/*
* This implements the Quadrilateralized Spherical Cube (QSC) projection.
*
* Copyright (c) 2011, 2012 Martin Lambers <marlam@marlam.de>
*
* The QSC projection was introduced in:
* [OL76]
* E.M. O'Neill and R.E. Laubscher, "Extended Studies of a Quadrilateralized
* Spherical Cube Earth Data Base", Naval Environmental Prediction Research
* Facility Tech. Report NEPRF 3-76 (CSC), May 1976.
*
* The preceding shift from an ellipsoid to a sphere, which allows to apply
* this projection to ellipsoids as used in the Ellipsoidal Cube Map model,
* is described in
* [LK12]
* M. Lambers and A. Kolb, "Ellipsoidal Cube Maps for Accurate Rendering of
* Planetary-Scale Terrain Data", Proc. Pacfic Graphics (Short Papers), Sep.
* 2012
*
* You have to choose one of the following projection centers,
* corresponding to the centers of the six cube faces:
* phi0 = 0.0, lam0 = 0.0 ("front" face)
* phi0 = 0.0, lam0 = 90.0 ("right" face)
* phi0 = 0.0, lam0 = 180.0 ("back" face)
* phi0 = 0.0, lam0 = -90.0 ("left" face)
* phi0 = 90.0 ("top" face)
* phi0 = -90.0 ("bottom" face)
* Other projection centers will not work!
*
* In the projection code below, each cube face is handled differently.
* See the computation of the face parameter in the ENTRY0(qsc) function
* and the handling of different face values (FACE_*) in the forward and
* inverse projections.
*
* Furthermore, the projection is originally only defined for theta angles
* between (-1/4 * PI) and (+1/4 * PI) on the current cube face. This area
* of definition is named AREA_0 in the projection code below. The other
* three areas of a cube face are handled by rotation of AREA_0.
*/
#define PROJ_PARMS__ \
int face; \
double a_squared; \
double b; \
double one_minus_f; \
double one_minus_f_squared;
#define PJ_LIB__
#include <projects.h>
PROJ_HEAD(qsc, "Quadrilateralized Spherical Cube") "\n\tAzi, Sph.";
#define EPS10 1.e-10
/* The six cube faces. */
#define FACE_FRONT 0
#define FACE_RIGHT 1
#define FACE_BACK 2
#define FACE_LEFT 3
#define FACE_TOP 4
#define FACE_BOTTOM 5
/* The four areas on a cube face. AREA_0 is the area of definition,
* the other three areas are counted counterclockwise. */
#define AREA_0 0
#define AREA_1 1
#define AREA_2 2
#define AREA_3 3
/* Helper function for forward projection: compute the theta angle
* and determine the area number. */
static double
qsc_fwd_equat_face_theta(double phi, double y, double x, int *area) {
double theta;
if (phi < EPS10) {
*area = AREA_0;
theta = 0.0;
} else {
theta = atan2(y, x);
if (fabs(theta) <= FORTPI) {
*area = AREA_0;
} else if (theta > FORTPI && theta <= HALFPI + FORTPI) {
*area = AREA_1;
theta -= HALFPI;
} else if (theta > HALFPI + FORTPI || theta <= -(HALFPI + FORTPI)) {
*area = AREA_2;
theta = (theta >= 0.0 ? theta - PI : theta + PI);
} else {
*area = AREA_3;
theta += HALFPI;
}
}
return (theta);
}
/* Helper function: shift the longitude. */
static double
qsc_shift_lon_origin(double lon, double offset) {
double slon = lon + offset;
if (slon < -PI) {
slon += TWOPI;
} else if (slon > +PI) {
slon -= TWOPI;
}
return slon;
}
/* Forward projection, ellipsoid */
FORWARD(e_forward);
double lat, lon;
double theta, phi;
double t, mu; /* nu; */
int area;
/* Convert the geodetic latitude to a geocentric latitude.
* This corresponds to the shift from the ellipsoid to the sphere
* described in [LK12]. */
if (P->es) {
lat = atan(P->one_minus_f_squared * tan(lp.phi));
} else {
lat = lp.phi;
}
/* Convert the input lat, lon into theta, phi as used by QSC.
* This depends on the cube face and the area on it.
* For the top and bottom face, we can compute theta and phi
* directly from phi, lam. For the other faces, we must use
* unit sphere cartesian coordinates as an intermediate step. */
lon = lp.lam;
if (P->face == FACE_TOP) {
phi = HALFPI - lat;
if (lon >= FORTPI && lon <= HALFPI + FORTPI) {
area = AREA_0;
theta = lon - HALFPI;
} else if (lon > HALFPI + FORTPI || lon <= -(HALFPI + FORTPI)) {
area = AREA_1;
theta = (lon > 0.0 ? lon - PI : lon + PI);
} else if (lon > -(HALFPI + FORTPI) && lon <= -FORTPI) {
area = AREA_2;
theta = lon + HALFPI;
} else {
area = AREA_3;
theta = lon;
}
} else if (P->face == FACE_BOTTOM) {
phi = HALFPI + lat;
if (lon >= FORTPI && lon <= HALFPI + FORTPI) {
area = AREA_0;
theta = -lon + HALFPI;
} else if (lon < FORTPI && lon >= -FORTPI) {
area = AREA_1;
theta = -lon;
} else if (lon < -FORTPI && lon >= -(HALFPI + FORTPI)) {
area = AREA_2;
theta = -lon - HALFPI;
} else {
area = AREA_3;
theta = (lon > 0.0 ? -lon + PI : -lon - PI);
}
} else {
double q, r, s;
double sinlat, coslat;
double sinlon, coslon;
if (P->face == FACE_RIGHT) {
lon = qsc_shift_lon_origin(lon, +HALFPI);
} else if (P->face == FACE_BACK) {
lon = qsc_shift_lon_origin(lon, +PI);
} else if (P->face == FACE_LEFT) {
lon = qsc_shift_lon_origin(lon, -HALFPI);
}
sinlat = sin(lat);
coslat = cos(lat);
sinlon = sin(lon);
coslon = cos(lon);
q = coslat * coslon;
r = coslat * sinlon;
s = sinlat;
if (P->face == FACE_FRONT) {
phi = acos(q);
theta = qsc_fwd_equat_face_theta(phi, s, r, &area);
} else if (P->face == FACE_RIGHT) {
phi = acos(r);
theta = qsc_fwd_equat_face_theta(phi, s, -q, &area);
} else if (P->face == FACE_BACK) {
phi = acos(-q);
theta = qsc_fwd_equat_face_theta(phi, s, -r, &area);
} else if (P->face == FACE_LEFT) {
phi = acos(-r);
theta = qsc_fwd_equat_face_theta(phi, s, q, &area);
} else {
/* Impossible */
phi = theta = 0.0;
area = AREA_0;
}
}
/* Compute mu and nu for the area of definition.
* For mu, see Eq. (3-21) in [OL76], but note the typos:
* compare with Eq. (3-14). For nu, see Eq. (3-38). */
mu = atan((12.0 / PI) * (theta + acos(sin(theta) * cos(FORTPI)) - HALFPI));
t = sqrt((1.0 - cos(phi)) / (cos(mu) * cos(mu)) / (1.0 - cos(atan(1.0 / cos(theta)))));
/* nu = atan(t); We don't really need nu, just t, see below. */
/* Apply the result to the real area. */
if (area == AREA_1) {
mu += HALFPI;
} else if (area == AREA_2) {
mu += PI;
} else if (area == AREA_3) {
mu += HALFPI + PI;
}
/* Now compute x, y from mu and nu */
/* t = tan(nu); */
xy.x = t * cos(mu);
xy.y = t * sin(mu);
return (xy);
}
/* Inverse projection, ellipsoid */
INVERSE(e_inverse);
double mu, nu, cosmu, tannu;
double tantheta, theta, cosphi, phi;
double t;
int area;
/* Convert the input x, y to the mu and nu angles as used by QSC.
* This depends on the area of the cube face. */
nu = atan(sqrt(xy.x * xy.x + xy.y * xy.y));
mu = atan2(xy.y, xy.x);
if (xy.x >= 0.0 && xy.x >= fabs(xy.y)) {
area = AREA_0;
} else if (xy.y >= 0.0 && xy.y >= fabs(xy.x)) {
area = AREA_1;
mu -= HALFPI;
} else if (xy.x < 0.0 && -xy.x >= fabs(xy.y)) {
area = AREA_2;
mu = (mu < 0.0 ? mu + PI : mu - PI);
} else {
area = AREA_3;
mu += HALFPI;
}
/* Compute phi and theta for the area of definition.
* The inverse projection is not described in the original paper, but some
* good hints can be found here (as of 2011-12-14):
* http://fits.gsfc.nasa.gov/fitsbits/saf.93/saf.9302
* (search for "Message-Id: <9302181759.AA25477 at fits.cv.nrao.edu>") */
t = (PI / 12.0) * tan(mu);
tantheta = sin(t) / (cos(t) - (1.0 / sqrt(2.0)));
theta = atan(tantheta);
cosmu = cos(mu);
tannu = tan(nu);
cosphi = 1.0 - cosmu * cosmu * tannu * tannu * (1.0 - cos(atan(1.0 / cos(theta))));
if (cosphi < -1.0) {
cosphi = -1.0;
} else if (cosphi > +1.0) {
cosphi = +1.0;
}
/* Apply the result to the real area on the cube face.
* For the top and bottom face, we can compute phi and lam directly.
* For the other faces, we must use unit sphere cartesian coordinates
* as an intermediate step. */
if (P->face == FACE_TOP) {
phi = acos(cosphi);
lp.phi = HALFPI - phi;
if (area == AREA_0) {
lp.lam = theta + HALFPI;
} else if (area == AREA_1) {
lp.lam = (theta < 0.0 ? theta + PI : theta - PI);
} else if (area == AREA_2) {
lp.lam = theta - HALFPI;
} else /* area == AREA_3 */ {
lp.lam = theta;
}
} else if (P->face == FACE_BOTTOM) {
phi = acos(cosphi);
lp.phi = phi - HALFPI;
if (area == AREA_0) {
lp.lam = -theta + HALFPI;
} else if (area == AREA_1) {
lp.lam = -theta;
} else if (area == AREA_2) {
lp.lam = -theta - HALFPI;
} else /* area == AREA_3 */ {
lp.lam = (theta < 0.0 ? -theta - PI : -theta + PI);
}
} else {
/* Compute phi and lam via cartesian unit sphere coordinates. */
double q, r, s, t;
q = cosphi;
t = q * q;
if (t >= 1.0) {
s = 0.0;
} else {
s = sqrt(1.0 - t) * sin(theta);
}
t += s * s;
if (t >= 1.0) {
r = 0.0;
} else {
r = sqrt(1.0 - t);
}
/* Rotate q,r,s into the correct area. */
if (area == AREA_1) {
t = r;
r = -s;
s = t;
} else if (area == AREA_2) {
r = -r;
s = -s;
} else if (area == AREA_3) {
t = r;
r = s;
s = -t;
}
/* Rotate q,r,s into the correct cube face. */
if (P->face == FACE_RIGHT) {
t = q;
q = -r;
r = t;
} else if (P->face == FACE_BACK) {
q = -q;
r = -r;
} else if (P->face == FACE_LEFT) {
t = q;
q = r;
r = -t;
}
/* Now compute phi and lam from the unit sphere coordinates. */
lp.phi = acos(-s) - HALFPI;
lp.lam = atan2(r, q);
if (P->face == FACE_RIGHT) {
lp.lam = qsc_shift_lon_origin(lp.lam, -HALFPI);
} else if (P->face == FACE_BACK) {
lp.lam = qsc_shift_lon_origin(lp.lam, -PI);
} else if (P->face == FACE_LEFT) {
lp.lam = qsc_shift_lon_origin(lp.lam, +HALFPI);
}
}
/* Apply the shift from the sphere to the ellipsoid as described
* in [LK12]. */
if (P->es) {
int invert_sign;
double tanphi, xa;
invert_sign = (lp.phi < 0.0 ? 1 : 0);
tanphi = tan(lp.phi);
xa = P->b / sqrt(tanphi * tanphi + P->one_minus_f_squared);
lp.phi = atan(sqrt(P->a * P->a - xa * xa) / (P->one_minus_f * xa));
if (invert_sign) {
lp.phi = -lp.phi;
}
}
return (lp);
}
FREEUP; if (P) pj_dalloc(P); }
ENTRY0(qsc)
P->inv = e_inverse;
P->fwd = e_forward;
/* Determine the cube face from the center of projection. */
if (P->phi0 >= HALFPI - FORTPI / 2.0) {
P->face = FACE_TOP;
} else if (P->phi0 <= -(HALFPI - FORTPI / 2.0)) {
P->face = FACE_BOTTOM;
} else if (fabs(P->lam0) <= FORTPI) {
P->face = FACE_FRONT;
} else if (fabs(P->lam0) <= HALFPI + FORTPI) {
P->face = (P->lam0 > 0.0 ? FACE_RIGHT : FACE_LEFT);
} else {
P->face = FACE_BACK;
}
/* Fill in useful values for the ellipsoid <-> sphere shift
* described in [LK12]. */
if (P->es) {
P->a_squared = P->a * P->a;
P->b = P->a * sqrt(1.0 - P->es);
P->one_minus_f = 1.0 - (P->a - P->b) / P->a;
P->one_minus_f_squared = P->one_minus_f * P->one_minus_f;
}
ENDENTRY(P)