5   455      1
            //    6    56      0
            //
            // Because omg12 is near pi, estimate work with omg12a = pi - omg12
            real k = Astroid(x, y);
            real omg12a =
                lamscale * (_f >= 0 ? -x * k / (1 + k) : -y * (1 + k) / k);
            somg12 = sin(omg12a);
            comg12 = -cos(omg12a);
            // Update spherical estimate of alp1 using omg12 instead of lam12
            salp1 = cbet2 * somg12;
            calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
        }
    }

    // Sanity check on starting guess.  Backwards check allows NaN through.
    if (!(salp1 <= 0))
    {
        Math::norm(salp1, calp1);
    }
    else
    {
        salp1 = 1;
        calp1 = 0;
    }

    return sig12;
}

Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1, real sbet2,
                              real cbet2, real dn2, real salp1, real calp1,
                              real slam120, real clam120, real &salp2,
                              real &calp2, real &sig12, real &ssig1,
                              real &csig1, real &ssig2, real &csig2, real &eps,
                              real &domg12, bool diffp, real &dlam12,
                              // Scratch area of the right size
                              real Ca[]) const
{
    if (sbet1 == 0 && calp1 == 0)
    // Break degeneracy of equatorial line.  This case has already been
    // handled.
    {
        calp1 = -tiny_;
    }

    real
        // sin(alp1) * cos(bet1) = sin(alp0)
        salp0 = salp1 * cbet1,
        calp0 = Math::hypot(calp1, salp1 * sbet1); // calp0 > 0
    real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
    // tan(bet1) = tan(sig1) * cos(alp1)
    // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
    ssig1 = sbet1;
    somg1 = salp0 * sbet1;
    csig1 = comg1 = calp1 * cbet1;
    Math::norm(ssig1, csig1);
    // Math::norm(somg1, comg1); -- don't need to normalize!
    // Enforce symmetries in the case abs(bet2) = -bet1.  Need to be careful
    // about this case, since this can yield singularities in the Newton
    // iteration.
    // sin(alp2) * cos(bet2) = sin(alp0)
    salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
    // calp2 = sqrt(1 - sq(salp2))
    //       = sqrt(sq(calp0) - sq(sbet2)) / cbet2
    // and subst for calp0 and rearrange to give (choose positive sqrt
    // to give alp2 in [0, pi/2]).
    calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1
                ? sqrt(Math::sq(calp1 * cbet1) +
                       (cbet1 < -sbet1 ? (cbet2 - cbet1) * (cbet1 + cbet2)
                                       : (sbet1 - sbet2) * (sbet1 + sbet2))) /
                      cbet2
                : abs(calp1);
    // tan(bet2) = tan(sig2) * cos(alp2)
    // tan(omg2) = sin(alp0) * tan(sig2).
    ssig2 = sbet2;
    somg2 = salp0 * sbet2;
    csig2 = comg2 = calp2 * cbet2;
    Math::norm(ssig2, csig2);
    // Math::norm(somg2, comg2); -- don't need to normalize!
    // sig12 = sig2 - sig1, limit to [0, pi]
    sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
                  csig1 * csig2 + ssig1 * ssig2);
    // omg12 = omg2 - omg1, limit to [0, pi]
    somg12 = max(real(0), comg1 * somg2 - somg1 * comg2);
    comg12 = comg1 * comg2 + somg1 * somg2;
    // eta = omg12 - lam120
    real eta = atan2(somg12 * clam120 - comg12 * slam120,
                     comg12 * clam120 + somg12 * slam120);
    real B312;
    real k2 = Math::sq(calp0) * _ep2;
    eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
    C3f(eps, Ca);
    B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_ - 1) -
            SinCosSeries(true, ssig1, csig1, Ca, nC3_ - 1));
    domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
    lam12 = eta + domg12;

    if (diffp)
    {
        if (calp2 == 0)
        {
            dlam12 = -2 * _f1 * dn1 / sbet1;
        }
        else
        {
            real dummy;
            Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1,
                    cbet2, REDUCEDLENGTH, dummy, dlam12, dummy, dummy, dummy,
                    Ca);
            dlam12 *= _f1 / (calp2 * cbet2);
        }
    }

    return lam12;
}

Math::real Geodesic::A3f(real eps) const
{
    // Evaluate A3
    return Math::polyval(nA3_ - 1, _A3x, eps);
}

void Geodesic::C3f(real eps, real c[]) const
{
    // Evaluate C3 coeffs
    // Elements c[1] thru c[nC3_ - 1] are set
    real mult = 1;
    int o = 0;

    for (int l = 1; l < nC3_; ++l) // l is index of C3[l]
    {
        int m = nC3_ - l - 1; // order of polynomial in eps
        mult *= eps;
        c[l] = mult * Math::polyval(m, _C3x + o, eps);
        o += m + 1;
    }

    // Post condition: o == nC3x_
}

void Geodesic::C4f(real eps, real c[]) const
{
    // Evaluate C4 coeffs
    // Elements c[0] thru c[nC4_ - 1] are set
    real mult = 1;
    int o = 0;

    for (int l = 0; l < nC4_; ++l) // l is index of C4[l]
    {
        int m = nC4_ - l - 1; // order of polynomial in eps
        c[l] = mult * Math::polyval(m, _C4x + o, eps);
        o += m + 1;
        mult *= eps;
    }

    // Post condition: o == nC4x_
}

// The static const coefficient arrays in the following functions are
// generated by Maxima and give the coefficients of the Taylor expansions for
// the geodesics.  The convention on the order of these coefficients is as
// follows:
//
//   ascending order in the trigonometric expansion,
//   then powers of eps in descending order,
//   finally powers of n in descending order.
//
// (For some expansions, only a subset of levels occur.)  For each polynomial
// of order n at the lowest level, the (n+1) coefficients of the polynomial
// are followed by a divisor which is applied to the whole polynomial.  In
// this way, the coefficients are expressible with no round off error.  The
// sizes of the coefficient arrays are:
//
//   A1m1f, A2m1f            = floor(N/2) + 2
//   C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
//   C3coeff       = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
//   C4coeff       = N * (N + 1) * (N + 5) / 6
//
// where N = GEOGRAPHICLIB_GEODESIC_ORDER
//         = nA1 = nA2 = nC1 = nC1p = nA3 = nC4

// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
Math::real Geodesic::A1m1f(real eps)
{
    static const real coeff[] = {
        // (1-eps)*A1-1, polynomial in eps2 of order 3
        1, 4, 64, 0, 256,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(sizeof(coeff) / sizeof(real) == nA1_ / 2 + 2,
                                "Coefficient array size mismatch in A1m1f");
    int m = nA1_ / 2;
    real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
    return (t + eps) / (1 - eps);
}

// The coefficients C1[l] in the Fourier expansion of B1
void Geodesic::C1f(real eps, real c[])
{
    static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 2
        -1,
        6,
        -16,
        32,
        // C1[2]/eps^2, polynomial in eps2 of order 2
        -9,
        64,
        -128,
        2048,
        // C1[3]/eps^3, polynomial in eps2 of order 1
        9,
        -16,
        768,
        // C1[4]/eps^4, polynomial in eps2 of order 1
        3,
        -5,
        512,
        // C1[5]/eps^5, polynomial in eps2 of order 0
        -7,
        1280,
        // C1[6]/eps^6, polynomial in eps2 of order 0
        -7,
        2048,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) ==
            (nC1_ * nC1_ + 7 * nC1_ - 2 * (nC1_ / 2)) / 4,
        "Coefficient array size mismatch in C1f");
    real eps2 = Math::sq(eps), d = eps;
    int o = 0;

    for (int l = 1; l <= nC1_; ++l) // l is index of C1p[l]
    {
        int m = (nC1_ - l) / 2; // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
    }

    // Post condition: o == sizeof(coeff) / sizeof(real)
}

// The coefficients C1p[l] in the Fourier expansion of B1p
void Geodesic::C1pf(real eps, real c[])
{
    static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 2
        205,
        -432,
        768,
        1536,
        // C1p[2]/eps^2, polynomial in eps2 of order 2
        4005,
        -4736,
        3840,
        12288,
        // C1p[3]/eps^3, polynomial in eps2 of order 1
        -225,
        116,
        384,
        // C1p[4]/eps^4, polynomial in eps2 of order 1
        -7173,
        2695,
        7680,
        // C1p[5]/eps^5, polynomial in eps2 of order 0
        3467,
        7680,
        // C1p[6]/eps^6, polynomial in eps2 of order 0
        38081,
        61440,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) ==
            (nC1p_ * nC1p_ + 7 * nC1p_ - 2 * (nC1p_ / 2)) / 4,
        "Coefficient array size mismatch in C1pf");
    real eps2 = Math::sq(eps), d = eps;
    int o = 0;

    for (int l = 1; l <= nC1p_; ++l) // l is index of C1p[l]
    {
        int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
    }

    // Post condition: o == sizeof(coeff) / sizeof(real)
}

// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
Math::real Geodesic::A2m1f(real eps)
{
    static const real coeff[] = {
        // (eps+1)*A2-1, polynomial in eps2 of order 3
        -11, -28, -192, 0, 256,
    }; // count = 5
    GEOGRAPHICLIB_STATIC_ASSERT(sizeof(coeff) / sizeof(real) == nA2_ / 2 + 2,
                                "Coefficient array size mismatch in A2m1f");
    int m = nA2_ / 2;
    real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
    return (t - eps) / (1 + eps);
}

// The coefficients C2[l] in the Fourier expansion of B2
void Geodesic::C2f(real eps, real c[])
{
    static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 2
        1,
        2,
        16,
        32,
        // C2[2]/eps^2, polynomial in eps2 of order 2
        35,
        64,
        384,
        2048,
        // C2[3]/eps^3, polynomial in eps2 of order 1
        15,
        80,
        768,
        // C2[4]/eps^4, polynomial in eps2 of order 1
        7,
        35,
        512,
        // C2[5]/eps^5, polynomial in eps2 of order 0
        63,
        1280,
        // C2[6]/eps^6, polynomial in eps2 of order 0
        77,
        2048,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) ==
            (nC2_ * nC2_ + 7 * nC2_ - 2 * (nC2_ / 2)) / 4,
        "Coefficient array size mismatch in C2f");
    real eps2 = Math::sq(eps), d = eps;
    int o = 0;

    for (int l = 1; l <= nC2_; ++l) // l is index of C2[l]
    {
        int m = (nC2_ - l) / 2; // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
    }

    // Post condition: o == sizeof(coeff) / sizeof(real)
}

// The scale factor A3 = mean value of (d/dsigma)I3
void Geodesic::A3coeff()
{
    static const real coeff[] = {
        // A3, coeff of eps^5, polynomial in n of order 0
        -3,
        128,
        // A3, coeff of eps^4, polynomial in n of order 1
        -2,
        -3,
        64,
        // A3, coeff of eps^3, polynomial in n of order 2
        -1,
        -3,
        -1,
        16,
        // A3, coeff of eps^2, polynomial in n of order 2
        3,
        -1,
        -2,
        8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1,
        -1,
        2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1,
        1,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) ==
            (nA3_ * nA3_ + 7 * nA3_ - 2 * (nA3_ / 2)) / 4,
        "Coefficient array size mismatch in A3f");
    int o = 0, k = 0;

    for (int j = nA3_ - 1; j >= 0; --j) // coeff of eps^j
    {
        int m = min(nA3_ - j - 1, j); // order of polynomial in n
        _A3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
        o += m + 2;
    }

    // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
}

// The coefficients C3[l] in the Fourier expansion of B3
void Geodesic::C3coeff()
{
    static const real coeff[] = {
        // C3[1], coeff of eps^5, polynomial in n of order 0
        3,
        128,
        // C3[1], coeff of eps^4, polynomial in n of order 1
        2,
        5,
        128,
        // C3[1], coeff of eps^3, polynomial in n of order 2
        -1,
        3,
        3,
        64,
        // C3[1], coeff of eps^2, polynomial in n of order 2
        -1,
        0,
        1,
        8,
        // C3[1], coeff of eps^1, polynomial in n of order 1
        -1,
        1,
        4,
        // C3[2], coeff of eps^5, polynomial in n of order 0
        5,
        256,
        // C3[2], coeff of eps^4, polynomial in n of order 1
        1,
        3,
        128,
        // C3[2], coeff of eps^3, polynomial in n of order 2
        -3,
        -2,
        3,
        64,
        // C3[2], coeff of eps^2, polynomial in n of order 2
        1,
        -3,
        2,
        32,
        // C3[3], coeff of eps^5, polynomial in n of order 0
        7,
        512,
        // C3[3], coeff of eps^4, polynomial in n of order 1
        -10,
        9,
        384,
        // C3[3], coeff of eps^3, polynomial in n of order 2
        5,
        -9,
        5,
        192,
        // C3[4], coeff of eps^5, polynomial in n of order 0
        7,
        512,
        // C3[4], coeff of eps^4, polynomial in n of order 1
        -14,
        7,
        512,
        // C3[5], coeff of eps^5, polynomial in n of order 0
        21,
        2560,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) ==
            ((nC3_ - 1) * (nC3_ * nC3_ + 7 * nC3_ - 2 * (nC3_ / 2))) / 8,
        "Coefficient array size mismatch in C3coeff");
    int o = 0, k = 0;

    for (int l = 1; l < nC3_; ++l) // l is index of C3[l]
    {
        for (int j = nC3_ - 1; j >= l; --j) // coeff of eps^j
        {
            int m = min(nC3_ - j - 1, j); // order of polynomial in n
            _C3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
            o += m + 2;
        }
    }

    // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
}

void Geodesic::C4coeff()
{
    // Generated by Maxima on 2015-05-05 18:08:13-04:00
    static const real coeff[] = {
        // C4[0], coeff of eps^5, polynomial in n of order 0
        97,
        15015,
        // C4[0], coeff of eps^4, polynomial in n of order 1
        1088,
        156,
        45045,
        // C4[0], coeff of eps^3, polynomial in n of order 2
        -224,
        -4784,
        1573,
        45045,
        // C4[0], coeff of eps^2, polynomial in n of order 3
        -10656,
        14144,
        -4576,
        -858,
        45045,
        // C4[0], coeff of eps^1, polynomial in n of order 4
        64,
        624,
        -4576,
        6864,
        -3003,
        15015,
        // C4[0], coeff of eps^0, polynomial in n of order 5
        100,
        208,
        572,
        3432,
        -12012,
        30030,
        45045,
        // C4[1], coeff of eps^5, polynomial in n of order 0
        1,
        9009,
        // C4[1], coeff of eps^4, polynomial in n of order 1
        -2944,
        468,
        135135,
        // C4[1], coeff of eps^3, polynomial in n of order 2
        5792,
        1040,
        -1287,
        135135,
        // C4[1], coeff of eps^2, polynomial in n of order 3
        5952,
        -11648,
        9152,
        -2574,
        135135,
        // C4[1], coeff of eps^1, polynomial in n of order 4
        -64,
        -624,
        4576,
        -6864,
        3003,
        135135,
        // C4[2], coeff of eps^5, polynomial in n of order 0
        8,
        10725,
        // C4[2], coeff of eps^4, polynomial in n of order 1
        1856,
        -936,
        225225,
        // C4[2], coeff of eps^3, polynomial in n of order 2
        -8448,
        4992,
        -1144,
        225225,
        // C4[2], coeff of eps^2, polynomial in n of order 3
        -1440,
        4160,
        -4576,
        1716,
        225225,
        // C4[3], coeff of eps^5, polynomial in n of order 0
        -136,
        63063,
        // C4[3], coeff of eps^4, polynomial in n of order 1
        1024,
        -208,
        105105,
        // C4[3], coeff of eps^3, polynomial in n of order 2
        3584,
        -3328,
        1144,
        315315,
        // C4[4], coeff of eps^5, polynomial in n of order 0
        -128,
        135135,
        // C4[4], coeff of eps^4, polynomial in n of order 1
        -2560,
        832,
        405405,
        // C4[5], coeff of eps^5, polynomial in n of order 0
        128,
        99099,
    };
    GEOGRAPHICLIB_STATIC_ASSERT(
        sizeof(coeff) / sizeof(real) == (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
        "Coefficient array size mismatch in C4coeff");
    int o = 0, k = 0;

    for (int l = 0; l < nC4_; ++l) // l is index of C4[l]
    {
        for (int j = nC4_ - 1; j >= l; --j) // coeff of eps^j
        {
            int m = nC4_ - j - 1; // order of polynomial in n
            _C4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
            o += m + 2;
        }
    }

    // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
}

} // namespace GeographicLib