Merge pull request #2361 from rouault/ortho_ellipsoidal

Implement ellipsoidal formulation of +proj=ortho (fixes #397)

1 files changed, 172 insertions, 6 deletions

diff --git a/src/projections/ortho.cpp b/src/projections/ortho.cpp
index 5f9366c3..75b8199d 100644
--- a/src/projections/ortho.cpp
+++ b/src/projections/ortho.cpp
@@ -4,7 +4,7 @@
 #include "proj_internal.h"
 #include <math.h>
 
-PROJ_HEAD(ortho, "Orthographic") "\n\tAzi, Sph";
+PROJ_HEAD(ortho, "Orthographic") "\n\tAzi, Sph&Ell";
 
 namespace { // anonymous namespace
 enum Mode {
@@ -19,6 +19,9 @@ namespace { // anonymous namespace
 struct pj_opaque {
     double  sinph0;
     double  cosph0;
+    double  nu0;
+    double  y_shift;
+    double  y_scale;
     enum Mode mode;
 };
 } // anonymous namespace
@@ -49,6 +52,13 @@ static PJ_XY ortho_s_forward (PJ_LP lp, PJ *P) {           /* Spheroidal, forwar
         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;
@@ -121,6 +131,152 @@ static PJ_LP ortho_s_inverse (PJ_XY xy, PJ *P) {           /* Spheroidal, invers
 }
 
 
+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, PJD_ERR_TOLERANCE_CONDITION);
+                proj_log_trace(P, "Point (%.3f, %.3f) is outside the projection boundary");
+                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, PJD_ERR_TOLERANCE_CONDITION);
+            proj_log_trace(P, "Point (%.3f, %.3f) is outside the projection boundary");
+            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
+    // substition 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, PJD_ERR_TOLERANCE_CONDITION);
+        proj_log_trace(P, "Point (%.3f, %.3f) is outside the projection boundary");
+        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;
+        }
+    }
+    pj_ctx_set_errno(P->ctx, PJD_ERR_NON_CONVERGENT);
+    return lp;
+}
+
 
 PJ *PROJECTION(ortho) {
     struct pj_opaque *Q = static_cast<struct pj_opaque*>(pj_calloc (1, sizeof (struct pj_opaque)));
@@ -128,17 +284,27 @@ PJ *PROJECTION(ortho) {
         return pj_default_destructor(P, 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;
-        Q->sinph0 = sin(P->phi0);
-        Q->cosph0 = cos(P->phi0);
     } else
         Q->mode = EQUIT;
-    P->inv = ortho_s_inverse;
-    P->fwd = ortho_s_forward;
-    P->es = 0.;
+    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;
 }