static PJ_LP e_inverse (PJ_XY xy, PJ *P) { /* Ellipsoidal/spheroidal, inverse */
PJ_LP lp = {0.0,0.0};
struct pj_opaque *Q = static_cast<struct pj_opaque*>(P->opaque);
double yc, y2, y6;
int i;
/* Adjusting x and y for authalic radius */
xy.x /= Q->rqda;
xy.y /= Q->rqda;
/* Make sure y is inside valid range */
if (xy.y > MAX_Y)
xy.y = MAX_Y;
else if (xy.y < -MAX_Y)
xy.y = -MAX_Y;
yc = xy.y;
/* Newton-Raphson */
for (i = MAX_ITER; i ; --i) {
double f, fder, tol;
y2 = yc * yc;
y6 = y2 * y2 * y2;
f = yc * (A1 + A2 * y2 + y6 * (A3 + A4 * y2)) - xy.y;
fder = A1 + 3 * A2 * y2 + y6 * (7 * A3 + 9 * A4 * y2);
tol = f / fder;
yc -= tol;
if (fabs(tol) < EPS)
break;
}
if( i == 0 ) {
pj_ctx_set_errno( P->ctx, PJD_ERR_NON_CONVERGENT );
return lp;
}
/* Longitude */
y2 = yc * yc;
y6 = y2 * y2 * y2;
lp.lam = M * xy.x * (A1 + 3 * A2 * y2 + y6 * (7 * A3 + 9 * A4 * y2)) / cos(yc);
/* Latitude (for spheroidal case, this is latitude */
lp.phi = asin(sin(yc) / M);
/* Ellipsoidal case, converting auth. latitude */
if (P->es != 0.0)
lp.phi = pj_authlat(lp.phi, Q->apa);
return lp;
}
/**
* Return the authalic latitude of latitude alpha (if inverse=0) or
* return the approximate latitude of authalic latitude alpha (if inverse=1).
* P contains the relavent ellipsoid parameters.
**/
double auth_lat(PJ *P, double alpha, int inverse) {
if (inverse == 0) {
/* Authalic latitude. */
double q = pj_qsfn(sin(alpha), P->e, 1.0 - P->es);
double qp = P->qp;
double ratio = q/qp;
if (fabsl(ratio) > 1) {
/* Rounding error. */
ratio = pj_sign(ratio);
}
return asin(ratio);
} else {
/* Approximation to inverse authalic latitude. */
return pj_authlat(alpha, P->apa);
}
}
static LP e_inverse (XY xy, PJ *P) { /* Ellipsoidal, inverse */
LP lp = {0.0,0.0};
struct pj_opaque *Q = P->opaque;
double cCe, sCe, q, rho, ab=0.0;
switch (Q->mode) {
case EQUIT:
case OBLIQ:
xy.x /= Q->dd;
xy.y *= Q->dd;
rho = hypot(xy.x, xy.y);
if (rho < EPS10) {
lp.lam = 0.;
lp.phi = P->phi0;
return lp;
}
sCe = 2. * asin(.5 * rho / Q->rq);
cCe = cos(sCe);
sCe = sin(sCe);
xy.x *= sCe;
if (Q->mode == OBLIQ) {
ab = cCe * Q->sinb1 + xy.y * sCe * Q->cosb1 / rho;
xy.y = rho * Q->cosb1 * cCe - xy.y * Q->sinb1 * sCe;
} else {
ab = xy.y * sCe / rho;
xy.y = rho * cCe;
}
break;
case N_POLE:
xy.y = -xy.y;
/*-fallthrough*/
case S_POLE:
q = (xy.x * xy.x + xy.y * xy.y);
if (q == 0.0) {
lp.lam = 0.;
lp.phi = P->phi0;
return (lp);
}
ab = 1. - q / Q->qp;
if (Q->mode == S_POLE)
ab = - ab;
break;
}
lp.lam = atan2(xy.x, xy.y);
lp.phi = pj_authlat(asin(ab), Q->apa);
return lp;
}
inline T auth_lat(const Parameters& par, const par_healpix<T>& proj_parm, T const& alpha, int inverse)
{
if (inverse == 0) {
/* Authalic latitude. */
T q = pj_qsfn(sin(alpha), par.e, 1.0 - par.es);
T qp = proj_parm.qp;
T ratio = q/qp;
if (math::abs(ratio) > 1) {
/* Rounding error. */
ratio = pj_sign(ratio);
}
return asin(ratio);
} else {
/* Approximation to inverse authalic latitude. */
return pj_authlat(alpha, proj_parm.apa);
}
}
inline void inv(cartesian_type& xy_x, cartesian_type& xy_y, geographic_type& lp_lon, geographic_type& lp_lat) const
{
double cCe, sCe, q, rho, ab=0.0;
switch (this->m_proj_parm.mode) {
case EQUIT:
case OBLIQ:
if ((rho = boost::math::hypot(xy_x /= this->m_proj_parm.dd, xy_y *= this->m_proj_parm.dd)) < EPS10) {
lp_lon = 0.;
lp_lat = this->m_par.phi0;
return;
}
cCe = cos(sCe = 2. * asin(.5 * rho / this->m_proj_parm.rq));
xy_x *= (sCe = sin(sCe));
if (this->m_proj_parm.mode == OBLIQ) {
q = this->m_proj_parm.qp * (ab = cCe * this->m_proj_parm.sinb1 + xy_y * sCe * this->m_proj_parm.cosb1 / rho);
xy_y = rho * this->m_proj_parm.cosb1 * cCe - xy_y * this->m_proj_parm.sinb1 * sCe;
} else {
q = this->m_proj_parm.qp * (ab = xy_y * sCe / rho);
xy_y = rho * cCe;
}
break;
case N_POLE:
xy_y = -xy_y;
case S_POLE:
if (!(q = (xy_x * xy_x + xy_y * xy_y)) ) {
lp_lon = 0.;
lp_lat = this->m_par.phi0;
return;
}
/*
q = this->m_proj_parm.qp - q;
*/
ab = 1. - q / this->m_proj_parm.qp;
if (this->m_proj_parm.mode == S_POLE)
ab = - ab;
break;
}
lp_lon = atan2(xy_x, xy_y);
lp_lat = pj_authlat(asin(ab), this->m_proj_parm.apa);
}
inline void inv(cartesian_type& xy_x, cartesian_type& xy_y, geographic_type& lp_lon, geographic_type& lp_lat) const
{
lp_lat = pj_authlat(asin( 2. * xy_y * this->m_par.k0 / this->m_proj_parm.qp), this->m_proj_parm.apa);
lp_lon = xy_x / this->m_par.k0;
}