/**
* Return the inverse of healpix_sphere().
**/
LP healpix_sphere_inverse(XY xy) {
LP lp;
double x = xy.x;
double y = xy.y;
double y0 = M_PI/4.0;
/* Equatorial region. */
if (fabsl(y) <= y0) {
lp.lam = x;
lp.phi = asin(8.0*y/(3.0*M_PI));
} else if (fabsl(y) < M_PI/2.0) {
double cn = floor(2.0*x/M_PI + 2.0);
double xc, tau;
if (cn >= 4) {
cn = 3;
}
xc = -3.0*M_PI/4.0 + (M_PI/2.0)*cn;
tau = 2.0 - 4.0*fabsl(y)/M_PI;
lp.lam = xc + (x - xc)/tau;
lp.phi = pj_sign(y)*asin(1.0 - pow(tau , 2.0)/3.0);
} else {
lp.lam = -1.0*M_PI;
lp.phi = pj_sign(y)*M_PI/2.0;
}
return (lp);
}
inline void healpix_sphere_inverse(T const& xy_x, T const& xy_y, T& lp_lam, T& lp_phi)
{
static const T pi = detail::pi<T>();
static const T half_pi = detail::half_pi<T>();
static const T fourth_pi = detail::fourth_pi<T>();
T x = xy_x;
T y = xy_y;
T y0 = fourth_pi;
/* Equatorial region. */
if (math::abs(y) <= y0) {
lp_lam = x;
lp_phi = asin(8.0*y/(3.0*pi));
} else if (fabsl(y) < half_pi) {
T cn = floor(2.0*x/pi + 2.0);
T xc, tau;
if (cn >= 4) {
cn = 3;
}
xc = -3.0*fourth_pi + (half_pi)*cn;
tau = 2.0 - 4.0*fabsl(y)/pi;
lp_lam = xc + (x - xc)/tau;
lp_phi = pj_sign(y)*asin(1.0 - math::pow(tau, 2)/3.0);
} else {
lp_lam = -1.0*pi;
lp_phi = pj_sign(y)*half_pi;
}
return;
}
inline void healpix_sphere(T const& lp_lam, T const& lp_phi, T& xy_x, T& xy_y)
{
static const T pi = detail::pi<T>();
static const T half_pi = detail::half_pi<T>();
static const T fourth_pi = detail::fourth_pi<T>();
T lam = lp_lam;
T phi = lp_phi;
T phi0 = asin(T(2.0)/T(3.0));
/* equatorial region */
if ( fabsl(phi) <= phi0) {
xy_x = lam;
xy_y = 3.0*pi/8.0*sin(phi);
} else {
T lamc;
T sigma = sqrt(3.0*(1 - math::abs(sin(phi))));
T cn = floor(2*lam / pi + 2);
if (cn >= 4) {
cn = 3;
}
lamc = -3*fourth_pi + half_pi*cn;
xy_x = lamc + (lam - lamc)*sigma;
xy_y = pj_sign(phi)*fourth_pi*(2 - sigma);
}
return;
}
/**
* 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);
}
}
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);
}
}
/**
* Return the HEALPix projection of the longitude-latitude point lp on
* the unit sphere.
**/
XY healpix_sphere(LP lp) {
double lam = lp.lam;
double phi = lp.phi;
double phi0 = asin(2.0/3.0);
XY xy;
/* equatorial region */
if ( fabsl(phi) <= phi0) {
xy.x = lam;
xy.y = 3.0*PI/8.0*sin(phi);
} else {
double lamc;
double sigma = sqrt(3.0*(1 - fabsl(sin(phi))));
double cn = floor(2*lam / PI + 2);
if (cn >= 4) {
cn = 3;
}
lamc = -3*PI/4 + (PI/2)*cn;
xy.x = lamc + (lam - lamc)*sigma;
xy.y = pj_sign(phi)*PI/4*(2 - sigma);
}
return xy;
}