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#define PJ_LIB__
#include <errno.h>
#include "proj.h"
#include "proj_internal.h"
#include <math.h>
PROJ_HEAD(ortho, "Orthographic") "\n\tAzi, Sph&Ell";
namespace { // anonymous namespace
enum Mode {
N_POLE = 0,
S_POLE = 1,
EQUIT = 2,
OBLIQ = 3
};
} // anonymous namespace
namespace { // anonymous namespace
struct pj_opaque {
double sinph0;
double cosph0;
double nu0;
double y_shift;
double y_scale;
enum Mode mode;
};
} // anonymous namespace
#define EPS10 1.e-10
static PJ_XY forward_error(PJ *P, PJ_LP lp, PJ_XY xy) {
proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
proj_log_trace(P, "Coordinate (%.3f, %.3f) is on the unprojected hemisphere",
proj_todeg(lp.lam), proj_todeg(lp.phi));
return xy;
}
static PJ_XY ortho_s_forward (PJ_LP lp, PJ *P) { /* Spheroidal, forward */
PJ_XY xy;
struct pj_opaque *Q = static_cast<struct pj_opaque*>(P->opaque);
double coslam, cosphi, sinphi;
xy.x = HUGE_VAL; xy.y = HUGE_VAL;
cosphi = cos(lp.phi);
coslam = cos(lp.lam);
switch (Q->mode) {
case EQUIT:
if (cosphi * coslam < - EPS10)
return forward_error(P, lp, xy);
xy.y = sin(lp.phi);
break;
case OBLIQ:
sinphi = sin(lp.phi);
// Is the point visible from the projection plane ?
// From https://lists.osgeo.org/pipermail/proj/2020-September/009831.html
// this is the dot product of the normal of the ellipsoid at the center of
// the projection and at the point considered for projection.
// [cos(phi)*cos(lambda), cos(phi)*sin(lambda), sin(phi)]
// Also from Snyder's Map Projection - A working manual, equation (5-3), page 149
if (Q->sinph0 * sinphi + Q->cosph0 * cosphi * coslam < - EPS10)
return forward_error(P, lp, xy);
xy.y = Q->cosph0 * sinphi - Q->sinph0 * cosphi * coslam;
break;
case N_POLE:
coslam = - coslam;
/*-fallthrough*/
case S_POLE:
if (fabs(lp.phi - P->phi0) - EPS10 > M_HALFPI)
return forward_error(P, lp, xy);
xy.y = cosphi * coslam;
break;
}
xy.x = cosphi * sin(lp.lam);
return xy;
}
static PJ_LP ortho_s_inverse (PJ_XY xy, PJ *P) { /* Spheroidal, inverse */
PJ_LP lp;
struct pj_opaque *Q = static_cast<struct pj_opaque*>(P->opaque);
double sinc;
lp.lam = HUGE_VAL; lp.phi = HUGE_VAL;
const double rh = hypot(xy.x, xy.y);
sinc = rh;
if (sinc > 1.) {
if ((sinc - 1.) > EPS10) {
proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
return lp;
}
sinc = 1.;
}
const double cosc = sqrt(1. - sinc * sinc); /* in this range OK */
if (fabs(rh) <= EPS10) {
lp.phi = P->phi0;
lp.lam = 0.0;
} else {
switch (Q->mode) {
case N_POLE:
xy.y = -xy.y;
lp.phi = acos(sinc);
break;
case S_POLE:
lp.phi = - acos(sinc);
break;
case EQUIT:
lp.phi = xy.y * sinc / rh;
xy.x *= sinc;
xy.y = cosc * rh;
goto sinchk;
case OBLIQ:
lp.phi = cosc * Q->sinph0 + xy.y * sinc * Q->cosph0 /rh;
xy.y = (cosc - Q->sinph0 * lp.phi) * rh;
xy.x *= sinc * Q->cosph0;
sinchk:
if (fabs(lp.phi) >= 1.)
lp.phi = lp.phi < 0. ? -M_HALFPI : M_HALFPI;
else
lp.phi = asin(lp.phi);
break;
}
lp.lam = (xy.y == 0. && (Q->mode == OBLIQ || Q->mode == EQUIT))
? (xy.x == 0. ? 0. : xy.x < 0. ? -M_HALFPI : M_HALFPI)
: atan2(xy.x, xy.y);
}
return lp;
}
static PJ_XY ortho_e_forward (PJ_LP lp, PJ *P) { /* Ellipsoidal, forward */
PJ_XY xy;
struct pj_opaque *Q = static_cast<struct pj_opaque*>(P->opaque);
// From EPSG guidance note 7.2, March 2020, §3.3.5 Orthographic
const double cosphi = cos(lp.phi);
const double sinphi = sin(lp.phi);
const double coslam = cos(lp.lam);
const double sinlam = sin(lp.lam);
// Is the point visible from the projection plane ?
// Same condition as in spherical case
if (Q->sinph0 * sinphi + Q->cosph0 * cosphi * coslam < - EPS10) {
xy.x = HUGE_VAL; xy.y = HUGE_VAL;
return forward_error(P, lp, xy);
}
const double nu = 1.0 / sqrt(1.0 - P->es * sinphi * sinphi);
xy.x = nu * cosphi * sinlam;
xy.y = nu * (sinphi * Q->cosph0 - cosphi * Q->sinph0 * coslam) +
P->es * (Q->nu0 * Q->sinph0 - nu * sinphi) * Q->cosph0;
return xy;
}
static PJ_LP ortho_e_inverse (PJ_XY xy, PJ *P) { /* Ellipsoidal, inverse */
PJ_LP lp;
struct pj_opaque *Q = static_cast<struct pj_opaque*>(P->opaque);
const auto SQ = [](double x) { return x*x; };
if( Q->mode == N_POLE || Q->mode == S_POLE )
{
// Polar case. Forward case equations can be simplified as:
// xy.x = nu * cosphi * sinlam
// xy.y = nu * -cosphi * coslam * sign(phi0)
// ==> lam = atan2(xy.x, -xy.y * sign(phi0))
// ==> xy.x^2 + xy.y^2 = nu^2 * cosphi^2
// rh^2 = cosphi^2 / (1 - es * sinphi^2)
// ==> cosphi^2 = rh^2 * (1 - es) / (1 - es * rh^2)
const double rh2 = SQ(xy.x) + SQ(xy.y);
if (rh2 >= 1. - 1e-15) {
if ((rh2 - 1.) > EPS10) {
proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
lp.lam = HUGE_VAL; lp.phi = HUGE_VAL;
return lp;
}
lp.phi = 0;
}
else {
lp.phi = acos(sqrt(rh2 * P->one_es / (1 - P->es * rh2))) * (Q->mode == N_POLE ? 1 : -1);
}
lp.lam = atan2(xy.x, xy.y * (Q->mode == N_POLE ? -1 : 1));
return lp;
}
if( Q->mode == EQUIT )
{
// Equatorial case. Forward case equations can be simplified as:
// xy.x = nu * cosphi * sinlam
// xy.y = nu * sinphi * (1 - P->es)
// x^2 * (1 - es * sinphi^2) = (1 - sinphi^2) * sinlam^2
// y^2 / ((1 - es)^2 + y^2 * es) = sinphi^2
// Equation of the ellipse
if( SQ(xy.x) + SQ(xy.y * (P->a / P->b)) > 1 + 1e-11 ) {
proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
lp.lam = HUGE_VAL; lp.phi = HUGE_VAL;
return lp;
}
const double sinphi2 = xy.y == 0 ? 0 : 1.0 / (SQ((1 - P->es) / xy.y) + P->es);
if( sinphi2 > 1 - 1e-11 ) {
lp.phi = M_PI_2 * (xy.y > 0 ? 1 : -1);
lp.lam = 0;
return lp;
}
lp.phi = asin(sqrt(sinphi2)) * (xy.y > 0 ? 1 : -1);
const double sinlam = xy.x * sqrt((1 - P->es * sinphi2) / (1 - sinphi2));
if( fabs(sinlam) - 1 > -1e-15 )
lp.lam = M_PI_2 * (xy.x > 0 ? 1: -1);
else
lp.lam = asin(sinlam);
return lp;
}
// Using Q->sinph0 * sinphi + Q->cosph0 * cosphi * coslam == 0 (visibity
// condition of the forward case) in the forward equations, and a lot of
// substitution games...
PJ_XY xy_recentered;
xy_recentered.x = xy.x;
xy_recentered.y = (xy.y - Q->y_shift) / Q->y_scale;
if( SQ(xy.x) + SQ(xy_recentered.y) > 1 + 1e-11 ) {
proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
lp.lam = HUGE_VAL; lp.phi = HUGE_VAL;
return lp;
}
// From EPSG guidance note 7.2, March 2020, §3.3.5 Orthographic
// It suggests as initial guess:
// lp.lam = 0;
// lp.phi = P->phi0;
// But for poles, this will not converge well. Better use:
lp = ortho_s_inverse(xy_recentered, P);
for( int i = 0; i < 20; i++ )
{
const double cosphi = cos(lp.phi);
const double sinphi = sin(lp.phi);
const double coslam = cos(lp.lam);
const double sinlam = sin(lp.lam);
const double one_minus_es_sinphi2 = 1.0 - P->es * sinphi * sinphi;
const double nu = 1.0 / sqrt(one_minus_es_sinphi2);
PJ_XY xy_new;
xy_new.x = nu * cosphi * sinlam;
xy_new.y = nu * (sinphi * Q->cosph0 - cosphi * Q->sinph0 * coslam) +
P->es * (Q->nu0 * Q->sinph0 - nu * sinphi) * Q->cosph0;
const double rho = (1.0 - P->es) * nu / one_minus_es_sinphi2;
const double J11 = -rho * sinphi * sinlam;
const double J12 = nu * cosphi * coslam;
const double J21 = rho * (cosphi * Q->cosph0 + sinphi * Q->sinph0 * coslam);
const double J22 = nu * Q->sinph0 * Q->cosph0 * sinlam;
const double D = J11 * J22 - J12 * J21;
const double dx = xy.x - xy_new.x;
const double dy = xy.y - xy_new.y;
const double dphi = (J22 * dx - J12 * dy) / D;
const double dlam = (-J21 * dx + J11 * dy) / D;
lp.phi += dphi;
if( lp.phi > M_PI_2) lp.phi = M_PI_2;
else if( lp.phi < -M_PI_2) lp.phi = -M_PI_2;
lp.lam += dlam;
if( fabs(dphi) < 1e-12 && fabs(dlam) < 1e-12 )
{
return lp;
}
}
proj_context_errno_set(P->ctx, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
return lp;
}
PJ *PROJECTION(ortho) {
struct pj_opaque *Q = static_cast<struct pj_opaque*>(calloc (1, sizeof (struct pj_opaque)));
if (nullptr==Q)
return pj_default_destructor(P, PROJ_ERR_OTHER /*ENOMEM*/);
P->opaque = Q;
Q->sinph0 = sin(P->phi0);
Q->cosph0 = cos(P->phi0);
if (fabs(fabs(P->phi0) - M_HALFPI) <= EPS10)
Q->mode = P->phi0 < 0. ? S_POLE : N_POLE;
else if (fabs(P->phi0) > EPS10) {
Q->mode = OBLIQ;
} else
Q->mode = EQUIT;
if( P->es == 0 )
{
P->inv = ortho_s_inverse;
P->fwd = ortho_s_forward;
}
else
{
Q->nu0 = 1.0 / sqrt(1.0 - P->es * Q->sinph0 * Q->sinph0);
Q->y_shift = P->es * Q->nu0 * Q->sinph0 * Q->cosph0;
Q->y_scale = 1.0 / sqrt(1.0 - P->es * Q->cosph0 * Q->cosph0);
P->inv = ortho_e_inverse;
P->fwd = ortho_e_forward;
}
return P;
}
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