: _a(a)

, _f(f <= 1 ? f : 1/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 :

(_e2 > 0 ? Math::atanh(sqrt(_e2)) : atan(sqrt(-_e2))) /

sqrt(abs(_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))

throw GeographicErr("Major radius is not positive");

if (!(Math::isfinite(_b) && _b > 0))

throw GeographicErr("Minor radius is not positive");

A3coeff();

C3coeff();

C4coeff();

real sinx, real cosx,

const real c[], int n) throw() {

// 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)

bool arcmode, real s12_a12, unsigned outmask,

real& lat2, real& lon2, real& azi2,

real& s12, real& m12, real& M12, real& M21,

real& S12) const throw() {

return GeodesicLine(*this, lat1, lon1, azi1,

// Automatically supply DISTANCE_IN if necessary

outmask | (arcmode ? NONE : DISTANCE_IN))

. // Note the dot!

GenPosition(arcmode, s12_a12, outmask,

lat2, lon2, azi2, s12, m12, M12, M21, S12);

unsigned outmask,

real& s12, real& azi1, real& azi2,

real& m12, real& M12, real& M21, real& S12)

const throw() {

outmask &= OUT_ALL;

// 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 lon12 = Math::AngDiff(Math::AngNormalize(lon1),

Math::AngNormalize(lon2));

// If very close to being on the same half-meridian, then make it so.

lon12 = AngRound(lon12);

// Make longitude difference positive.

int lonsign = lon12 >= 0 ? 1 : -1;

lon12 *= lonsign;

// If really close to the equator, treat as on equator.

lat1 = AngRound(lat1);

lat2 = AngRound(lat2);

// Swap points so that point with higher (abs) latitude is point 1

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 phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x;

phi = lat1 * Math::degree<real>();

// Ensure cbet1 = +epsilon at poles

sbet1 = _f1 * sin(phi);

cbet1 = lat1 == -90 ? tiny_ : cos(phi);

SinCosNorm(sbet1, cbet1);

phi = lat2 * Math::degree<real>();

// Ensure cbet2 = +epsilon at poles

sbet2 = _f1 * sin(phi);

cbet2 = abs(lat2) == 90 ? tiny_ : cos(phi);

SinCosNorm(sbet2, 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

lam12 = lon12 * Math::degree<real>(),

slam12 = abs(lon12) == 180 ? 0 : sin(lam12),

clam12 = cos(lam12); // lon12 == 90 isn't interesting

real a12, sig12, calp1, salp1, calp2, salp2;

// index zero elements of these arrays are unused

real C1a[nC1_ + 1], C2a[nC2_ + 1], C3a[nC3_];

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(csig1 * ssig2 - ssig1 * csig2, real(0)),

csig1 * csig2 + ssig1 * ssig2);

{

real dummy;

Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,

cbet1, cbet2, s12x, m12x, dummy,

(outmask & GEODESICSCALE) != 0U, M12, M21, C1a, C2a);

}

// Add the check for sig12 since zero length geodesics might yield m12 <

// 0. Test case was

//

// echo 20.001 0 20.001 0 | Geod -i

//

// In fact, we will have sig12 > pi/2 for meridional geodesic which is

// not a shortest path.

if (sig12 < 1 || m12x >= 0) {

m12x *= _b;

s12x *= _b;

a12 = sig12 / Math::degree<real>();

} else

// m12 < 0, i.e., prolate and too close to anti-podal

meridian = false;

}

real omg12;

if (!meridian &&

sbet1 == 0 && // and sbet2 == 0

// Mimic the way Lambda12 works with calp1 = 0

(_f <= 0 || lam12 <= Math::pi<real>() - _f * Math::pi<real>())) {

// 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,

salp1, calp1, salp2, calp2, dnm,

C1a, C2a);

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<real>();

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.

real ssig1, csig1, ssig2, csig2, eps;

unsigned numit = 0;

// Bracketing range

real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;

for (bool tripn = false, tripb = false; numit < maxit2_; ++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,

salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,

eps, omg12, numit < maxit1_, dv, C1a, C2a, C3a)

- lam12;

// 2 * tol0 is approximately 1 ulp for a number in [0, pi].

// Reversed test to allow escape with NaNs

if (tripb || !(abs(v) >= (tripn ? 8 : 2) * 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<real>()) {

calp1 = calp1 * cdalp1 - salp1 * sdalp1;

salp1 = nsalp1;

SinCosNorm(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 postive 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;

SinCosNorm(salp1, calp1);

tripn = false;

tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||

abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);

}

{

real dummy;

Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,

cbet1, cbet2, s12x, m12x, dummy,

(outmask & GEODESICSCALE) != 0U, M12, M21, C1a, C2a);

}

m12x *= _b;

s12x *= _b;

a12 = sig12 / Math::degree<real>();

omg12 = lam12 - omg12;

}

}

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;

SinCosNorm(ssig1, csig1);

SinCosNorm(ssig2, csig2);

real C4a[nC4_];

C4f(eps, C4a);

real

B41 = SinCosSeries(false, ssig1, csig1, C4a, nC4_),

B42 = SinCosSeries(false, ssig2, csig2, C4a, nC4_);

S12 = A4 * (B42 - B41);

} else

// Avoid problems with indeterminate sig1, sig2 on equator

S12 = 0;

if (!meridian &&

omg12 < real(0.75) * Math::pi<real>() && // Long difference too big

sbet2 - sbet1 < real(1.75)) { // Lat difference 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

somg12 = sin(omg12), domg12 = 1 + cos(omg12),

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;

if (outmask & AZIMUTH) {

// minus signs give range [-180, 180). 0- converts -0 to +0.

azi1 = 0 - atan2(-salp1, calp1) / Math::degree<real>();

azi2 = 0 - atan2(-salp2, calp2) / Math::degree<real>();

}

// Returned value in [0, 180]

return a12;

real ssig1, real csig1, real dn1,

real ssig2, real csig2, real dn2,

real cbet1, real cbet2,

real& s12b, real& m12b, real& m0,

bool scalep, real& M12, real& M21,

// Scratch areas of the right size

real C1a[], real C2a[]) const throw() {

// Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,

// and m0 = coefficient of secular term in expression for reduced length.

C1f(eps, C1a);

C2f(eps, C2a);

real

A1m1 = A1m1f(eps),

AB1 = (1 + A1m1) * (SinCosSeries(true, ssig2, csig2, C1a, nC1_) -

SinCosSeries(true, ssig1, csig1, C1a, nC1_)),

A2m1 = A2m1f(eps),

AB2 = (1 + A2m1) * (SinCosSeries(true, ssig2, csig2, C2a, nC2_) -

SinCosSeries(true, ssig1, csig1, C2a, nC2_));

m0 = A1m1 - A2m1;

real J12 = m0 * sig12 + (AB1 - AB2);

// 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;

// Missing a factor of _b

s12b = (1 + A1m1) * sig12 + AB1;

if (scalep) {

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;

}

// 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 discrimant 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 != 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) + 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;

real sbet2, real cbet2, real dn2,

real lam12,

real& salp1, real& calp1,

// Only updated if return val >= 0

real& salp2, real& calp2,

// Only updated for short lines

real& dnm,

// Scratch areas of the right size

real C1a[], real C2a[]) const throw() {

// 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;

{

volatile real xx1 = sbet2 * cbet1;

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 omg12 = lam12;

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);

omg12 /= _f1 * dnm;

}

real somg12 = sin(omg12), comg12 = cos(omg12);

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);

SinCosNorm(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<real>() * 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

volatile real x;

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<real>();

}

betscale = lamscale * cbet1;

x = (lam12 - Math::pi<real>()) / 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<real>() + bet12a,

sbet1, -cbet1, dn1, sbet2, cbet2, dn2,

cbet1, cbet2, dummy, m12b, m0, false,

dummy, dummy, C1a, C2a);

x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi<real>());

betscale = x < -real(0.01) ? sbet12a / x :

-_f * Math::sq(cbet1) * Math::pi<real>();

lamscale = betscale / cbet1;

y = (lam12 - Math::pi<real>()) / 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);

}

}

if (salp1 > 0) // Sanity check on starting guess

SinCosNorm(salp1, calp1);

else {

salp1 = 1; calp1 = 0;

}

return sig12;

real sbet2, real cbet2, real dn2,

real salp1, real calp1,

real& salp2, real& calp2,

real& sig12,

real& ssig1, real& csig1,

real& ssig2, real& csig2,

real& eps, real& domg12,

bool diffp, real& dlam12,

// Scratch areas of the right size

real C1a[], real C2a[], real C3a[]) const

throw() {

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, omg12, 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;

SinCosNorm(ssig1, csig1);

// SinCosNorm(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;

SinCosNorm(ssig2, csig2);

// SinCosNorm(somg2, comg2); -- don't need to normalize!

// sig12 = sig2 - sig1, limit to [0, pi]

sig12 = atan2(max(csig1 * ssig2 - ssig1 * csig2, real(0)),

csig1 * csig2 + ssig1 * ssig2);

// omg12 = omg2 - omg1, limit to [0, pi]

omg12 = atan2(max(comg1 * somg2 - somg1 * comg2, real(0)),

comg1 * comg2 + somg1 * somg2);

real B312, h0;

real k2 = Math::sq(calp0) * _ep2;

eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);

C3f(eps, C3a);

B312 = (SinCosSeries(true, ssig2, csig2, C3a, nC3_-1) -

SinCosSeries(true, ssig1, csig1, C3a, nC3_-1));

h0 = -_f * A3f(eps);

domg12 = salp0 * h0 * (sig12 + B312);

lam12 = omg12 + 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, dummy, dlam12, dummy,

false, dummy, dummy, C1a, C2a);

dlam12 *= _f1 / (calp2 * cbet2);

}

}

return lam12;

// Evaluate sum(_A3x[k] * eps^k, k, 0, nA3x_-1) by Horner's method

real v = 0;

for (int i = nA3x_; i; )

v = eps * v + _A3x[--i];

return v;

// Evaluate C3 coeffs by Horner's method

// Elements c[1] thru c[nC3_ - 1] are set

for (int j = nC3x_, k = nC3_ - 1; k; ) {

real t = 0;

for (int i = nC3_ - k; i; --i)

t = eps * t + _C3x[--j];

c[k--] = t;

}

real mult = 1;

for (int k = 1; k < nC3_; ) {

mult *= eps;

c[k++] *= mult;

}

// Evaluate C4 coeffs by Horner's method

// Elements c[0] thru c[nC4_ - 1] are set

for (int j = nC4x_, k = nC4_; k; ) {

real t = 0;

for (int i = nC4_ - k + 1; i; --i)

t = eps * t + _C4x[--j];

c[--k] = t;

}

real mult = 1;

for (int k = 1; k < nC4_; ) {

mult *= eps;

c[k++] *= mult;

}

real

eps2 = Math::sq(eps),

t;

switch (nA1_/2) {

case 0:

t = 0;

break;

case 1:

t = eps2/4;

break;

case 2:

t = eps2*(eps2+16)/64;

break;

case 3:

t = eps2*(eps2*(eps2+4)+64)/256;

break;

case 4:

t = eps2*(eps2*(eps2*(25*eps2+64)+256)+4096)/16384;

break;

default:

STATIC_ASSERT(nA1_ >= 0 && nA1_ <= 8, "Bad value of nA1_");

t = 0;

}

return (t + eps) / (1 - eps);

real

eps2 = Math::sq(eps),

d = eps;

switch (nC1_) {

case 0:

break;

case 1:

c[1] = -d/2;

break;

case 2:

c[1] = -d/2;

d *= eps;

c[2] = -d/16;

break;

case 3:

c[1] = d*(3*eps2-8)/16;

d *= eps;

c[2] = -d/16;

d *= eps;

c[3] = -d/48;

break;

case 4:

c[1] = d*(3*eps2-8)/16;

d *= eps;

c[2] = d*(eps2-2)/32;

d *= eps;

c[3] = -d/48;

d *= eps;

c[4] = -5*d/512;

break;

case 5:

c[1] = d*((6-eps2)*eps2-16)/32;

d *= eps;

c[2] = d*(eps2-2)/32;

d *= eps;

c[3] = d*(9*eps2-16)/768;

d *= eps;

c[4] = -5*d/512;

d *= eps;

c[5] = -7*d/1280;

break;

case 6:

c[1] = d*((6-eps2)*eps2-16)/32;

d *= eps;

c[2] = d*((64-9*eps2)*eps2-128)/2048;

d *= eps;

c[3] = d*(9*eps2-16)/768;

d *= eps;

c[4] = d*(3*eps2-5)/512;

d *= eps;

c[5] = -7*d/1280;

d *= eps;

c[6] = -7*d/2048;

break;

case 7:

c[1] = d*(eps2*(eps2*(19*eps2-64)+384)-1024)/2048;

d *= eps;

c[2] = d*((64-9*eps2)*eps2-128)/2048;

d *= eps;

c[3] = d*((72-9*eps2)*eps2-128)/6144;

d *= eps;

c[4] = d*(3*eps2-5)/512;

d *= eps;

c[5] = d*(35*eps2-56)/10240;

d *= eps;

c[6] = -7*d/2048;

d *= eps;

c[7] = -33*d/14336;

break;

case 8:

c[1] = d*(eps2*(eps2*(19*eps2-64)+384)-1024)/2048;

d *= eps;

c[2] = d*(eps2*(eps2*(7*eps2-18)+128)-256)/4096;

d *= eps;

c[3] = d*((72-9*eps2)*eps2-128)/6144;

d *= eps;

c[4] = d*((96-11*eps2)*eps2-160)/16384;

d *= eps;

c[5] = d*(35*eps2-56)/10240;

d *= eps;

c[6] = d*(9*eps2-14)/4096;

d *= eps;

c[7] = -33*d/14336;

d *= eps;

c[8] = -429*d/262144;

break;

default:

STATIC_ASSERT(nC1_ >= 0 && nC1_ <= 8, "Bad value of nC1_");

}

real

eps2 = Math::sq(eps),

d = eps;

switch (nC1p_) {

case 0:

break;

case 1:

c[1] = d/2;

break;

case 2:

c[1] = d/2;

d *= eps;

c[2] = 5*d/16;

break;

case 3:

c[1] = d*(16-9*eps2)/32;

d *= eps;

c[2] = 5*d/16;

d *= eps;

c[3] = 29*d/96;

break;

case 4:

c[1] = d*(16-9*eps2)/32;

d *= eps;

c[2] = d*(30-37*eps2)/96;

d *= eps;

c[3] = 29*d/96;

d *= eps;

c[4] = 539*d/1536;

break;

case 5:

c[1] = d*(eps2*(205*eps2-432)+768)/1536;

d *= eps;

c[2] = d*(30-37*eps2)/96;