source: trunk/liblwgeom/lwgeodetic.c

/**********************************************************************
 *
 * PostGIS - Spatial Types for PostgreSQL
 * http://postgis.net
 *
 * PostGIS is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 2 of the License, or
 * (at your option) any later version.
 *
 * PostGIS is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with PostGIS.  If not, see <http://www.gnu.org/licenses/>.
 *
 **********************************************************************
 *
 * Copyright 2009 Paul Ramsey <pramsey@cleverelephant.ca>
 * Copyright 2009 David Skea <David.Skea@gov.bc.ca>
 *
 **********************************************************************/


#include "liblwgeom_internal.h"
#include "lwgeodetic.h"
#include "lwgeom_log.h"

/**
* For testing geodetic bounding box, we have a magic global variable.
* When this is true (when the cunit tests set it), use the slow, but
* guaranteed correct, algorithm. Otherwise use the regular one.
*/
int gbox_geocentric_slow = LW_FALSE;

/**
* Utility function for ptarray_contains_point_sphere()
*/
static int
point3d_equals(const POINT3D *p1, const POINT3D *p2)
{
        return FP_EQUALS(p1->x, p2->x) && FP_EQUALS(p1->y, p2->y) && FP_EQUALS(p1->z, p2->z);
}

/**
* Convert a longitude to the range of -PI,PI
*/
double longitude_radians_normalize(double lon)
{
        if ( lon == -1.0 * M_PI )
                return M_PI;
        if ( lon == -2.0 * M_PI )
                return 0.0;

        if ( lon > 2.0 * M_PI )
                lon = remainder(lon, 2.0 * M_PI);

        if ( lon < -2.0 * M_PI )
                lon = remainder(lon, -2.0 * M_PI);

        if ( lon > M_PI )
                lon = -2.0 * M_PI + lon;

        if ( lon < -1.0 * M_PI )
                lon = 2.0 * M_PI + lon;

        if ( lon == -2.0 * M_PI )
                lon *= -1.0;

        return lon;
}

/**
* Convert a latitude to the range of -PI/2,PI/2
*/
double latitude_radians_normalize(double lat)
{

        if ( lat > 2.0 * M_PI )
                lat = remainder(lat, 2.0 * M_PI);

        if ( lat < -2.0 * M_PI )
                lat = remainder(lat, -2.0 * M_PI);

        if ( lat > M_PI )
                lat = M_PI - lat;

        if ( lat < -1.0 * M_PI )
                lat = -1.0 * M_PI - lat;

        if ( lat > M_PI_2 )
                lat = M_PI - lat;

        if ( lat < -1.0 * M_PI_2 )
                lat = -1.0 * M_PI - lat;

        return lat;
}

/**
* Convert a longitude to the range of -180,180
* @param lon longitude in degrees
*/
double longitude_degrees_normalize(double lon)
{
        if ( lon > 360.0 )
                lon = remainder(lon, 360.0);

        if ( lon < -360.0 )
                lon = remainder(lon, -360.0);

        if ( lon > 180.0 )
                lon = -360.0 + lon;

        if ( lon < -180.0 )
                lon = 360 + lon;

        if ( lon == -180.0 )
                return 180.0;

        if ( lon == -360.0 )
                return 0.0;

        return lon;
}

/**
* Convert a latitude to the range of -90,90
* @param lat latitude in degrees
*/
double latitude_degrees_normalize(double lat)
{

        if ( lat > 360.0 )
                lat = remainder(lat, 360.0);

        if ( lat < -360.0 )
                lat = remainder(lat, -360.0);

        if ( lat > 180.0 )
                lat = 180.0 - lat;

        if ( lat < -180.0 )
                lat = -180.0 - lat;

        if ( lat > 90.0 )
                lat = 180.0 - lat;

        if ( lat < -90.0 )
                lat = -180.0 - lat;

        return lat;
}

/**
* Shift a point around by a number of radians
*/
void point_shift(GEOGRAPHIC_POINT *p, double shift)
{
        double lon = p->lon + shift;
        if ( lon > M_PI )
                p->lon = -1.0 * M_PI + (lon - M_PI);
        else
                p->lon = lon;
        return;
}

int geographic_point_equals(const GEOGRAPHIC_POINT *g1, const GEOGRAPHIC_POINT *g2)
{
        return FP_EQUALS(g1->lat, g2->lat) && FP_EQUALS(g1->lon, g2->lon);
}

/**
* Initialize a geographic point
* @param lon longitude in degrees
* @param lat latitude in degrees
*/
void geographic_point_init(double lon, double lat, GEOGRAPHIC_POINT *g)
{
        g->lat = latitude_radians_normalize(deg2rad(lat));
        g->lon = longitude_radians_normalize(deg2rad(lon));
}

/** Returns the angular height (latitudinal span) of the box in radians */
double
gbox_angular_height(const GBOX* gbox)
{
        double d[6];
        int i;
        double zmin = FLT_MAX;
        double zmax = -1 * FLT_MAX;
        POINT3D pt;

        /* Take a copy of the box corners so we can treat them as a list */
        /* Elements are xmin, xmax, ymin, ymax, zmin, zmax */
        memcpy(d, &(gbox->xmin), 6*sizeof(double));

        /* Generate all 8 corner vectors of the box */
        for ( i = 0; i < 8; i++ )
        {
                pt.x = d[i / 4];
                pt.y = d[2 + (i % 4) / 2];
                pt.z = d[4 + (i % 2)];
                normalize(&pt);
                if ( pt.z < zmin ) zmin = pt.z;
                if ( pt.z > zmax ) zmax = pt.z;
        }
        return asin(zmax) - asin(zmin);
}

/** Returns the angular width (longitudinal span) of the box in radians */
double
gbox_angular_width(const GBOX* gbox)
{
        double d[6];
        int i, j;
        POINT3D pt[3];
        double maxangle;
        double magnitude;

        /* Take a copy of the box corners so we can treat them as a list */
        /* Elements are xmin, xmax, ymin, ymax, zmin, zmax */
        memcpy(d, &(gbox->xmin), 6*sizeof(double));

        /* Start with the bottom corner */
        pt[0].x = gbox->xmin;
        pt[0].y = gbox->ymin;
        magnitude = sqrt(pt[0].x*pt[0].x + pt[0].y*pt[0].y);
        pt[0].x /= magnitude;
        pt[0].y /= magnitude;

        /* Generate all 8 corner vectors of the box */
        /* Find the vector furthest from our seed vector */
        for ( j = 0; j < 2; j++ )
        {
                maxangle = -1 * FLT_MAX;
                for ( i = 0; i < 4; i++ )
                {
                        double angle, dotprod;
                        POINT3D pt_n;

                        pt_n.x = d[i / 2];
                        pt_n.y = d[2 + (i % 2)];
                        magnitude = sqrt(pt_n.x*pt_n.x + pt_n.y*pt_n.y);
                        pt_n.x /= magnitude;
                        pt_n.y /= magnitude;
                        pt_n.z = 0.0;

                        dotprod = pt_n.x*pt[j].x + pt_n.y*pt[j].y;
                        angle = acos(dotprod > 1.0 ? 1.0 : dotprod);
                        if ( angle > maxangle )
                        {
                                pt[j+1] = pt_n;
                                maxangle = angle;
                        }
                }
        }

        /* Return the distance between the two furthest vectors */
        return maxangle;
}

/** Computes the average(ish) center of the box and returns success. */
int
gbox_centroid(const GBOX* gbox, POINT2D* out)
{
        double d[6];
        GEOGRAPHIC_POINT g;
        POINT3D pt;
        int i;

        /* Take a copy of the box corners so we can treat them as a list */
        /* Elements are xmin, xmax, ymin, ymax, zmin, zmax */
        memcpy(d, &(gbox->xmin), 6*sizeof(double));

        /* Zero out our return vector */
        pt.x = pt.y = pt.z = 0.0;

        for ( i = 0; i < 8; i++ )
        {
                POINT3D pt_n;

                pt_n.x = d[i / 4];
                pt_n.y = d[2 + ((i % 4) / 2)];
                pt_n.z = d[4 + (i % 2)];
                normalize(&pt_n);

                pt.x += pt_n.x;
                pt.y += pt_n.y;
                pt.z += pt_n.z;
        }

        pt.x /= 8.0;
        pt.y /= 8.0;
        pt.z /= 8.0;
        normalize(&pt);

        cart2geog(&pt, &g);
        out->x = longitude_degrees_normalize(rad2deg(g.lon));
        out->y = latitude_degrees_normalize(rad2deg(g.lat));

        return LW_SUCCESS;
}

/**
* Check to see if this geocentric gbox is wrapped around a pole.
* Only makes sense if this gbox originated from a polygon, as it's assuming
* the box is generated from external edges and there's an "interior" which
* contains the pole.
*
* This function is overdetermined, for very large polygons it might add an
* unwarranted pole. STILL NEEDS WORK!
*/
static int gbox_check_poles(GBOX *gbox)
{
        int rv = LW_FALSE;
#if POSTGIS_DEBUG_LEVEL >= 4
        char *gbox_str = gbox_to_string(gbox);
        LWDEBUG(4, "checking poles");
        LWDEBUGF(4, "gbox %s", gbox_str);
        lwfree(gbox_str);
#endif
        /* Z axis */
        if (gbox->xmin < 0.0 && gbox->xmax > 0.0 &&
            gbox->ymin < 0.0 && gbox->ymax > 0.0)
        {
                /* Extrema lean positive */
                if ((gbox->zmin > 0.0) && (gbox->zmax > 0.0))
                {
                        LWDEBUG(4, "enclosed positive z axis");
                        gbox->zmax = 1.0;
                }
                /* Extrema lean negative */
                else if ((gbox->zmin < 0.0) && (gbox->zmax < 0.0))
                {
                        LWDEBUG(4, "enclosed negative z axis");
                        gbox->zmin = -1.0;
                }
                /* Extrema both sides! */
                else
                {
                        LWDEBUG(4, "enclosed both z axes");
                        gbox->zmin = -1.0;
                        gbox->zmax = 1.0;
                }
                rv = LW_TRUE;
        }

        /* Y axis */
        if (gbox->xmin < 0.0 && gbox->xmax > 0.0 &&
            gbox->zmin < 0.0 && gbox->zmax > 0.0)
        {
                if ((gbox->ymin > 0.0) && (gbox->ymax > 0.0))
                {
                        LWDEBUG(4, "enclosed positive y axis");
                        gbox->ymax = 1.0;
                }
                else if ((gbox->ymin < 0.0) && (gbox->ymax < 0.0))
                {
                        LWDEBUG(4, "enclosed negative y axis");
                        gbox->ymax = -1.0;
                }
                else
                {
                        LWDEBUG(4, "enclosed both y axes");
                        gbox->ymax = 1.0;
                        gbox->ymin = -1.0;
                }
                rv = LW_TRUE;
        }

        /* X axis */
        if (gbox->ymin < 0.0 && gbox->ymax > 0.0 &&
            gbox->zmin < 0.0 && gbox->zmax > 0.0)
        {
                if ((gbox->xmin > 0.0) && (gbox->xmax > 0.0))
                {
                        LWDEBUG(4, "enclosed positive x axis");
                        gbox->xmax = 1.0;
                }
                else if ((gbox->xmin < 0.0) && (gbox->xmax < 0.0))
                {
                        LWDEBUG(4, "enclosed negative x axis");
                        gbox->xmin = -1.0;
                }
                else
                {
                        LWDEBUG(4, "enclosed both x axes");
                        gbox->xmax = 1.0;
                        gbox->xmin = -1.0;
                }

                rv = LW_TRUE;
        }

        return rv;
}

/**
* Convert spherical coordinates to cartesian coordinates on unit sphere
*/
void geog2cart(const GEOGRAPHIC_POINT *g, POINT3D *p)
{
        p->x = cos(g->lat) * cos(g->lon);
        p->y = cos(g->lat) * sin(g->lon);
        p->z = sin(g->lat);
}

/**
* Convert cartesian coordinates on unit sphere to spherical coordinates
*/
void cart2geog(const POINT3D *p, GEOGRAPHIC_POINT *g)
{
        g->lon = atan2(p->y, p->x);
        g->lat = asin(p->z);
}

/**
* Convert lon/lat coordinates to cartesian coordinates on unit sphere
*/
void ll2cart(const POINT2D *g, POINT3D *p)
{
        double x_rad = M_PI * g->x / 180.0;
        double y_rad = M_PI * g->y / 180.0;
        double cos_y_rad = cos(y_rad);
        p->x = cos_y_rad * cos(x_rad);
        p->y = cos_y_rad * sin(x_rad);
        p->z = sin(y_rad);
}

/**
* Convert cartesian coordinates on unit sphere to lon/lat coordinates
static void cart2ll(const POINT3D *p, POINT2D *g)
{
        g->x = longitude_degrees_normalize(180.0 * atan2(p->y, p->x) / M_PI);
        g->y = latitude_degrees_normalize(180.0 * asin(p->z) / M_PI);
}
*/

/**
* Calculate the dot product of two unit vectors
* (-1 == opposite, 0 == orthogonal, 1 == identical)
*/
static double dot_product(const POINT3D *p1, const POINT3D *p2)
{
        return (p1->x*p2->x) + (p1->y*p2->y) + (p1->z*p2->z);
}

/**
* Calculate the cross product of two vectors
*/
static void cross_product(const POINT3D *a, const POINT3D *b, POINT3D *n)
{
        n->x = a->y * b->z - a->z * b->y;
        n->y = a->z * b->x - a->x * b->z;
        n->z = a->x * b->y - a->y * b->x;
        return;
}

/**
* Calculate the sum of two vectors
*/
void vector_sum(const POINT3D *a, const POINT3D *b, POINT3D *n)
{
        n->x = a->x + b->x;
        n->y = a->y + b->y;
        n->z = a->z + b->z;
        return;
}

/**
* Calculate the difference of two vectors
*/
static void vector_difference(const POINT3D *a, const POINT3D *b, POINT3D *n)
{
        n->x = a->x - b->x;
        n->y = a->y - b->y;
        n->z = a->z - b->z;
        return;
}

/**
* Scale a vector out by a factor
*/
void vector_scale(POINT3D *n, double scale)
{
        n->x *= scale;
        n->y *= scale;
        n->z *= scale;
        return;
}

/*
* static inline double vector_magnitude(const POINT3D* v)
* {
*       return sqrt(v->x*v->x + v->y*v->y + v->z*v->z);
* }
*/

/**
* Angle between two unit vectors
*/
double vector_angle(const POINT3D* v1, const POINT3D* v2)
{
        POINT3D v3, normal;
        double angle, x, y;

        cross_product(v1, v2, &normal);
        normalize(&normal);
        cross_product(&normal, v1, &v3);

        x = dot_product(v1, v2);
        y = dot_product(v2, &v3);

        angle = atan2(y, x);
        return angle;
}

/**
* Normalize to a unit vector.
*/
static void normalize2d(POINT2D *p)
{
        double d = sqrt(p->x*p->x + p->y*p->y);
        if (FP_IS_ZERO(d))
        {
                p->x = p->y = 0.0;
                return;
        }
        p->x = p->x / d;
        p->y = p->y / d;
        return;
}

/**
* Calculates the unit normal to two vectors, trying to avoid
* problems with over-narrow or over-wide cases.
*/
void unit_normal(const POINT3D *P1, const POINT3D *P2, POINT3D *normal)
{
        double p_dot = dot_product(P1, P2);
        POINT3D P3;

        /* If edge is really large, calculate a narrower equivalent angle A1/A3. */
        if ( p_dot < 0 )
        {
                vector_sum(P1, P2, &P3);
                normalize(&P3);
        }
        /* If edge is narrow, calculate a wider equivalent angle A1/A3. */
        else if ( p_dot > 0.95 )
        {
                vector_difference(P2, P1, &P3);
                normalize(&P3);
        }
        /* Just keep the current angle in A1/A3. */
        else
        {
                P3 = *P2;
        }

        /* Normals to the A-plane and B-plane */
        cross_product(P1, &P3, normal);
        normalize(normal);
}

/**
* Rotates v1 through an angle (in radians) within the plane defined by v1/v2, returns
* the rotated vector in n.
*/
void vector_rotate(const POINT3D* v1, const POINT3D* v2, double angle, POINT3D* n)
{
        POINT3D u;
        double cos_a = cos(angle);
        double sin_a = sin(angle);
        double uxuy, uyuz, uxuz;
        double ux2, uy2, uz2;
        double rxx, rxy, rxz, ryx, ryy, ryz, rzx, rzy, rzz;

        /* Need a unit vector normal to rotate around */
        unit_normal(v1, v2, &u);

        uxuy = u.x * u.y;
        uxuz = u.x * u.z;
        uyuz = u.y * u.z;

        ux2 = u.x * u.x;
        uy2 = u.y * u.y;
        uz2 = u.z * u.z;

        rxx = cos_a + ux2 * (1 - cos_a);
        rxy = uxuy * (1 - cos_a) - u.z * sin_a;
        rxz = uxuz * (1 - cos_a) + u.y * sin_a;

        ryx = uxuy * (1 - cos_a) + u.z * sin_a;
        ryy = cos_a + uy2 * (1 - cos_a);
        ryz = uyuz * (1 - cos_a) - u.x * sin_a;

        rzx = uxuz * (1 - cos_a) - u.y * sin_a;
        rzy = uyuz * (1 - cos_a) + u.x * sin_a;
        rzz = cos_a + uz2 * (1 - cos_a);

        n->x = rxx * v1->x + rxy * v1->y + rxz * v1->z;
        n->y = ryx * v1->x + ryy * v1->y + ryz * v1->z;
        n->z = rzx * v1->x + rzy * v1->y + rzz * v1->z;

        normalize(n);
}

/**
* Normalize to a unit vector.
*/
void normalize(POINT3D *p)
{
        double d = sqrt(p->x*p->x + p->y*p->y + p->z*p->z);
        if (FP_IS_ZERO(d))
        {
                p->x = p->y = p->z = 0.0;
                return;
        }
        p->x = p->x / d;
        p->y = p->y / d;
        p->z = p->z / d;
        return;
}


/**
* Computes the cross product of two vectors using their lat, lng representations.
* Good even for small distances between p and q.
*/
void robust_cross_product(const GEOGRAPHIC_POINT *p, const GEOGRAPHIC_POINT *q, POINT3D *a)
{
        double lon_qpp = (q->lon + p->lon) / -2.0;
        double lon_qmp = (q->lon - p->lon) / 2.0;
        double sin_p_lat_minus_q_lat = sin(p->lat-q->lat);
        double sin_p_lat_plus_q_lat = sin(p->lat+q->lat);
        double sin_lon_qpp = sin(lon_qpp);
        double sin_lon_qmp = sin(lon_qmp);
        double cos_lon_qpp = cos(lon_qpp);
        double cos_lon_qmp = cos(lon_qmp);
        a->x = sin_p_lat_minus_q_lat * sin_lon_qpp * cos_lon_qmp -
               sin_p_lat_plus_q_lat * cos_lon_qpp * sin_lon_qmp;
        a->y = sin_p_lat_minus_q_lat * cos_lon_qpp * cos_lon_qmp +
               sin_p_lat_plus_q_lat * sin_lon_qpp * sin_lon_qmp;
        a->z = cos(p->lat) * cos(q->lat) * sin(q->lon-p->lon);
}

void x_to_z(POINT3D *p)
{
        double tmp = p->z;
        p->z = p->x;
        p->x = tmp;
}

void y_to_z(POINT3D *p)
{
        double tmp = p->z;
        p->z = p->y;
        p->y = tmp;
}


int crosses_dateline(const GEOGRAPHIC_POINT *s, const GEOGRAPHIC_POINT *e)
{
        double sign_s = SIGNUM(s->lon);
        double sign_e = SIGNUM(e->lon);
        double ss = fabs(s->lon);
        double ee = fabs(e->lon);
        if ( sign_s == sign_e )
        {
                return LW_FALSE;
        }
        else
        {
                double dl = ss + ee;
                if ( dl < M_PI )
                        return LW_FALSE;
                else if ( FP_EQUALS(dl, M_PI) )
                        return LW_FALSE;
                else
                        return LW_TRUE;
        }
}

/**
* Returns -1 if the point is to the left of the plane formed
* by the edge, 1 if the point is to the right, and 0 if the
* point is on the plane.
*/
static int
edge_point_side(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *p)
{
        POINT3D normal, pt;
        double w;
        /* Normal to the plane defined by e */
        robust_cross_product(&(e->start), &(e->end), &normal);
        normalize(&normal);
        geog2cart(p, &pt);
        /* We expect the dot product of with normal with any vector in the plane to be zero */
        w = dot_product(&normal, &pt);
        LWDEBUGF(4,"dot product %.9g",w);
        if ( FP_IS_ZERO(w) )
        {
                LWDEBUG(4, "point is on plane (dot product is zero)");
                return 0;
        }

        if ( w < 0 )
                return -1;
        else
                return 1;
}

/**
* Returns the angle in radians at point B of the triangle formed by A-B-C
*/
static double
sphere_angle(const GEOGRAPHIC_POINT *a, const GEOGRAPHIC_POINT *b,  const GEOGRAPHIC_POINT *c)
{
        POINT3D normal1, normal2;
        robust_cross_product(b, a, &normal1);
        robust_cross_product(b, c, &normal2);
        normalize(&normal1);
        normalize(&normal2);
        return sphere_distance_cartesian(&normal1, &normal2);
}

/**
* Computes the spherical area of a triangle. If C is to the left of A/B,
* the area is negative. If C is to the right of A/B, the area is positive.
*
* @param a The first triangle vertex.
* @param b The second triangle vertex.
* @param c The last triangle vertex.
* @return the signed area in radians.
*/
static double
sphere_signed_area(const GEOGRAPHIC_POINT *a, const GEOGRAPHIC_POINT *b, const GEOGRAPHIC_POINT *c)
{
        double angle_a, angle_b, angle_c;
        double area_radians = 0.0;
        int side;
        GEOGRAPHIC_EDGE e;

        angle_a = sphere_angle(b,a,c);
        angle_b = sphere_angle(a,b,c);
        angle_c = sphere_angle(b,c,a);

        area_radians = angle_a + angle_b + angle_c - M_PI;

        /* What's the direction of the B/C edge? */
        e.start = *a;
        e.end = *b;
        side = edge_point_side(&e, c);

        /* Co-linear points implies no area */
        if ( side == 0 )
                return 0.0;

        /* Add the sign to the area */
        return side * area_radians;
}



/**
* Returns true if the point p is on the great circle plane.
* Forms the scalar triple product of A,B,p and if the volume of the
* resulting parallelepiped is near zero the point p is on the
* great circle plane.
*/
int edge_point_on_plane(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *p)
{
        int side = edge_point_side(e, p);
        if ( side == 0 )
                return LW_TRUE;

        return LW_FALSE;
}

/**
* Returns true if the point p is inside the cone defined by the
* two ends of the edge e.
*/
int edge_point_in_cone(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *p)
{
        POINT3D vcp, vs, ve, vp;
        double vs_dot_vcp, vp_dot_vcp;
        geog2cart(&(e->start), &vs);
        geog2cart(&(e->end), &ve);
        /* Antipodal case, everything is inside. */
        if ( vs.x == -1.0 * ve.x && vs.y == -1.0 * ve.y && vs.z == -1.0 * ve.z )
                return LW_TRUE;
        geog2cart(p, &vp);
        /* The normalized sum bisects the angle between start and end. */
        vector_sum(&vs, &ve, &vcp);
        normalize(&vcp);
        /* The projection of start onto the center defines the minimum similarity */
        vs_dot_vcp = dot_product(&vs, &vcp);
        LWDEBUGF(4,"vs_dot_vcp %.19g",vs_dot_vcp);
        /* The projection of candidate p onto the center */
        vp_dot_vcp = dot_product(&vp, &vcp);
        LWDEBUGF(4,"vp_dot_vcp %.19g",vp_dot_vcp);
        /* If p is more similar than start then p is inside the cone */
        LWDEBUGF(4,"fabs(vp_dot_vcp - vs_dot_vcp) %.39g",fabs(vp_dot_vcp - vs_dot_vcp));

        /*
        ** We want to test that vp_dot_vcp is >= vs_dot_vcp but there are
        ** numerical stability issues for values that are very very nearly
        ** equal. Unfortunately there are also values of vp_dot_vcp that are legitimately
        ** very close to but still less than vs_dot_vcp which we also need to catch.
        ** The tolerance of 10-17 seems to do the trick on 32-bit and 64-bit architectures,
        ** for the test cases here.
        ** However, tuning the tolerance value feels like a dangerous hack.
        ** Fundamentally, the problem is that this test is so sensitive.
        */

        /* 1.1102230246251565404236316680908203125e-16 */

        if ( vp_dot_vcp > vs_dot_vcp || fabs(vp_dot_vcp - vs_dot_vcp) < 2e-16 )
        {
                LWDEBUG(4, "point is in cone");
                return LW_TRUE;
        }
        LWDEBUG(4, "point is not in cone");
        return LW_FALSE;
}

/**
* True if the longitude of p is within the range of the longitude of the ends of e
*/
int edge_contains_coplanar_point(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *p)
{
        GEOGRAPHIC_EDGE g;
        GEOGRAPHIC_POINT q;
        double slon = fabs((e->start).lon) + fabs((e->end).lon);
        double dlon = fabs(fabs((e->start).lon) - fabs((e->end).lon));
        double slat = (e->start).lat + (e->end).lat;

        LWDEBUGF(4, "e.start == GPOINT(%.6g %.6g) ", (e->start).lat, (e->start).lon);
        LWDEBUGF(4, "e.end == GPOINT(%.6g %.6g) ", (e->end).lat, (e->end).lon);
        LWDEBUGF(4, "p == GPOINT(%.6g %.6g) ", p->lat, p->lon);

        /* Copy values into working registers */
        g = *e;
        q = *p;

        /* Vertical plane, we need to do this calculation in latitude */
        if ( FP_EQUALS( g.start.lon, g.end.lon ) )
        {
                LWDEBUG(4, "vertical plane, we need to do this calculation in latitude");
                /* Supposed to be co-planar... */
                if ( ! FP_EQUALS( q.lon, g.start.lon ) )
                        return LW_FALSE;

                if ( ( g.start.lat <= q.lat && q.lat <= g.end.lat ) ||
                     ( g.end.lat <= q.lat && q.lat <= g.start.lat ) )
                {
                        return LW_TRUE;
                }
                else
                {
                        return LW_FALSE;
                }
        }

        /* Over the pole, we need normalize latitude and do this calculation in latitude */
        if ( FP_EQUALS( slon, M_PI ) && ( SIGNUM(g.start.lon) != SIGNUM(g.end.lon) || FP_EQUALS(dlon, M_PI) ) )
        {
                LWDEBUG(4, "over the pole...");
                /* Antipodal, everything (or nothing?) is inside */
                if ( FP_EQUALS( slat, 0.0 ) )
                        return LW_TRUE;

                /* Point *is* the north pole */
                if ( slat > 0.0 && FP_EQUALS(q.lat, M_PI_2 ) )
                        return LW_TRUE;

                /* Point *is* the south pole */
                if ( slat < 0.0 && FP_EQUALS(q.lat, -1.0 * M_PI_2) )
                        return LW_TRUE;

                LWDEBUG(4, "coplanar?...");

                /* Supposed to be co-planar... */
                if ( ! FP_EQUALS( q.lon, g.start.lon ) )
                        return LW_FALSE;

                LWDEBUG(4, "north or south?...");

                /* Over north pole, test based on south pole */
                if ( slat > 0.0 )
                {
                        LWDEBUG(4, "over the north pole...");
                        if ( q.lat > FP_MIN(g.start.lat, g.end.lat) )
                                return LW_TRUE;
                        else
                                return LW_FALSE;
                }
                else
                        /* Over south pole, test based on north pole */
                {
                        LWDEBUG(4, "over the south pole...");
                        if ( q.lat < FP_MAX(g.start.lat, g.end.lat) )
                                return LW_TRUE;
                        else
                                return LW_FALSE;
                }
        }

        /* Dateline crossing, flip everything to the opposite hemisphere */
        else if ( slon > M_PI && ( SIGNUM(g.start.lon) != SIGNUM(g.end.lon) ) )
        {
                LWDEBUG(4, "crosses dateline, flip longitudes...");
                if ( g.start.lon > 0.0 )
                        g.start.lon -= M_PI;
                else
                        g.start.lon += M_PI;
                if ( g.end.lon > 0.0 )
                        g.end.lon -= M_PI;
                else
                        g.end.lon += M_PI;

                if ( q.lon > 0.0 )
                        q.lon -= M_PI;
                else
                        q.lon += M_PI;
        }

        if ( ( g.start.lon <= q.lon && q.lon <= g.end.lon ) ||
             ( g.end.lon <= q.lon && q.lon <= g.start.lon ) )
        {
                LWDEBUG(4, "true, this edge contains point");
                return LW_TRUE;
        }

        LWDEBUG(4, "false, this edge does not contain point");
        return LW_FALSE;
}


/**
* Given two points on a unit sphere, calculate their distance apart in radians.
*/
double sphere_distance(const GEOGRAPHIC_POINT *s, const GEOGRAPHIC_POINT *e)
{
        double d_lon = e->lon - s->lon;
        double cos_d_lon = cos(d_lon);
        double cos_lat_e = cos(e->lat);
        double sin_lat_e = sin(e->lat);
        double cos_lat_s = cos(s->lat);
        double sin_lat_s = sin(s->lat);

        double a1 = POW2(cos_lat_e * sin(d_lon));
        double a2 = POW2(cos_lat_s * sin_lat_e - sin_lat_s * cos_lat_e * cos_d_lon);
        double a = sqrt(a1 + a2);
        double b = sin_lat_s * sin_lat_e + cos_lat_s * cos_lat_e * cos_d_lon;
        return atan2(a, b);
}

/**
* Given two unit vectors, calculate their distance apart in radians.
*/
double sphere_distance_cartesian(const POINT3D *s, const POINT3D *e)
{
        return acos(FP_MIN(1.0, dot_product(s, e)));
}

/**
* Given two points on a unit sphere, calculate the direction from s to e.
*/
double sphere_direction(const GEOGRAPHIC_POINT *s, const GEOGRAPHIC_POINT *e, double d)
{
        double heading = 0.0;
        double f;

        /* Starting from the poles? Special case. */
        if ( FP_IS_ZERO(cos(s->lat)) )
                return (s->lat > 0.0) ? M_PI : 0.0;

        f = (sin(e->lat) - sin(s->lat) * cos(d)) / (sin(d) * cos(s->lat));
        if ( FP_EQUALS(f, 1.0) )
                heading = 0.0;
        else if ( FP_EQUALS(f, -1.0) )
                heading = M_PI;
        else if ( fabs(f) > 1.0 )
        {
                LWDEBUGF(4, "f = %g", f);
                heading = acos(f);
        }
        else
                heading = acos(f);

        if ( sin(e->lon - s->lon) < 0.0 )
                heading = -1 * heading;

        return heading;
}

#if 0 /* unused */
/**
* Computes the spherical excess of a spherical triangle defined by
* the three vertices A, B, C. Computes on the unit sphere (i.e., divides
* edge lengths by the radius, even if the radius is 1.0). The excess is
* signed based on the sign of the delta longitude of A and B.
*
* @param a The first triangle vertex.
* @param b The second triangle vertex.
* @param c The last triangle vertex.
* @return the signed spherical excess.
*/
static double sphere_excess(const GEOGRAPHIC_POINT *a, const GEOGRAPHIC_POINT *b, const GEOGRAPHIC_POINT *c)
{
        double a_dist = sphere_distance(b, c);
        double b_dist = sphere_distance(c, a);
        double c_dist = sphere_distance(a, b);
        double hca = sphere_direction(c, a, b_dist);
        double hcb = sphere_direction(c, b, a_dist);
        double sign = SIGNUM(hcb-hca);
        double ss = (a_dist + b_dist + c_dist) / 2.0;
        double E = tan(ss/2.0)*tan((ss-a_dist)/2.0)*tan((ss-b_dist)/2.0)*tan((ss-c_dist)/2.0);
        return 4.0 * atan(sqrt(fabs(E))) * sign;
}
#endif


/**
* Returns true if the point p is on the minor edge defined by the
* end points of e.
*/
int edge_contains_point(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *p)
{
        if ( edge_point_in_cone(e, p) && edge_point_on_plane(e, p) )
                /*      if ( edge_contains_coplanar_point(e, p) && edge_point_on_plane(e, p) ) */
        {
                LWDEBUG(4, "point is on edge");
                return LW_TRUE;
        }
        LWDEBUG(4, "point is not on edge");
        return LW_FALSE;
}

/**
* Used in great circle to compute the pole of the great circle.
*/
double z_to_latitude(double z, int top)
{
        double sign = SIGNUM(z);
        double tlat = acos(z);
        LWDEBUGF(4, "inputs: z(%.8g) sign(%.8g) tlat(%.8g)", z, sign, tlat);
        if (FP_IS_ZERO(z))
        {
                if (top) return M_PI_2;
                else return -1.0 * M_PI_2;
        }
        if (fabs(tlat) > M_PI_2 )
        {
                tlat = sign * (M_PI - fabs(tlat));
        }
        else
        {
                tlat = sign * tlat;
        }
        LWDEBUGF(4, "output: tlat(%.8g)", tlat);
        return tlat;
}

/**
* Computes the pole of the great circle disk which is the intersection of
* the great circle with the line of maximum/minimum gradient that lies on
* the great circle plane.
*/
int clairaut_cartesian(const POINT3D *start, const POINT3D *end, GEOGRAPHIC_POINT *g_top, GEOGRAPHIC_POINT *g_bottom)
{
        POINT3D t1, t2;
        GEOGRAPHIC_POINT vN1, vN2;
        LWDEBUG(4,"entering function");
        unit_normal(start, end, &t1);
        unit_normal(end, start, &t2);
        LWDEBUGF(4, "unit normal t1 == POINT(%.8g %.8g %.8g)", t1.x, t1.y, t1.z);
        LWDEBUGF(4, "unit normal t2 == POINT(%.8g %.8g %.8g)", t2.x, t2.y, t2.z);
        cart2geog(&t1, &vN1);
        cart2geog(&t2, &vN2);
        g_top->lat = z_to_latitude(t1.z,LW_TRUE);
        g_top->lon = vN2.lon;
        g_bottom->lat = z_to_latitude(t2.z,LW_FALSE);
        g_bottom->lon = vN1.lon;
        LWDEBUGF(4, "clairaut top == GPOINT(%.6g %.6g)", g_top->lat, g_top->lon);
        LWDEBUGF(4, "clairaut bottom == GPOINT(%.6g %.6g)", g_bottom->lat, g_bottom->lon);
        return LW_SUCCESS;
}

/**
* Computes the pole of the great circle disk which is the intersection of
* the great circle with the line of maximum/minimum gradient that lies on
* the great circle plane.
*/
int clairaut_geographic(const GEOGRAPHIC_POINT *start, const GEOGRAPHIC_POINT *end, GEOGRAPHIC_POINT *g_top, GEOGRAPHIC_POINT *g_bottom)
{
        POINT3D t1, t2;
        GEOGRAPHIC_POINT vN1, vN2;
        LWDEBUG(4,"entering function");
        robust_cross_product(start, end, &t1);
        normalize(&t1);
        robust_cross_product(end, start, &t2);
        normalize(&t2);
        LWDEBUGF(4, "unit normal t1 == POINT(%.8g %.8g %.8g)", t1.x, t1.y, t1.z);
        LWDEBUGF(4, "unit normal t2 == POINT(%.8g %.8g %.8g)", t2.x, t2.y, t2.z);
        cart2geog(&t1, &vN1);
        cart2geog(&t2, &vN2);
        g_top->lat = z_to_latitude(t1.z,LW_TRUE);
        g_top->lon = vN2.lon;
        g_bottom->lat = z_to_latitude(t2.z,LW_FALSE);
        g_bottom->lon = vN1.lon;
        LWDEBUGF(4, "clairaut top == GPOINT(%.6g %.6g)", g_top->lat, g_top->lon);
        LWDEBUGF(4, "clairaut bottom == GPOINT(%.6g %.6g)", g_bottom->lat, g_bottom->lon);
        return LW_SUCCESS;
}

/**
* Returns true if an intersection can be calculated, and places it in *g.
* Returns false otherwise.
*/
int edge_intersection(const GEOGRAPHIC_EDGE *e1, const GEOGRAPHIC_EDGE *e2, GEOGRAPHIC_POINT *g)
{
        POINT3D ea, eb, v;
        LWDEBUGF(4, "e1 start(%.20g %.20g) end(%.20g %.20g)", e1->start.lat, e1->start.lon, e1->end.lat, e1->end.lon);
        LWDEBUGF(4, "e2 start(%.20g %.20g) end(%.20g %.20g)", e2->start.lat, e2->start.lon, e2->end.lat, e2->end.lon);

        LWDEBUGF(4, "e1 start(%.20g %.20g) end(%.20g %.20g)", rad2deg(e1->start.lon), rad2deg(e1->start.lat), rad2deg(e1->end.lon), rad2deg(e1->end.lat));
        LWDEBUGF(4, "e2 start(%.20g %.20g) end(%.20g %.20g)", rad2deg(e2->start.lon), rad2deg(e2->start.lat), rad2deg(e2->end.lon), rad2deg(e2->end.lat));

        if ( geographic_point_equals(&(e1->start), &(e2->start)) )
        {
                *g = e1->start;
                return LW_TRUE;
        }
        if ( geographic_point_equals(&(e1->end), &(e2->end)) )
        {
                *g = e1->end;
                return LW_TRUE;
        }
        if ( geographic_point_equals(&(e1->end), &(e2->start)) )
        {
                *g = e1->end;
                return LW_TRUE;
        }
        if ( geographic_point_equals(&(e1->start), &(e2->end)) )
        {
                *g = e1->start;
                return LW_TRUE;
        }

        robust_cross_product(&(e1->start), &(e1->end), &ea);
        normalize(&ea);
        robust_cross_product(&(e2->start), &(e2->end), &eb);
        normalize(&eb);
        LWDEBUGF(4, "e1 cross product == POINT(%.12g %.12g %.12g)", ea.x, ea.y, ea.z);
        LWDEBUGF(4, "e2 cross product == POINT(%.12g %.12g %.12g)", eb.x, eb.y, eb.z);
        LWDEBUGF(4, "fabs(dot_product(ea, eb)) == %.14g", fabs(dot_product(&ea, &eb)));
        if ( FP_EQUALS(fabs(dot_product(&ea, &eb)), 1.0) )
        {
                LWDEBUGF(4, "parallel edges found! dot_product = %.12g", dot_product(&ea, &eb));
                /* Parallel (maybe equal) edges! */
                /* Hack alert, only returning ONE end of the edge right now, most do better later. */
                /* Hack alert #2, returning a value of 2 to indicate a co-linear crossing event. */
                if ( edge_contains_point(e1, &(e2->start)) )
                {
                        *g = e2->start;
                        return 2;
                }
                if ( edge_contains_point(e1, &(e2->end)) )
                {
                        *g = e2->end;
                        return 2;
                }
                if ( edge_contains_point(e2, &(e1->start)) )
                {
                        *g = e1->start;
                        return 2;
                }
                if ( edge_contains_point(e2, &(e1->end)) )
                {
                        *g = e1->end;
                        return 2;
                }
        }
        unit_normal(&ea, &eb, &v);
        LWDEBUGF(4, "v == POINT(%.12g %.12g %.12g)", v.x, v.y, v.z);
        g->lat = atan2(v.z, sqrt(v.x * v.x + v.y * v.y));
        g->lon = atan2(v.y, v.x);
        LWDEBUGF(4, "g == GPOINT(%.12g %.12g)", g->lat, g->lon);
        LWDEBUGF(4, "g == POINT(%.12g %.12g)", rad2deg(g->lon), rad2deg(g->lat));
        if ( edge_contains_point(e1, g) && edge_contains_point(e2, g) )
        {
                return LW_TRUE;
        }
        else
        {
                LWDEBUG(4, "flipping point to other side of sphere");
                g->lat = -1.0 * g->lat;
                g->lon = g->lon + M_PI;
                if ( g->lon > M_PI )
                {
                        g->lon = -1.0 * (2.0 * M_PI - g->lon);
                }
                if ( edge_contains_point(e1, g) && edge_contains_point(e2, g) )
                {
                        return LW_TRUE;
                }
        }
        return LW_FALSE;
}

double edge_distance_to_point(const GEOGRAPHIC_EDGE *e, const GEOGRAPHIC_POINT *gp, GEOGRAPHIC_POINT *closest)
{
        double d1 = 1000000000.0, d2, d3, d_nearest;
        POINT3D n, p, k;
        GEOGRAPHIC_POINT gk, g_nearest;

        /* Zero length edge, */
        if ( geographic_point_equals(&(e->start), &(e->end)) )
        {
                *closest = e->start;
                return sphere_distance(&(e->start), gp);
        }

        robust_cross_product(&(e->start), &(e->end), &n);
        normalize(&n);
        geog2cart(gp, &p);
        vector_scale(&n, dot_product(&p, &n));
        vector_difference(&p, &n, &k);
        normalize(&k);
        cart2geog(&k, &gk);
        if ( edge_contains_point(e, &gk) )
        {
                d1 = sphere_distance(gp, &gk);
        }
        d2 = sphere_distance(gp, &(e->start));
        d3 = sphere_distance(gp, &(e->end));

        d_nearest = d1;
        g_nearest = gk;

        if ( d2 < d_nearest )
        {
                d_nearest = d2;
                g_nearest = e->start;
        }
        if ( d3 < d_nearest )
        {
                d_nearest = d3;
                g_nearest = e->end;
        }
        if (closest)
                *closest = g_nearest;

        return d_nearest;
}

/**
* Calculate the distance between two edges.
* IMPORTANT: this test does not check for edge intersection!!! (distance == 0)
* You have to check for intersection before calling this function.
*/
double edge_distance_to_edge(const GEOGRAPHIC_EDGE *e1, const GEOGRAPHIC_EDGE *e2, GEOGRAPHIC_POINT *closest1, GEOGRAPHIC_POINT *closest2)
{
        double d;
        GEOGRAPHIC_POINT gcp1s, gcp1e, gcp2s, gcp2e, c1, c2;
        double d1s = edge_distance_to_point(e1, &(e2->start), &gcp1s);
        double d1e = edge_distance_to_point(e1, &(e2->end), &gcp1e);
        double d2s = edge_distance_to_point(e2, &(e1->start), &gcp2s);
        double d2e = edge_distance_to_point(e2, &(e1->end), &gcp2e);

        d = d1s;
        c1 = gcp1s;
        c2 = e2->start;

        if ( d1e < d )
        {
                d = d1e;
                c1 = gcp1e;
                c2 = e2->end;
        }

        if ( d2s < d )
        {
                d = d2s;
                c1 = e1->start;
                c2 = gcp2s;
        }

        if ( d2e < d )
        {
                d = d2e;
                c1 = e1->end;
                c2 = gcp2e;
        }

        if ( closest1 ) *closest1 = c1;
        if ( closest2 ) *closest2 = c2;

        return d;
}


/**
* Given a starting location r, a distance and an azimuth
* to the new point, compute the location of the projected point on the unit sphere.
*/
int sphere_project(const GEOGRAPHIC_POINT *r, double distance, double azimuth, GEOGRAPHIC_POINT *n)
{
        double d = distance;
        double lat1 = r->lat;
        double lon1 = r->lon;
        double lat2, lon2;

        lat2 = asin(sin(lat1)*cos(d) + cos(lat1)*sin(d)*cos(azimuth));

        /* If we're going straight up or straight down, we don't need to calculate the longitude */
        /* TODO: this isn't quite true, what if we're going over the pole? */
        if ( FP_EQUALS(azimuth, M_PI) || FP_EQUALS(azimuth, 0.0) )
        {
                lon2 = r->lon;
        }
        else
        {
                lon2 = lon1 + atan2(sin(azimuth)*sin(d)*cos(lat1), cos(d)-sin(lat1)*sin(lat2));
        }

        if ( isnan(lat2) || isnan(lon2) )
                return LW_FAILURE;

        n->lat = lat2;
        n->lon = lon2;

        return LW_SUCCESS;
}


int edge_calculate_gbox_slow(const GEOGRAPHIC_EDGE *e, GBOX *gbox)
{
        int steps = 1000000;
        int i;
        double dx, dy, dz;
        double distance = sphere_distance(&(e->start), &(e->end));
        POINT3D pn, p, start, end;

        /* Edge is zero length, just return the naive box */
        if ( FP_IS_ZERO(distance) )
        {
                LWDEBUG(4, "edge is zero length. returning");
                geog2cart(&(e->start), &start);
                geog2cart(&(e->end), &end);
                gbox_init_point3d(&start, gbox);
                gbox_merge_point3d(&end, gbox);
                return LW_SUCCESS;
        }

        /* Edge is antipodal (one point on each side of the globe),
           set the box to contain the whole world and return */
        if ( FP_EQUALS(distance, M_PI) )
        {
                LWDEBUG(4, "edge is antipodal. setting to maximum size box, and returning");
                gbox->xmin = gbox->ymin = gbox->zmin = -1.0;
                gbox->xmax = gbox->ymax = gbox->zmax = 1.0;
                return LW_SUCCESS;
        }

        /* Walk along the chord between start and end incrementally,
           normalizing at each step. */
        geog2cart(&(e->start), &start);
        geog2cart(&(e->end), &end);
        dx = (end.x - start.x)/steps;
        dy = (end.y - start.y)/steps;
        dz = (end.z - start.z)/steps;
        p = start;
        gbox->xmin = gbox->xmax = p.x;
        gbox->ymin = gbox->ymax = p.y;
        gbox->zmin = gbox->zmax = p.z;
        for ( i = 0; i < steps; i++ )
        {
                p.x += dx;
                p.y += dy;
                p.z += dz;
                pn = p;
                normalize(&pn);
                gbox_merge_point3d(&pn, gbox);
        }
        return LW_SUCCESS;
}

/**
* The magic function, given an edge in spherical coordinates, calculate a
* 3D bounding box that fully contains it, taking into account the curvature
* of the sphere on which it is inscribed.
*
* Any arc on the sphere defines a plane that bisects the sphere. In this plane,
* the arc is a portion of a unit circle.
* Projecting the end points of the axes (1,0,0), (-1,0,0) etc, into the plane
* and normalizing yields potential extrema points. Those points on the
* side of the plane-dividing line formed by the end points that is opposite
* the origin of the plane are extrema and should be added to the bounding box.
*/
int edge_calculate_gbox(const POINT3D *A1, const POINT3D *A2, GBOX *gbox)
{
        POINT2D R1, R2, RX, O;
        POINT3D AN, A3;
        POINT3D X[6];
        int i, o_side;

        /* Initialize the box with the edge end points */
        gbox_init_point3d(A1, gbox);
        gbox_merge_point3d(A2, gbox);

        /* Zero length edge, just return! */
        if ( p3d_same(A1, A2) )
                return LW_SUCCESS;

        /* Error out on antipodal edge */
        if ( FP_EQUALS(A1->x, -1*A2->x) && FP_EQUALS(A1->y, -1*A2->y) && FP_EQUALS(A1->z, -1*A2->z) )
        {
                lwerror("Antipodal (180 degrees long) edge detected!");
                return LW_FAILURE;
        }

        /* Create A3, a vector in the plane of A1/A2, orthogonal to A1  */
        unit_normal(A1, A2, &AN);
        unit_normal(&AN, A1, &A3);

        /* Project A1 and A2 into the 2-space formed by the plane A1/A3 */
        R1.x = 1.0;
        R1.y = 0.0;
        R2.x = dot_product(A2, A1);
        R2.y = dot_product(A2, &A3);

        /* Initialize our 3-space axis points (x+, x-, y+, y-, z+, z-) */
        memset(X, 0, sizeof(POINT3D) * 6);
        X[0].x = X[2].y = X[4].z =  1.0;
        X[1].x = X[3].y = X[5].z = -1.0;

        /* Initialize a 2-space origin point. */
        O.x = O.y = 0.0;
        /* What side of the line joining R1/R2 is O? */
        o_side = lw_segment_side(&R1, &R2, &O);

        /* Add any extrema! */
        for ( i = 0; i < 6; i++ )
        {
                /* Convert 3-space axis points to 2-space unit vectors */
                RX.x = dot_product(&(X[i]), A1);
                RX.y = dot_product(&(X[i]), &A3);
                normalize2d(&RX);

                /* Any axis end on the side of R1/R2 opposite the origin */
                /* is an extreme point in the arc, so we add the 3-space */
                /* version of the point on R1/R2 to the gbox */
                if ( lw_segment_side(&R1, &R2, &RX) != o_side )
                {
                        POINT3D Xn;
                        Xn.x = RX.x * A1->x + RX.y * A3.x;
                        Xn.y = RX.x * A1->y + RX.y * A3.y;
                        Xn.z = RX.x * A1->z + RX.y * A3.z;

                        gbox_merge_point3d(&Xn, gbox);
                }
        }

        return LW_SUCCESS;
}

/*
* When we have a globe-covering gbox but we still want an outside
* point, we do this Very Bad Hack, which is look at the first two points
* in the ring and then nudge a point to the left of that arc.
* There is an assumption of convexity built in there, as well as that
* the shape doesn't have a sharp reversal in it. It's ugly, but
* it fixes some common cases (large selection polygons) that users
* are generating. At some point all of geodetic needs a clean-room
* rewrite.
* There is also an assumption of CCW exterior ring, which is how the
* GeoJSON spec defined geographic ring orientation.
*/
static int lwpoly_pt_outside_hack(const LWPOLY *poly, POINT2D *pt_outside)
{
        GEOGRAPHIC_POINT g1, g2, gSum;
        POINT4D p1, p2;
        POINT3D q1, q2, qMid, qCross, qSum;
        POINTARRAY *pa;
        if (lwgeom_is_empty((LWGEOM*)poly))
                return LW_FAILURE;
        if (poly->nrings < 1)
                return LW_FAILURE;
        pa = poly->rings[0];
        if (pa->npoints < 2)
                return LW_FAILURE;

        /* First two points of ring */
        getPoint4d_p(pa, 0, &p1);
        getPoint4d_p(pa, 1, &p2);
        /* Convert to XYZ unit vectors */
        geographic_point_init(p1.x, p1.y, &g1);
        geographic_point_init(p2.x, p2.y, &g2);
        geog2cart(&g1, &q1);
        geog2cart(&g2, &q2);
        /* Mid-point of first two points */
        vector_sum(&q1, &q2, &qMid);
        normalize(&qMid);
        /* Cross product of first two points (perpendicular) */
        cross_product(&q1, &q2, &qCross);
        normalize(&qCross);
        /* Invert it to put it outside, and scale down */
        vector_scale(&qCross, -0.2);
        /* Project midpoint to the right */
        vector_sum(&qMid, &qCross, &qSum);
        normalize(&qSum);
        /* Convert back to lon/lat */
        cart2geog(&qSum, &gSum);
        pt_outside->x = rad2deg(gSum.lon);
        pt_outside->y = rad2deg(gSum.lat);
        return LW_SUCCESS;
}

int lwpoly_pt_outside(const LWPOLY *poly, POINT2D *pt_outside)
{
        int rv;
        /* Make sure we have boxes */
        if ( poly->bbox )
        {
                rv = gbox_pt_outside(poly->bbox, pt_outside);
        }
        else
        {
                GBOX gbox;
                lwgeom_calculate_gbox_geodetic((LWGEOM*)poly, &gbox);
                rv = gbox_pt_outside(&gbox, pt_outside);
        }

        if (rv == LW_FALSE)
                return lwpoly_pt_outside_hack(poly, pt_outside);

        return rv;
}

/**
* Given a unit geocentric gbox, return a lon/lat (degrees) coordinate point point that is
* guaranteed to be outside the box (and therefore anything it contains).
*/
int gbox_pt_outside(const GBOX *gbox, POINT2D *pt_outside)
{
        double grow = M_PI / 180.0 / 60.0; /* one arc-minute */
        int i;
        GBOX ge;
        POINT3D corners[8];
        POINT3D pt;
        GEOGRAPHIC_POINT g;

        while ( grow < M_PI )
        {
                /* Assign our box and expand it slightly. */
                ge = *gbox;
                if ( ge.xmin > -1 ) ge.xmin -= grow;
                if ( ge.ymin > -1 ) ge.ymin -= grow;
                if ( ge.zmin > -1 ) ge.zmin -= grow;
                if ( ge.xmax < 1 )  ge.xmax += grow;
                if ( ge.ymax < 1 )  ge.ymax += grow;
                if ( ge.zmax < 1 )  ge.zmax += grow;

                /* Build our eight corner points */
                corners[0].x = ge.xmin;
                corners[0].y = ge.ymin;
                corners[0].z = ge.zmin;

                corners[1].x = ge.xmin;
                corners[1].y = ge.ymax;
                corners[1].z = ge.zmin;

                corners[2].x = ge.xmin;
                corners[2].y = ge.ymin;
                corners[2].z = ge.zmax;

                corners[3].x = ge.xmax;
                corners[3].y = ge.ymin;
                corners[3].z = ge.zmin;

                corners[4].x = ge.xmax;
                corners[4].y = ge.ymax;
                corners[4].z = ge.zmin;

                corners[5].x = ge.xmax;
                corners[5].y = ge.ymin;
                corners[5].z = ge.zmax;

                corners[6].x = ge.xmin;
                corners[6].y = ge.ymax;
                corners[6].z = ge.zmax;

                corners[7].x = ge.xmax;
                corners[7].y = ge.ymax;
                corners[7].z = ge.zmax;

                LWDEBUG(4, "trying to use a box corner point...");
                for ( i = 0; i < 8; i++ )
                {
                        normalize(&(corners[i]));
                        LWDEBUGF(4, "testing corner %d: POINT(%.8g %.8g %.8g)", i, corners[i].x, corners[i].y, corners[i].z);
                        if ( ! gbox_contains_point3d(gbox, &(corners[i])) )
                        {
                                LWDEBUGF(4, "corner %d is outside our gbox", i);
                                pt = corners[i];
                                normalize(&pt);
                                cart2geog(&pt, &g);
                                pt_outside->x = rad2deg(g.lon);
                                pt_outside->y = rad2deg(g.lat);
                                LWDEBUGF(4, "returning POINT(%.8g %.8g) as outside point", pt_outside->x, pt_outside->y);
                                return LW_SUCCESS;
                        }
                }

                /* Try a wider growth to push the corners outside the original box. */
                grow *= 2.0;
        }

        /* This should never happen! */
        // lwerror("BOOM! Could not generate outside point!");
        return LW_FAILURE;
}


static int ptarray_segmentize_sphere_edge_recursive (
        const POINT3D *p1, const POINT3D *p2, /* 3-space points we are interpolating between */
        const POINT4D *v1, const POINT4D *v2, /* real values and z/m values */
        double d, double max_seg_length, /* current segment length and segment limit */
        POINTARRAY *pa) /* write out results here */
{
        GEOGRAPHIC_POINT g;
        /* Reached the terminal leaf in recursion. Add */
        /* the left-most point to the pointarray here */
        /* We recurse down the left side first, so outputs should */
        /* end up added to the array in order this way */
        if (d <= max_seg_length)
        {
                POINT4D p;
                cart2geog(p1, &g);
                p.x = v1->x;
                p.y = v1->y;
                p.z = v1->z;
                p.m = v1->m;
                return ptarray_append_point(pa, &p, LW_FALSE);
        }
        /* Find the mid-point and recurse on the left and then the right */
        else
        {
                /* Calculate mid-point */
                POINT3D mid;
                mid.x = (p1->x + p2->x) / 2.0;
                mid.y = (p1->y + p2->y) / 2.0;
                mid.z = (p1->z + p2->z) / 2.0;
                normalize(&mid);

                /* Calculate z/m mid-values */
                POINT4D midv;
                cart2geog(&mid, &g);
                midv.x = rad2deg(g.lon);
                midv.y = rad2deg(g.lat);
                midv.z = (v1->z + v2->z) / 2.0;
                midv.m = (v1->m + v2->m) / 2.0;
                /* Recurse on the left first */
                ptarray_segmentize_sphere_edge_recursive(p1, &mid, v1, &midv, d/2.0, max_seg_length, pa);
                ptarray_segmentize_sphere_edge_recursive(&mid, p2, &midv, v2, d/2.0, max_seg_length, pa);
                return LW_SUCCESS;
        }
}

/**
* Create a new point array with no segment longer than the input segment length (expressed in radians!)
* @param pa_in - input point array pointer
* @param max_seg_length - maximum output segment length in radians
*/
static POINTARRAY*
ptarray_segmentize_sphere(const POINTARRAY *pa_in, double max_seg_length)
{
        POINTARRAY *pa_out;
        int hasz = ptarray_has_z(pa_in);
        int hasm = ptarray_has_m(pa_in);
        POINT4D p1, p2;
        POINT3D q1, q2;
        GEOGRAPHIC_POINT g1, g2;
        uint32_t i;

        /* Just crap out on crazy input */
        if ( ! pa_in )
                lwerror("%s: null input pointarray", __func__);
        if ( max_seg_length <= 0.0 )
                lwerror("%s: maximum segment length must be positive", __func__);

        /* Empty starting array */
        pa_out = ptarray_construct_empty(hasz, hasm, pa_in->npoints);

        /* Simple loop per edge */
        for (i = 1; i < pa_in->npoints; i++)
        {
                getPoint4d_p(pa_in, i-1, &p1);
                getPoint4d_p(pa_in, i, &p2);
                geographic_point_init(p1.x, p1.y, &g1);
                geographic_point_init(p2.x, p2.y, &g2);

                /* Skip duplicate points (except in case of 2-point lines!) */
                if ((pa_in->npoints > 2) && p4d_same(&p1, &p2))
                        continue;

                /* How long is this edge? */
                double d = sphere_distance(&g1, &g2);

                if (d > max_seg_length)
                {
                        geog2cart(&g1, &q1);
                        geog2cart(&g2, &q2);
                        /* 3-d end points, XYZM end point, current edge size, min edge size */
                        ptarray_segmentize_sphere_edge_recursive(&q1, &q2, &p1, &p2, d, max_seg_length, pa_out);
                }
                /* If we don't segmentize, we need to add first point manually */
                else
                {
                        ptarray_append_point(pa_out, &p1, LW_TRUE);
                }
        }
        /* Always add the last point */
        ptarray_append_point(pa_out, &p2, LW_TRUE);
        return pa_out;
}

/**
* Create a new, densified geometry where no segment is longer than max_seg_length.
* Input geometry is not altered, output geometry must be freed by caller.
* @param lwg_in = input geometry
* @param max_seg_length = maximum segment length in radians
*/
LWGEOM*
lwgeom_segmentize_sphere(const LWGEOM *lwg_in, double max_seg_length)
{
        POINTARRAY *pa_out;
        LWLINE *lwline;
        LWPOLY *lwpoly_in, *lwpoly_out;
        LWCOLLECTION *lwcol_in, *lwcol_out;
        uint32_t i;

        /* Reflect NULL */
        if ( ! lwg_in )
                return NULL;

        /* Clone empty */
        if ( lwgeom_is_empty(lwg_in) )
                return lwgeom_clone(lwg_in);

        switch (lwg_in->type)
        {
        case MULTIPOINTTYPE:
        case POINTTYPE:
                return lwgeom_clone_deep(lwg_in);
                break;
        case LINETYPE:
                lwline = lwgeom_as_lwline(lwg_in);
                pa_out = ptarray_segmentize_sphere(lwline->points, max_seg_length);
                return lwline_as_lwgeom(lwline_construct(lwg_in->srid, NULL, pa_out));
                break;
        case POLYGONTYPE:
                lwpoly_in = lwgeom_as_lwpoly(lwg_in);
                lwpoly_out = lwpoly_construct_empty(lwg_in->srid, lwgeom_has_z(lwg_in), lwgeom_has_m(lwg_in));
                for ( i = 0; i < lwpoly_in->nrings; i++ )
                {
                        pa_out = ptarray_segmentize_sphere(lwpoly_in->rings[i], max_seg_length);
                        lwpoly_add_ring(lwpoly_out, pa_out);
                }
                return lwpoly_as_lwgeom(lwpoly_out);
                break;
        case MULTILINETYPE:
        case MULTIPOLYGONTYPE:
        case COLLECTIONTYPE:
                lwcol_in = lwgeom_as_lwcollection(lwg_in);
                lwcol_out = lwcollection_construct_empty(lwg_in->type, lwg_in->srid, lwgeom_has_z(lwg_in), lwgeom_has_m(lwg_in));
                for ( i = 0; i < lwcol_in->ngeoms; i++ )
                {
                        lwcollection_add_lwgeom(lwcol_out, lwgeom_segmentize_sphere(lwcol_in->geoms[i], max_seg_length));
                }
                return lwcollection_as_lwgeom(lwcol_out);
                break;
        default:
                lwerror("lwgeom_segmentize_sphere: unsupported input geometry type: %d - %s",
                        lwg_in->type, lwtype_name(lwg_in->type));
                break;
        }

        lwerror("lwgeom_segmentize_sphere got to the end of the function, should not happen");
        return NULL;
}


/**
* Returns the area of the ring (ring must be closed) in square radians (surface of
* the sphere is 4*PI).
*/
double
ptarray_area_sphere(const POINTARRAY *pa)
{
        uint32_t i;
        const POINT2D *p;
        GEOGRAPHIC_POINT a, b, c;
        double area = 0.0;

        /* Return zero on nonsensical inputs */
        if ( ! pa || pa->npoints < 4 )
                return 0.0;

        p = getPoint2d_cp(pa, 0);
        geographic_point_init(p->x, p->y, &a);
        p = getPoint2d_cp(pa, 1);
        geographic_point_init(p->x, p->y, &b);

        for ( i = 2; i < pa->npoints-1; i++ )
        {
                p = getPoint2d_cp(pa, i);
                geographic_point_init(p->x, p->y, &c);
                area += sphere_signed_area(&a, &b, &c);
                b = c;
        }

        return fabs(area);
}


static double ptarray_distance_spheroid(const POINTARRAY *pa1, const POINTARRAY *pa2, const SPHEROID *s, double tolerance, int check_intersection)
{
        GEOGRAPHIC_EDGE e1, e2;
        GEOGRAPHIC_POINT g1, g2;
        GEOGRAPHIC_POINT nearest1, nearest2;
        POINT3D A1, A2, B1, B2;
        const POINT2D *p;
        double distance;
        uint32_t i, j;
        int use_sphere = (s->a == s->b ? 1 : 0);

        /* Make result really big, so that everything will be smaller than it */
        distance = FLT_MAX;

        /* Empty point arrays? Return negative */
        if ( pa1->npoints == 0 || pa2->npoints == 0 )
                return -1.0;

        /* Handle point/point case here */
        if ( pa1->npoints == 1 && pa2->npoints == 1 )
        {
                p = getPoint2d_cp(pa1, 0);
                geographic_point_init(p->x, p->y, &g1);
                p = getPoint2d_cp(pa2, 0);
                geographic_point_init(p->x, p->y, &g2);
                /* Sphere special case, axes equal */
                distance = s->radius * sphere_distance(&g1, &g2);
                if ( use_sphere )
                        return distance;
                /* Below tolerance, actual distance isn't of interest */
                else if ( distance < 0.95 * tolerance )
                        return distance;
                /* Close or greater than tolerance, get the real answer to be sure */
                else
                        return spheroid_distance(&g1, &g2, s);
        }

        /* Handle point/line case here */
        if ( pa1->npoints == 1 || pa2->npoints == 1 )
        {
                /* Handle one/many case here */
                uint32_t i;
                const POINTARRAY *pa_one;
                const POINTARRAY *pa_many;

                if ( pa1->npoints == 1 )
                {
                        pa_one = pa1;
                        pa_many = pa2;
                }
                else
                {
                        pa_one = pa2;
                        pa_many = pa1;
                }

                /* Initialize our point */
                p = getPoint2d_cp(pa_one, 0);
                geographic_point_init(p->x, p->y, &g1);

                /* Initialize start of line */
                p = getPoint2d_cp(pa_many, 0);
                geographic_point_init(p->x, p->y, &(e1.start));

                /* Iterate through the edges in our line */
                for ( i = 1; i < pa_many->npoints; i++ )
                {
                        double d;
                        p = getPoint2d_cp(pa_many, i);
                        geographic_point_init(p->x, p->y, &(e1.end));
                        /* Get the spherical distance between point and edge */
                        d = s->radius * edge_distance_to_point(&e1, &g1, &g2);
                        /* New shortest distance! Record this distance / location */
                        if ( d < distance )
                        {
                                distance = d;
                                nearest2 = g2;
                        }
                        /* We've gotten closer than the tolerance... */
                        if ( d < tolerance )
                        {
                                /* Working on a sphere? The answer is correct, return */
                                if ( use_sphere )
                                {
                                        return d;
                                }
                                /* Far enough past the tolerance that the spheroid calculation won't change things */
                                else if ( d < tolerance * 0.95 )
                                {
                                        return d;
                                }
                                /* On a spheroid and near the tolerance? Confirm that we are *actually* closer than tolerance */
                                else
                                {
                                        d = spheroid_distance(&g1, &nearest2, s);
                                        /* Yes, closer than tolerance, return! */
                                        if ( d < tolerance )
                                                return d;
                                }
                        }
                        e1.start = e1.end;
                }

                /* On sphere, return answer */
                if ( use_sphere )
                        return distance;
                /* On spheroid, calculate final answer based on closest approach */
                else
                        return spheroid_distance(&g1, &nearest2, s);

        }

        /* Initialize start of line 1 */
        p = getPoint2d_cp(pa1, 0);
        geographic_point_init(p->x, p->y, &(e1.start));
        geog2cart(&(e1.start), &A1);


        /* Handle line/line case */
        for ( i = 1; i < pa1->npoints; i++ )
        {
                p = getPoint2d_cp(pa1, i);
                geographic_point_init(p->x, p->y, &(e1.end));
                geog2cart(&(e1.end), &A2);

                /* Initialize start of line 2 */
                p = getPoint2d_cp(pa2, 0);
                geographic_point_init(p->x, p->y, &(e2.start));
                geog2cart(&(e2.start), &B1);

                for ( j = 1; j < pa2->npoints; j++ )
                {
                        double d;

                        p = getPoint2d_cp(pa2, j);
                        geographic_point_init(p->x, p->y, &(e2.end));
                        geog2cart(&(e2.end), &B2);

                        LWDEBUGF(4, "e1.start == GPOINT(%.6g %.6g) ", e1.start.lat, e1.start.lon);
                        LWDEBUGF(4, "e1.end == GPOINT(%.6g %.6g) ", e1.end.lat, e1.end.lon);
                        LWDEBUGF(4, "e2.start == GPOINT(%.6g %.6g) ", e2.start.lat, e2.start.lon);
                        LWDEBUGF(4, "e2.end == GPOINT(%.6g %.6g) ", e2.end.lat, e2.end.lon);

                        if ( check_intersection && edge_intersects(&A1, &A2, &B1, &B2) )
                        {
                                LWDEBUG(4,"edge intersection! returning 0.0");
                                return 0.0;
                        }
                        d = s->radius * edge_distance_to_edge(&e1, &e2, &g1, &g2);
                        LWDEBUGF(4,"got edge_distance_to_edge %.8g", d);

                        if ( d < distance )
                        {
                                distance = d;
                                nearest1 = g1;
                                nearest2 = g2;
                        }
                        if ( d < tolerance )
                        {
                                if ( use_sphere )
                                {
                                        return d;
                                }
                                else
                                {
                                        d = spheroid_distance(&nearest1, &nearest2, s);
                                        if ( d < tolerance )
                                                return d;
                                }
                        }

                        /* Copy end to start to allow a new end value in next iteration */
                        e2.start = e2.end;
                        B1 = B2;
                }

                /* Copy end to start to allow a new end value in next iteration */
                e1.start = e1.end;
                A1 = A2;
                LW_ON_INTERRUPT(return -1.0);
        }
        LWDEBUGF(4,"finished all loops, returning %.8g", distance);

        if ( use_sphere )
                return distance;
        else
                return spheroid_distance(&nearest1, &nearest2, s);
}


/**
* Calculate the area of an LWGEOM. Anything except POLYGON, MULTIPOLYGON
* and GEOMETRYCOLLECTION return zero immediately. Multi's recurse, polygons
* calculate external ring area and subtract internal ring area. A GBOX is
* required to calculate an outside point.
*/
double lwgeom_area_sphere(const LWGEOM *lwgeom, const SPHEROID *spheroid)
{
        int type;
        double radius2 = spheroid->radius * spheroid->radius;

        assert(lwgeom);

        /* No area in nothing */
        if ( lwgeom_is_empty(lwgeom) )
                return 0.0;

        /* Read the geometry type number */
        type = lwgeom->type;

        /* Anything but polygons and collections returns zero */
        if ( ! ( type == POLYGONTYPE || type == MULTIPOLYGONTYPE || type == COLLECTIONTYPE ) )
                return 0.0;

        /* Actually calculate area */
        if ( type == POLYGONTYPE )
        {
                LWPOLY *poly = (LWPOLY*)lwgeom;
                uint32_t i;
                double area = 0.0;

                /* Just in case there's no rings */
                if ( poly->nrings < 1 )
                        return 0.0;

                /* First, the area of the outer ring */
                area += radius2 * ptarray_area_sphere(poly->rings[0]);

                /* Subtract areas of inner rings */
                for ( i = 1; i < poly->nrings; i++ )
                {
                        area -= radius2 * ptarray_area_sphere(poly->rings[i]);
                }
                return area;
        }

        /* Recurse into sub-geometries to get area */
        if ( type == MULTIPOLYGONTYPE || type == COLLECTIONTYPE )
        {
                LWCOLLECTION *col = (LWCOLLECTION*)lwgeom;
                uint32_t i;
                double area = 0.0;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        area += lwgeom_area_sphere(col->geoms[i], spheroid);
                }
                return area;
        }

        /* Shouldn't get here. */
        return 0.0;
}


/**
* Calculate a projected point given a source point, a distance and a bearing.
* @param r - location of first point.
* @param spheroid - spheroid definition.
* @param distance - distance, in units of the spheroid def'n.
* @param azimuth - azimuth in radians.
* @return s - location of projected point.
*
*/
LWPOINT* lwgeom_project_spheroid(const LWPOINT *r, const SPHEROID *spheroid, double distance, double azimuth)
{
        GEOGRAPHIC_POINT geo_source, geo_dest;
        POINT4D pt_dest;
        double x, y;
        POINTARRAY *pa;
        LWPOINT *lwp;

        /* Normalize distance to be positive*/
        if ( distance < 0.0 ) {
                distance = -distance;
                azimuth += M_PI;
        }

        /* Normalize azimuth */
        azimuth -= 2.0 * M_PI * floor(azimuth / (2.0 * M_PI));

        /* Check the distance validity */
        if ( distance > (M_PI * spheroid->radius) )
        {
                lwerror("Distance must not be greater than %g", M_PI * spheroid->radius);
                return NULL;
        }

        /* Convert to ta geodetic point */
        x = lwpoint_get_x(r);
        y = lwpoint_get_y(r);
        geographic_point_init(x, y, &geo_source);

        /* Try the projection */
        if( spheroid_project(&geo_source, spheroid, distance, azimuth, &geo_dest) == LW_FAILURE )
        {
                LWDEBUGF(3, "Unable to project from (%g %g) with azimuth %g and distance %g", x, y, azimuth, distance);
                lwerror("Unable to project from (%g %g) with azimuth %g and distance %g", x, y, azimuth, distance);
                return NULL;
        }

        /* Build the output LWPOINT */
        pa = ptarray_construct(0, 0, 1);
        pt_dest.x = rad2deg(longitude_radians_normalize(geo_dest.lon));
        pt_dest.y = rad2deg(latitude_radians_normalize(geo_dest.lat));
        pt_dest.z = pt_dest.m = 0.0;
        ptarray_set_point4d(pa, 0, &pt_dest);
        lwp = lwpoint_construct(r->srid, NULL, pa);
        lwgeom_set_geodetic(lwpoint_as_lwgeom(lwp), LW_TRUE);
        return lwp;
}


/**
* Calculate a bearing (azimuth) given a source and destination point.
* @param r - location of first point.
* @param s - location of second point.
* @param spheroid - spheroid definition.
* @return azimuth - azimuth in radians.
*
*/
double lwgeom_azumith_spheroid(const LWPOINT *r, const LWPOINT *s, const SPHEROID *spheroid)
{
        GEOGRAPHIC_POINT g1, g2;
        double x1, y1, x2, y2;

        /* Convert r to a geodetic point */
        x1 = lwpoint_get_x(r);
        y1 = lwpoint_get_y(r);
        geographic_point_init(x1, y1, &g1);

        /* Convert s to a geodetic point */
        x2 = lwpoint_get_x(s);
        y2 = lwpoint_get_y(s);
        geographic_point_init(x2, y2, &g2);

        /* Same point, return NaN */
        if ( FP_EQUALS(x1, x2) && FP_EQUALS(y1, y2) )
        {
                return NAN;
        }

        /* Do the direction calculation */
        return spheroid_direction(&g1, &g2, spheroid);
}

/**
* Calculate the distance between two LWGEOMs, using the coordinates are
* longitude and latitude. Return immediately when the calculated distance drops
* below the tolerance (useful for dwithin calculations).
* Return a negative distance for incalculable cases.
*/
double lwgeom_distance_spheroid(const LWGEOM *lwgeom1, const LWGEOM *lwgeom2, const SPHEROID *spheroid, double tolerance)
{
        uint8_t type1, type2;
        int check_intersection = LW_FALSE;
        GBOX gbox1, gbox2;

        gbox_init(&gbox1);
        gbox_init(&gbox2);

        assert(lwgeom1);
        assert(lwgeom2);

        LWDEBUGF(4, "entered function, tolerance %.8g", tolerance);

        /* What's the distance to an empty geometry? We don't know.
           Return a negative number so the caller can catch this case. */
        if ( lwgeom_is_empty(lwgeom1) || lwgeom_is_empty(lwgeom2) )
        {
                return -1.0;
        }

        type1 = lwgeom1->type;
        type2 = lwgeom2->type;

        /* Make sure we have boxes */
        if ( lwgeom1->bbox )
                gbox1 = *(lwgeom1->bbox);
        else
                lwgeom_calculate_gbox_geodetic(lwgeom1, &gbox1);

        /* Make sure we have boxes */
        if ( lwgeom2->bbox )
                gbox2 = *(lwgeom2->bbox);
        else
                lwgeom_calculate_gbox_geodetic(lwgeom2, &gbox2);

        /* If the boxes aren't disjoint, we have to check for edge intersections */
        if ( gbox_overlaps(&gbox1, &gbox2) )
                check_intersection = LW_TRUE;

        /* Point/line combinations can all be handled with simple point array iterations */
        if ( ( type1 == POINTTYPE || type1 == LINETYPE ) &&
             ( type2 == POINTTYPE || type2 == LINETYPE ) )
        {
                POINTARRAY *pa1, *pa2;

                if ( type1 == POINTTYPE )
                        pa1 = ((LWPOINT*)lwgeom1)->point;
                else
                        pa1 = ((LWLINE*)lwgeom1)->points;

                if ( type2 == POINTTYPE )
                        pa2 = ((LWPOINT*)lwgeom2)->point;
                else
                        pa2 = ((LWLINE*)lwgeom2)->points;

                return ptarray_distance_spheroid(pa1, pa2, spheroid, tolerance, check_intersection);
        }

        /* Point/Polygon cases, if point-in-poly, return zero, else return distance. */
        if ( ( type1 == POLYGONTYPE && type2 == POINTTYPE ) ||
             ( type2 == POLYGONTYPE && type1 == POINTTYPE ) )
        {
                const POINT2D *p;
                LWPOLY *lwpoly;
                LWPOINT *lwpt;
                double distance = FLT_MAX;
                uint32_t i;

                if ( type1 == POINTTYPE )
                {
                        lwpt = (LWPOINT*)lwgeom1;
                        lwpoly = (LWPOLY*)lwgeom2;
                }
                else
                {
                        lwpt = (LWPOINT*)lwgeom2;
                        lwpoly = (LWPOLY*)lwgeom1;
                }
                p = getPoint2d_cp(lwpt->point, 0);

                /* Point in polygon implies zero distance */
                if ( lwpoly_covers_point2d(lwpoly, p) )
                {
                        return 0.0;
                }

                /* Not inside, so what's the actual distance? */
                for ( i = 0; i < lwpoly->nrings; i++ )
                {
                        double ring_distance = ptarray_distance_spheroid(lwpoly->rings[i], lwpt->point, spheroid, tolerance, check_intersection);
                        if ( ring_distance < distance )
                                distance = ring_distance;
                        if ( distance < tolerance )
                                return distance;
                }
                return distance;
        }

        /* Line/polygon case, if start point-in-poly, return zero, else return distance. */
        if ( ( type1 == POLYGONTYPE && type2 == LINETYPE ) ||
             ( type2 == POLYGONTYPE && type1 == LINETYPE ) )
        {
                const POINT2D *p;
                LWPOLY *lwpoly;
                LWLINE *lwline;
                double distance = FLT_MAX;
                uint32_t i;

                if ( type1 == LINETYPE )
                {
                        lwline = (LWLINE*)lwgeom1;
                        lwpoly = (LWPOLY*)lwgeom2;
                }
                else
                {
                        lwline = (LWLINE*)lwgeom2;
                        lwpoly = (LWPOLY*)lwgeom1;
                }
                p = getPoint2d_cp(lwline->points, 0);

                LWDEBUG(4, "checking if a point of line is in polygon");

                /* Point in polygon implies zero distance */
                if ( lwpoly_covers_point2d(lwpoly, p) )
                        return 0.0;

                LWDEBUG(4, "checking ring distances");

                /* Not contained, so what's the actual distance? */
                for ( i = 0; i < lwpoly->nrings; i++ )
                {
                        double ring_distance = ptarray_distance_spheroid(lwpoly->rings[i], lwline->points, spheroid, tolerance, check_intersection);
                        LWDEBUGF(4, "ring[%d] ring_distance = %.8g", i, ring_distance);
                        if ( ring_distance < distance )
                                distance = ring_distance;
                        if ( distance < tolerance )
                                return distance;
                }
                LWDEBUGF(4, "all rings checked, returning distance = %.8g", distance);
                return distance;

        }

        /* Polygon/polygon case, if start point-in-poly, return zero, else
         * return distance. */
        if (type1 == POLYGONTYPE && type2 == POLYGONTYPE)
        {
                const POINT2D* p;
                LWPOLY* lwpoly1 = (LWPOLY*)lwgeom1;
                LWPOLY* lwpoly2 = (LWPOLY*)lwgeom2;
                double distance = FLT_MAX;
                uint32_t i, j;

                /* Point of 2 in polygon 1 implies zero distance */
                p = getPoint2d_cp(lwpoly1->rings[0], 0);
                if (lwpoly_covers_point2d(lwpoly2, p)) return 0.0;

                /* Point of 1 in polygon 2 implies zero distance */
                p = getPoint2d_cp(lwpoly2->rings[0], 0);
                if (lwpoly_covers_point2d(lwpoly1, p)) return 0.0;

                /* Not contained, so what's the actual distance? */
                for (i = 0; i < lwpoly1->nrings; i++)
                {
                        for (j = 0; j < lwpoly2->nrings; j++)
                        {
                                double ring_distance =
                                    ptarray_distance_spheroid(
                                        lwpoly1->rings[i],
                                        lwpoly2->rings[j],
                                        spheroid,
                                        tolerance,
                                        check_intersection);
                                if (ring_distance < distance)
                                        distance = ring_distance;
                                if (distance < tolerance) return distance;
                        }
                }
                return distance;
        }

        /* Recurse into collections */
        if ( lwtype_is_collection(type1) )
        {
                uint32_t i;
                double distance = FLT_MAX;
                LWCOLLECTION *col = (LWCOLLECTION*)lwgeom1;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        double geom_distance = lwgeom_distance_spheroid(
                            col->geoms[i], lwgeom2, spheroid, tolerance);
                        if ( geom_distance < distance )
                                distance = geom_distance;
                        if ( distance < tolerance )
                                return distance;
                }
                return distance;
        }

        /* Recurse into collections */
        if ( lwtype_is_collection(type2) )
        {
                uint32_t i;
                double distance = FLT_MAX;
                LWCOLLECTION *col = (LWCOLLECTION*)lwgeom2;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        double geom_distance = lwgeom_distance_spheroid(lwgeom1, col->geoms[i], spheroid, tolerance);
                        if ( geom_distance < distance )
                                distance = geom_distance;
                        if ( distance < tolerance )
                                return distance;
                }
                return distance;
        }


        lwerror("arguments include unsupported geometry type (%s, %s)", lwtype_name(type1), lwtype_name(type1));
        return -1.0;

}


int lwgeom_covers_lwgeom_sphere(const LWGEOM *lwgeom1, const LWGEOM *lwgeom2)
{
        int type1, type2;
        GBOX gbox1, gbox2;
        gbox1.flags = gbox2.flags = 0;

        assert(lwgeom1);
        assert(lwgeom2);

        type1 = lwgeom1->type;
        type2 = lwgeom2->type;

        /* dim(geom2) > dim(geom1) always returns false (because geom2 is bigger) */
        if ( (type1 == POINTTYPE && type2 == LINETYPE)
                || (type1 == POINTTYPE && type2 == POLYGONTYPE)
                || (type1 == LINETYPE && type2 == POLYGONTYPE) )
        {
                LWDEBUG(4, "dimension of geom2 is bigger than geom1");
                return LW_FALSE;
        }

        /* Make sure we have boxes */
        if ( lwgeom1->bbox )
                gbox1 = *(lwgeom1->bbox);
        else
                lwgeom_calculate_gbox_geodetic(lwgeom1, &gbox1);

        /* Make sure we have boxes */
        if ( lwgeom2->bbox )
                gbox2 = *(lwgeom2->bbox);
        else
                lwgeom_calculate_gbox_geodetic(lwgeom2, &gbox2);


        /* Handle the polygon/point case */
        if ( type1 == POLYGONTYPE && type2 == POINTTYPE )
        {
                POINT2D pt_to_test;
                getPoint2d_p(((LWPOINT*)lwgeom2)->point, 0, &pt_to_test);
                return lwpoly_covers_point2d((LWPOLY*)lwgeom1, &pt_to_test);
        }
        else if ( type1 == POLYGONTYPE && type2 == LINETYPE)
        {
                return lwpoly_covers_lwline((LWPOLY*)lwgeom1, (LWLINE*)lwgeom2);
        }
        else if ( type1 == POLYGONTYPE && type2 == POLYGONTYPE)
        {
                return lwpoly_covers_lwpoly((LWPOLY*)lwgeom1, (LWPOLY*)lwgeom2);
        }
        else if ( type1 == LINETYPE && type2 == POINTTYPE)
        {
                return lwline_covers_lwpoint((LWLINE*)lwgeom1, (LWPOINT*)lwgeom2);
        }
        else if ( type1 == LINETYPE && type2 == LINETYPE)
        {
                return lwline_covers_lwline((LWLINE*)lwgeom1, (LWLINE*)lwgeom2);
        }
        else if ( type1 == POINTTYPE && type2 == POINTTYPE)
        {
                return lwpoint_same((LWPOINT*)lwgeom1, (LWPOINT*)lwgeom2);
        }

        /* If any of the first argument parts covers the second argument, it's true */
        if ( lwtype_is_collection( type1 ) )
        {
                uint32_t i;
                LWCOLLECTION *col = (LWCOLLECTION*)lwgeom1;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        if ( lwgeom_covers_lwgeom_sphere(col->geoms[i], lwgeom2) )
                        {
                                return LW_TRUE;
                        }
                }
                return LW_FALSE;
        }

        /* Only if all of the second arguments are covered by the first argument is the condition true */
        if ( lwtype_is_collection( type2 ) )
        {
                uint32_t i;
                LWCOLLECTION *col = (LWCOLLECTION*)lwgeom2;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        if ( ! lwgeom_covers_lwgeom_sphere(lwgeom1, col->geoms[i]) )
                        {
                                return LW_FALSE;
                        }
                }
                return LW_TRUE;
        }

        /* Don't get here */
        lwerror("lwgeom_covers_lwgeom_sphere: reached end of function without resolution");
        return LW_FALSE;

}

/**
* Given a polygon (lon/lat decimal degrees) and point (lon/lat decimal degrees) and
* a guaranteed outside point (lon/lat decimal degrees) (calculate with gbox_pt_outside())
* return LW_TRUE if point is inside or on edge of polygon.
*/
int lwpoly_covers_point2d(const LWPOLY *poly, const POINT2D *pt_to_test)
{
        uint32_t i;
        int in_hole_count = 0;
        POINT3D p;
        GEOGRAPHIC_POINT gpt_to_test;
        POINT2D pt_outside;
        GBOX gbox;
#if POSTGIS_DEBUG_LEVEL >= 4
        char *geom_ewkt;
#endif
        gbox.flags = 0;

        /* Nulls and empties don't contain anything! */
        if ( ! poly || lwgeom_is_empty((LWGEOM*)poly) )
        {
                LWDEBUG(4,"returning false, geometry is empty or null");
                return LW_FALSE;
        }

        /* Make sure we have boxes */
        if ( poly->bbox )
                gbox = *(poly->bbox);
        else
                lwgeom_calculate_gbox_geodetic((LWGEOM*)poly, &gbox);

        /* Point not in box? Done! */
        geographic_point_init(pt_to_test->x, pt_to_test->y, &gpt_to_test);
        geog2cart(&gpt_to_test, &p);
        if ( ! gbox_contains_point3d(&gbox, &p) )
        {
                LWDEBUG(4, "the point is not in the box!");
                return LW_FALSE;
        }

        /* Calculate our outside point from the gbox */
        lwpoly_pt_outside(poly, &pt_outside);

        LWDEBUGF(4, "pt_outside POINT(%.18g %.18g)", pt_outside.x, pt_outside.y);
        LWDEBUGF(4, "pt_to_test POINT(%.18g %.18g)", pt_to_test->x, pt_to_test->y);
#if POSTGIS_DEBUG_LEVEL >= 4
        geom_ewkt = lwgeom_to_ewkt((LWGEOM*)poly);
        LWDEBUGF(4, "polygon %s", geom_ewkt);
        lwfree(geom_ewkt);
        geom_ewkt = gbox_to_string(&gbox);
        LWDEBUGF(4, "gbox %s", geom_ewkt);
        lwfree(geom_ewkt);
#endif

        /* Not in outer ring? We're done! */
        if ( ! ptarray_contains_point_sphere(poly->rings[0], &pt_outside, pt_to_test) )
        {
                LWDEBUG(4,"returning false, point is outside ring");
                return LW_FALSE;
        }

        LWDEBUGF(4, "testing %d rings", poly->nrings);

        /* But maybe point is in a hole... */
        for ( i = 1; i < poly->nrings; i++ )
        {
                LWDEBUGF(4, "ring test loop %d", i);
                /* Count up hole containment. Odd => outside boundary. */
                if ( ptarray_contains_point_sphere(poly->rings[i], &pt_outside, pt_to_test) )
                        in_hole_count++;
        }

        LWDEBUGF(4, "in_hole_count == %d", in_hole_count);

        if ( in_hole_count % 2 )
        {
                LWDEBUG(4,"returning false, inner ring containment count is odd");
                return LW_FALSE;
        }

        LWDEBUG(4,"returning true, inner ring containment count is even");
        return LW_TRUE;
}

/**
 * Given a polygon1 check if all points of polygon2 are inside polygon1 and no
 * intersections of the polygon edges occur.
 * return LW_TRUE if polygon is inside or on edge of polygon.
 */
int lwpoly_covers_lwpoly(const LWPOLY *poly1, const LWPOLY *poly2)
{
        uint32_t i;

        /* Nulls and empties don't contain anything! */
        if ( ! poly1 || lwgeom_is_empty((LWGEOM*)poly1) )
        {
                LWDEBUG(4,"returning false, geometry1 is empty or null");
                return LW_FALSE;
        }

        /* Nulls and empties don't contain anything! */
        if ( ! poly2 || lwgeom_is_empty((LWGEOM*)poly2) )
        {
                LWDEBUG(4,"returning false, geometry2 is empty or null");
                return LW_FALSE;
        }

        /* check if all vertices of poly2 are inside poly1 */
        for (i = 0; i < poly2->nrings; i++)
        {

                /* every other ring is a hole, check if point is inside the actual polygon */
                if ( i % 2 == 0)
                {
                        if (LW_FALSE == lwpoly_covers_pointarray(poly1, poly2->rings[i]))
                        {
                                LWDEBUG(4,"returning false, geometry2 has point outside of geometry1");
                                return LW_FALSE;
                        }
                }
                else
                {
                        if (LW_TRUE == lwpoly_covers_pointarray(poly1, poly2->rings[i]))
                        {
                                LWDEBUG(4,"returning false, geometry2 has point inside a hole of geometry1");
                                return LW_FALSE;
                        }
                }
        }

        /* check for any edge intersections, so nothing is partially outside of poly1 */
        for (i = 0; i < poly2->nrings; i++)
        {
                if (LW_TRUE == lwpoly_intersects_line(poly1, poly2->rings[i]))
                {
                        LWDEBUG(4,"returning false, geometry2 is partially outside of geometry1");
                        return LW_FALSE;
                }
        }

        /* no abort condition found, so the poly2 should be completly inside poly1 */
        return LW_TRUE;
}

/**
 *
 */
int lwpoly_covers_lwline(const LWPOLY *poly, const LWLINE *line)
{
   /* Nulls and empties don't contain anything! */
   if ( ! poly || lwgeom_is_empty((LWGEOM*)poly) )
   {
           LWDEBUG(4,"returning false, geometry1 is empty or null");
           return LW_FALSE;
   }

   /* Nulls and empties don't contain anything! */
   if ( ! line || lwgeom_is_empty((LWGEOM*)line) )
   {
           LWDEBUG(4,"returning false, geometry2 is empty or null");
           return LW_FALSE;
   }

   if (LW_FALSE == lwpoly_covers_pointarray(poly, line->points))
   {
           LWDEBUG(4,"returning false, geometry2 has point outside of geometry1");
           return LW_FALSE;
   }

   /* check for any edge intersections, so nothing is partially outside of poly1 */
   if (LW_TRUE == lwpoly_intersects_line(poly, line->points))
   {
           LWDEBUG(4,"returning false, geometry2 is partially outside of geometry1");
           return LW_FALSE;
   }

   /* no abort condition found, so the poly2 should be completely inside poly1 */
   return LW_TRUE;
}

/**
 * return LW_TRUE if all points are inside the polygon
 */
int lwpoly_covers_pointarray(const LWPOLY* lwpoly, const POINTARRAY* pta)
{
        uint32_t i;
        for (i = 0; i < pta->npoints; i++) {
                const POINT2D* pt_to_test = getPoint2d_cp(pta, i);

                if ( LW_FALSE == lwpoly_covers_point2d(lwpoly, pt_to_test) ) {
                        LWDEBUG(4,"returning false, geometry2 has point outside of geometry1");
                        return LW_FALSE;
                }
        }

        return LW_TRUE;
}

/**
 * Checks if any edges of lwpoly intersect with the line formed by the pointarray
 * return LW_TRUE if any intersection between the given polygon and the line
 */
int lwpoly_intersects_line(const LWPOLY* lwpoly, const POINTARRAY* line)
{
        uint32_t i, j, k;
        POINT3D pa1, pa2, pb1, pb2;
        for (i = 0; i < lwpoly->nrings; i++)
        {
                for (j = 0; j < lwpoly->rings[i]->npoints - 1; j++)
                {
                        const POINT2D* a1 = getPoint2d_cp(lwpoly->rings[i], j);
                        const POINT2D* a2 = getPoint2d_cp(lwpoly->rings[i], j+1);

                        /* Set up our stab line */
                        ll2cart(a1, &pa1);
                        ll2cart(a2, &pa2);

                        for (k = 0; k < line->npoints - 1; k++)
                        {
                                const POINT2D* b1 = getPoint2d_cp(line, k);
                                const POINT2D* b2 = getPoint2d_cp(line, k+1);

                                /* Set up our stab line */
                                ll2cart(b1, &pb1);
                                ll2cart(b2, &pb2);

                                int inter = edge_intersects(&pa1, &pa2, &pb1, &pb2);

                                /* ignore same edges */
                                if (inter & PIR_INTERSECTS
                                        && !(inter & PIR_B_TOUCH_RIGHT || inter & PIR_COLINEAR) )
                                {
                                        return LW_TRUE;
                                }
                        }
                }
        }

        return LW_FALSE;
}

/**
 * return LW_TRUE if any of the line segments covers the point
 */
int lwline_covers_lwpoint(const LWLINE* lwline, const LWPOINT* lwpoint)
{
        uint32_t i;
        GEOGRAPHIC_POINT p;
        GEOGRAPHIC_EDGE e;

        for ( i = 0; i < lwline->points->npoints - 1; i++)
        {
                const POINT2D* a1 = getPoint2d_cp(lwline->points, i);
                const POINT2D* a2 = getPoint2d_cp(lwline->points, i+1);

                geographic_point_init(a1->x, a1->y, &(e.start));
                geographic_point_init(a2->x, a2->y, &(e.end));

                geographic_point_init(lwpoint_get_x(lwpoint), lwpoint_get_y(lwpoint), &p);

                if ( edge_contains_point(&e, &p) ) {
                        return LW_TRUE;
                }
        }

        return LW_FALSE;
}

/**
 * Check if first and last point of line2 are covered by line1 and then each
 * point in between has to be one line1 in the exact same order
 * return LW_TRUE if all edge points of line2 are on line1
 */
int lwline_covers_lwline(const LWLINE* lwline1, const LWLINE* lwline2)
{
        uint32_t i, j;
        GEOGRAPHIC_EDGE e1, e2;
        GEOGRAPHIC_POINT p1, p2;
        int start = LW_FALSE;
        int changed = LW_FALSE;

        /* first point on line */
        if ( ! lwline_covers_lwpoint(lwline1, lwline_get_lwpoint(lwline2, 0)))
        {
                LWDEBUG(4,"returning false, first point of line2 is not covered by line1");
                return LW_FALSE;
        }

        /* last point on line */
        if ( ! lwline_covers_lwpoint(lwline1, lwline_get_lwpoint(lwline2, lwline2->points->npoints - 1)))
        {
                LWDEBUG(4,"returning false, last point of line2 is not covered by line1");
                return LW_FALSE;
        }

        j = 0;
        i = 0;
        while (i < lwline1->points->npoints - 1 && j < lwline2->points->npoints - 1)
        {
                changed = LW_FALSE;
                const POINT2D* a1 = getPoint2d_cp(lwline1->points, i);
                const POINT2D* a2 = getPoint2d_cp(lwline1->points, i+1);
                const POINT2D* b1 = getPoint2d_cp(lwline2->points, j);
                const POINT2D* b2 = getPoint2d_cp(lwline2->points, j+1);

                geographic_point_init(a1->x, a1->y, &(e1.start));
                geographic_point_init(a2->x, a2->y, &(e1.end));
                geographic_point_init(b1->x, b1->y, &p2);

                /* we already know, that the last point is on line1, so we're done */
                if ( j == lwline2->points->npoints - 1)
                {
                        return LW_TRUE;
                }
                else if (start == LW_TRUE)
                {
                        /* point is on current line1 edge, check next point in line2 */
                        if ( edge_contains_point(&e1, &p2)) {
                                j++;
                                changed = LW_TRUE;
                        }

                        geographic_point_init(a1->x, a1->y, &(e2.start));
                        geographic_point_init(a2->x, b2->y, &(e2.end));
                        geographic_point_init(a1->x, a1->y, &p1);

                        /* point is on current line2 edge, check next point in line1 */
                        if ( edge_contains_point(&e2, &p1)) {
                                i++;
                                changed = LW_TRUE;
                        }

                        /* no edge progressed -> point left one line */
                        if ( changed == LW_FALSE )
                        {
                                LWDEBUG(4,"returning false, found point not covered by both lines");
                                return LW_FALSE;
                        }
                        else
                        {
                                continue;
                        }
                }

                /* find first edge to cover line2 */
                if (edge_contains_point(&e1, &p2))
                {
                        start = LW_TRUE;
                }

                /* next line1 edge */
                i++;
        }

        /* no uncovered point found */
        return LW_TRUE;
}

/**
* This function can only be used on LWGEOM that is built on top of
* GSERIALIZED, otherwise alignment errors will ensue.
*/
int getPoint2d_p_ro(const POINTARRAY *pa, uint32_t n, POINT2D **point)
{
        uint8_t *pa_ptr = NULL;
        assert(pa);
        assert(n < pa->npoints);

        pa_ptr = getPoint_internal(pa, n);
        /* printf( "pa_ptr[0]: %g\n", *((double*)pa_ptr)); */
        *point = (POINT2D*)pa_ptr;

        return LW_SUCCESS;
}


int ptarray_calculate_gbox_geodetic(const POINTARRAY *pa, GBOX *gbox)
{
        uint32_t i;
        int first = LW_TRUE;
        const POINT2D *p;
        POINT3D A1, A2;
        GBOX edge_gbox;

        assert(gbox);
        assert(pa);

        gbox_init(&edge_gbox);
        edge_gbox.flags = gbox->flags;

        if ( pa->npoints == 0 ) return LW_FAILURE;

        if ( pa->npoints == 1 )
        {
                p = getPoint2d_cp(pa, 0);
                ll2cart(p, &A1);
                gbox->xmin = gbox->xmax = A1.x;
                gbox->ymin = gbox->ymax = A1.y;
                gbox->zmin = gbox->zmax = A1.z;
                return LW_SUCCESS;
        }

        p = getPoint2d_cp(pa, 0);
        ll2cart(p, &A1);

        for ( i = 1; i < pa->npoints; i++ )
        {

                p = getPoint2d_cp(pa, i);
                ll2cart(p, &A2);

                edge_calculate_gbox(&A1, &A2, &edge_gbox);

                /* Initialize the box */
                if ( first )
                {
                        gbox_duplicate(&edge_gbox, gbox);
                        first = LW_FALSE;
                }
                /* Expand the box where necessary */
                else
                {
                        gbox_merge(&edge_gbox, gbox);
                }

                A1 = A2;
        }

        return LW_SUCCESS;
}

static int lwpoint_calculate_gbox_geodetic(const LWPOINT *point, GBOX *gbox)
{
        assert(point);
        return ptarray_calculate_gbox_geodetic(point->point, gbox);
}

static int lwline_calculate_gbox_geodetic(const LWLINE *line, GBOX *gbox)
{
        assert(line);
        return ptarray_calculate_gbox_geodetic(line->points, gbox);
}

static int lwpolygon_calculate_gbox_geodetic(const LWPOLY *poly, GBOX *gbox)
{
        GBOX ringbox;
        uint32_t i;
        int first = LW_TRUE;
        assert(poly);
        if ( poly->nrings == 0 )
                return LW_FAILURE;
        ringbox.flags = gbox->flags;
        for ( i = 0; i < poly->nrings; i++ )
        {
                if ( ptarray_calculate_gbox_geodetic(poly->rings[i], &ringbox) == LW_FAILURE )
                        return LW_FAILURE;
                if ( first )
                {
                        gbox_duplicate(&ringbox, gbox);
                        first = LW_FALSE;
                }
                else
                {
                        gbox_merge(&ringbox, gbox);
                }
        }

        /* If the box wraps a poly, push that axis to the absolute min/max as appropriate */
        gbox_check_poles(gbox);

        return LW_SUCCESS;
}

static int lwtriangle_calculate_gbox_geodetic(const LWTRIANGLE *triangle, GBOX *gbox)
{
        assert(triangle);
        return ptarray_calculate_gbox_geodetic(triangle->points, gbox);
}


static int lwcollection_calculate_gbox_geodetic(const LWCOLLECTION *coll, GBOX *gbox)
{
        GBOX subbox;
        uint32_t i;
        int result = LW_FAILURE;
        int first = LW_TRUE;
        assert(coll);
        if ( coll->ngeoms == 0 )
                return LW_FAILURE;

        subbox.flags = gbox->flags;

        for ( i = 0; i < coll->ngeoms; i++ )
        {
                if ( lwgeom_calculate_gbox_geodetic((LWGEOM*)(coll->geoms[i]), &subbox) == LW_SUCCESS )
                {
                        /* Keep a copy of the sub-bounding box for later */
                        if ( coll->geoms[i]->bbox )
                                lwfree(coll->geoms[i]->bbox);
                        coll->geoms[i]->bbox = gbox_copy(&subbox);
                        if ( first )
                        {
                                gbox_duplicate(&subbox, gbox);
                                first = LW_FALSE;
                        }
                        else
                        {
                                gbox_merge(&subbox, gbox);
                        }
                        result = LW_SUCCESS;
                }
        }
        return result;
}

int lwgeom_calculate_gbox_geodetic(const LWGEOM *geom, GBOX *gbox)
{
        int result = LW_FAILURE;
        LWDEBUGF(4, "got type %d", geom->type);

        /* Add a geodetic flag to the incoming gbox */
        gbox->flags = lwflags(FLAGS_GET_Z(geom->flags),FLAGS_GET_M(geom->flags),1);

        switch (geom->type)
        {
        case POINTTYPE:
                result = lwpoint_calculate_gbox_geodetic((LWPOINT*)geom, gbox);
                break;
        case LINETYPE:
                result = lwline_calculate_gbox_geodetic((LWLINE *)geom, gbox);
                break;
        case POLYGONTYPE:
                result = lwpolygon_calculate_gbox_geodetic((LWPOLY *)geom, gbox);
                break;
        case TRIANGLETYPE:
                result = lwtriangle_calculate_gbox_geodetic((LWTRIANGLE *)geom, gbox);
                break;
        case MULTIPOINTTYPE:
        case MULTILINETYPE:
        case MULTIPOLYGONTYPE:
        case POLYHEDRALSURFACETYPE:
        case TINTYPE:
        case COLLECTIONTYPE:
                result = lwcollection_calculate_gbox_geodetic((LWCOLLECTION *)geom, gbox);
                break;
        default:
                lwerror("lwgeom_calculate_gbox_geodetic: unsupported input geometry type: %d - %s",
                        geom->type, lwtype_name(geom->type));
                break;
        }
        return result;
}



static int ptarray_check_geodetic(const POINTARRAY *pa)
{
        uint32_t t;
        POINT2D pt;

        assert(pa);

        for (t=0; t<pa->npoints; t++)
        {
                getPoint2d_p(pa, t, &pt);
                /* printf( "%d (%g, %g)\n", t, pt.x, pt.y); */
                if ( pt.x < -180.0 || pt.y < -90.0 || pt.x > 180.0 || pt.y > 90.0 )
                        return LW_FALSE;
        }

        return LW_TRUE;
}

static int lwpoint_check_geodetic(const LWPOINT *point)
{
        assert(point);
        return ptarray_check_geodetic(point->point);
}

static int lwline_check_geodetic(const LWLINE *line)
{
        assert(line);
        return ptarray_check_geodetic(line->points);
}

static int lwpoly_check_geodetic(const LWPOLY *poly)
{
        uint32_t i = 0;
        assert(poly);

        for ( i = 0; i < poly->nrings; i++ )
        {
                if ( ptarray_check_geodetic(poly->rings[i]) == LW_FALSE )
                        return LW_FALSE;
        }
        return LW_TRUE;
}

static int lwtriangle_check_geodetic(const LWTRIANGLE *triangle)
{
        assert(triangle);
        return ptarray_check_geodetic(triangle->points);
}


static int lwcollection_check_geodetic(const LWCOLLECTION *col)
{
        uint32_t i = 0;
        assert(col);

        for ( i = 0; i < col->ngeoms; i++ )
        {
                if ( lwgeom_check_geodetic(col->geoms[i]) == LW_FALSE )
                        return LW_FALSE;
        }
        return LW_TRUE;
}

int lwgeom_check_geodetic(const LWGEOM *geom)
{
        if ( lwgeom_is_empty(geom) )
                return LW_TRUE;

        switch (geom->type)
        {
        case POINTTYPE:
                return lwpoint_check_geodetic((LWPOINT *)geom);
        case LINETYPE:
                return lwline_check_geodetic((LWLINE *)geom);
        case POLYGONTYPE:
                return lwpoly_check_geodetic((LWPOLY *)geom);
        case TRIANGLETYPE:
                return lwtriangle_check_geodetic((LWTRIANGLE *)geom);
        case MULTIPOINTTYPE:
        case MULTILINETYPE:
        case MULTIPOLYGONTYPE:
        case POLYHEDRALSURFACETYPE:
        case TINTYPE:
        case COLLECTIONTYPE:
                return lwcollection_check_geodetic((LWCOLLECTION *)geom);
        default:
                lwerror("lwgeom_check_geodetic: unsupported input geometry type: %d - %s",
                        geom->type, lwtype_name(geom->type));
        }
        return LW_FALSE;
}

static int ptarray_force_geodetic(POINTARRAY *pa)
{
        uint32_t t;
        int changed = LW_FALSE;
        POINT4D pt;

        assert(pa);

        for ( t=0; t < pa->npoints; t++ )
        {
                getPoint4d_p(pa, t, &pt);
                if ( pt.x < -180.0 || pt.x > 180.0 || pt.y < -90.0 || pt.y > 90.0 )
                {
                        pt.x = longitude_degrees_normalize(pt.x);
                        pt.y = latitude_degrees_normalize(pt.y);
                        ptarray_set_point4d(pa, t, &pt);
                        changed = LW_TRUE;
                }
        }
        return changed;
}

static int lwpoint_force_geodetic(LWPOINT *point)
{
        assert(point);
        return ptarray_force_geodetic(point->point);
}

static int lwline_force_geodetic(LWLINE *line)
{
        assert(line);
        return ptarray_force_geodetic(line->points);
}

static int lwpoly_force_geodetic(LWPOLY *poly)
{
        uint32_t i = 0;
        int changed = LW_FALSE;
        assert(poly);

        for ( i = 0; i < poly->nrings; i++ )
        {
                if ( ptarray_force_geodetic(poly->rings[i]) == LW_TRUE )
                        changed = LW_TRUE;
        }
        return changed;
}

static int lwcollection_force_geodetic(LWCOLLECTION *col)
{
        uint32_t i = 0;
        int changed = LW_FALSE;
        assert(col);

        for ( i = 0; i < col->ngeoms; i++ )
        {
                if ( lwgeom_force_geodetic(col->geoms[i]) == LW_TRUE )
                        changed = LW_TRUE;
        }
        return changed;
}

int lwgeom_force_geodetic(LWGEOM *geom)
{
        switch ( lwgeom_get_type(geom) )
        {
                case POINTTYPE:
                        return lwpoint_force_geodetic((LWPOINT *)geom);
                case LINETYPE:
                        return lwline_force_geodetic((LWLINE *)geom);
                case POLYGONTYPE:
                        return lwpoly_force_geodetic((LWPOLY *)geom);
                case MULTIPOINTTYPE:
                case MULTILINETYPE:
                case MULTIPOLYGONTYPE:
                case COLLECTIONTYPE:
                        return lwcollection_force_geodetic((LWCOLLECTION *)geom);
                default:
                        lwerror("unsupported input geometry type: %d", lwgeom_get_type(geom));
        }
        return LW_FALSE;
}


double ptarray_length_spheroid(const POINTARRAY *pa, const SPHEROID *s)
{
        GEOGRAPHIC_POINT a, b;
        double za = 0.0, zb = 0.0;
        POINT4D p;
        uint32_t i;
        int hasz = LW_FALSE;
        double length = 0.0;
        double seglength = 0.0;

        /* Return zero on non-sensical inputs */
        if ( ! pa || pa->npoints < 2 )
                return 0.0;

        /* See if we have a third dimension */
        hasz = FLAGS_GET_Z(pa->flags);

        /* Initialize first point */
        getPoint4d_p(pa, 0, &p);
        geographic_point_init(p.x, p.y, &a);
        if ( hasz )
                za = p.z;

        /* Loop and sum the length for each segment */
        for ( i = 1; i < pa->npoints; i++ )
        {
                seglength = 0.0;
                getPoint4d_p(pa, i, &p);
                geographic_point_init(p.x, p.y, &b);
                if ( hasz )
                        zb = p.z;

                /* Special sphere case */
                if ( s->a == s->b )
                        seglength = s->radius * sphere_distance(&a, &b);
                /* Spheroid case */
                else
                        seglength = spheroid_distance(&a, &b, s);

                /* Add in the vertical displacement if we're in 3D */
                if ( hasz )
                        seglength = sqrt( (zb-za)*(zb-za) + seglength*seglength );

                /* Add this segment length to the total */
                length += seglength;

                /* B gets incremented in the next loop, so we save the value here */
                a = b;
                za = zb;
        }
        return length;
}

double lwgeom_length_spheroid(const LWGEOM *geom, const SPHEROID *s)
{
        int type;
        uint32_t i = 0;
        double length = 0.0;

        assert(geom);

        /* No area in nothing */
        if ( lwgeom_is_empty(geom) )
                return 0.0;

        type = geom->type;

        if ( type == POINTTYPE || type == MULTIPOINTTYPE )
                return 0.0;

        if ( type == LINETYPE )
                return ptarray_length_spheroid(((LWLINE*)geom)->points, s);

        if ( type == POLYGONTYPE )
        {
                LWPOLY *poly = (LWPOLY*)geom;
                for ( i = 0; i < poly->nrings; i++ )
                {
                        length += ptarray_length_spheroid(poly->rings[i], s);
                }
                return length;
        }

        if ( type == TRIANGLETYPE )
                return ptarray_length_spheroid(((LWTRIANGLE*)geom)->points, s);

        if ( lwtype_is_collection( type ) )
        {
                LWCOLLECTION *col = (LWCOLLECTION*)geom;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        length += lwgeom_length_spheroid(col->geoms[i], s);
                }
                return length;
        }

        lwerror("unsupported type passed to lwgeom_length_sphere");
        return 0.0;
}

/**
* When features are snapped or sometimes they are just this way, they are very close to
* the geodetic bounds but slightly over. This routine nudges those points, and only
* those points, back over to the bounds.
* http://trac.osgeo.org/postgis/ticket/1292
*/
static int
ptarray_nudge_geodetic(POINTARRAY *pa)
{

        uint32_t i;
        POINT4D p;
        int altered = LW_FALSE;
        int rv = LW_FALSE;
        static double tolerance = 1e-10;

        if ( ! pa )
                lwerror("ptarray_nudge_geodetic called with null input");

        for(i = 0; i < pa->npoints; i++ )
        {
                getPoint4d_p(pa, i, &p);
                if ( p.x < -180.0 && (-180.0 - p.x < tolerance) )
                {
                        p.x = -180.0;
                        altered = LW_TRUE;
                }
                if ( p.x > 180.0 && (p.x - 180.0 < tolerance) )
                {
                        p.x = 180.0;
                        altered = LW_TRUE;
                }
                if ( p.y < -90.0 && (-90.0 - p.y < tolerance) )
                {
                        p.y = -90.0;
                        altered = LW_TRUE;
                }
                if ( p.y > 90.0 && (p.y - 90.0 < tolerance) )
                {
                        p.y = 90.0;
                        altered = LW_TRUE;
                }
                if ( altered == LW_TRUE )
                {
                        ptarray_set_point4d(pa, i, &p);
                        altered = LW_FALSE;
                        rv = LW_TRUE;
                }
        }
        return rv;
}

/**
* When features are snapped or sometimes they are just this way, they are very close to
* the geodetic bounds but slightly over. This routine nudges those points, and only
* those points, back over to the bounds.
* http://trac.osgeo.org/postgis/ticket/1292
*/
int
lwgeom_nudge_geodetic(LWGEOM *geom)
{
        int type;
        uint32_t i = 0;
        int rv = LW_FALSE;

        assert(geom);

        /* No points in nothing */
        if ( lwgeom_is_empty(geom) )
                return LW_FALSE;

        type = geom->type;

        if ( type == POINTTYPE )
                return ptarray_nudge_geodetic(((LWPOINT*)geom)->point);

        if ( type == LINETYPE )
                return ptarray_nudge_geodetic(((LWLINE*)geom)->points);

        if ( type == POLYGONTYPE )
        {
                LWPOLY *poly = (LWPOLY*)geom;
                for ( i = 0; i < poly->nrings; i++ )
                {
                        int n = ptarray_nudge_geodetic(poly->rings[i]);
                        rv = (rv == LW_TRUE ? rv : n);
                }
                return rv;
        }

        if ( type == TRIANGLETYPE )
                return ptarray_nudge_geodetic(((LWTRIANGLE*)geom)->points);

        if ( lwtype_is_collection( type ) )
        {
                LWCOLLECTION *col = (LWCOLLECTION*)geom;

                for ( i = 0; i < col->ngeoms; i++ )
                {
                        int n = lwgeom_nudge_geodetic(col->geoms[i]);
                        rv = (rv == LW_TRUE ? rv : n);
                }
                return rv;
        }

        lwerror("unsupported type (%s) passed to lwgeom_nudge_geodetic", lwtype_name(type));
        return rv;
}


/**
* Utility function for checking if P is within the cone defined by A1/A2.
*/
static int
point_in_cone(const POINT3D *A1, const POINT3D *A2, const POINT3D *P)
{
        POINT3D AC; /* Center point of A1/A2 */
        double min_similarity, similarity;

        /* Boundary case */
        if (point3d_equals(A1, P) || point3d_equals(A2, P))
                return LW_TRUE;

        /* The normalized sum bisects the angle between start and end. */
        vector_sum(A1, A2, &AC);
        normalize(&AC);

        /* The projection of start onto the center defines the minimum similarity */
        min_similarity = dot_product(A1, &AC);

        /* If the edge is sufficiently curved, use the dot product test */
        if (fabs(1.0 - min_similarity) > 1e-10)
        {
                /* The projection of candidate p onto the center */
                similarity = dot_product(P, &AC);

                /* If the projection of the candidate is larger than */
                /* the projection of the start point, the candidate */
                /* must be closer to the center than the start, so */
                /* therefor inside the cone */
                if (similarity > min_similarity)
                {
                        return LW_TRUE;
                }
                else
                {
                        return LW_FALSE;
                }
        }
        else
        {
                /* Where the edge is very narrow, the dot product test */
                /* fails, but we can use the almost-planar nature of the */
                /* problem space then to test if the vector from the */
                /* candidate to the start point in a different direction */
                /* to the vector from candidate to end point */
                /* If so, then candidate is between start and end */
                POINT3D PA1, PA2;
                vector_difference(P, A1, &PA1);
                vector_difference(P, A2, &PA2);
                normalize(&PA1);
                normalize(&PA2);
                if (dot_product(&PA1, &PA2) < 0.0)
                {
                        return LW_TRUE;
                }
                else
                {
                        return LW_FALSE;
                }
        }
        return LW_FALSE;
}



/**
* Utility function for edge_intersects(), signum with a tolerance
* in determining if the value is zero.
*/
static int
dot_product_side(const POINT3D *p, const POINT3D *q)
{
        double dp = dot_product(p, q);

        if ( FP_IS_ZERO(dp) )
                return 0;

        return dp < 0.0 ? -1 : 1;
}

/**
* Returns non-zero if edges A and B interact. The type of interaction is given in the
* return value with the bitmask elements defined above.
*/
uint32_t
edge_intersects(const POINT3D *A1, const POINT3D *A2, const POINT3D *B1, const POINT3D *B2)
{
        POINT3D AN, BN, VN;  /* Normals to plane A and plane B */
        double ab_dot;
        int a1_side, a2_side, b1_side, b2_side;
        int rv = PIR_NO_INTERACT;

        /* Normals to the A-plane and B-plane */
        unit_normal(A1, A2, &AN);
        unit_normal(B1, B2, &BN);

        /* Are A-plane and B-plane basically the same? */
        ab_dot = dot_product(&AN, &BN);

        if ( FP_EQUALS(fabs(ab_dot), 1.0) )
        {
                /* Co-linear case */
                if ( point_in_cone(A1, A2, B1) || point_in_cone(A1, A2, B2) ||
                     point_in_cone(B1, B2, A1) || point_in_cone(B1, B2, A2) )
                {
                        rv |= PIR_INTERSECTS;
                        rv |= PIR_COLINEAR;
                }
                return rv;
        }

        /* What side of plane-A and plane-B do the end points */
        /* of A and B fall? */
        a1_side = dot_product_side(&BN, A1);
        a2_side = dot_product_side(&BN, A2);
        b1_side = dot_product_side(&AN, B1);
        b2_side = dot_product_side(&AN, B2);

        /* Both ends of A on the same side of plane B. */
        if ( a1_side == a2_side && a1_side != 0 )
        {
                /* No intersection. */
                return PIR_NO_INTERACT;
        }

        /* Both ends of B on the same side of plane A. */
        if ( b1_side == b2_side && b1_side != 0 )
        {
                /* No intersection. */
                return PIR_NO_INTERACT;
        }

        /* A straddles B and B straddles A, so... */
        if ( a1_side != a2_side && (a1_side + a2_side) == 0 &&
             b1_side != b2_side && (b1_side + b2_side) == 0 )
        {
                /* Have to check if intersection point is inside both arcs */
                unit_normal(&AN, &BN, &VN);
                if ( point_in_cone(A1, A2, &VN) && point_in_cone(B1, B2, &VN) )
                {
                        return PIR_INTERSECTS;
                }

                /* Have to check if intersection point is inside both arcs */
                vector_scale(&VN, -1);
                if ( point_in_cone(A1, A2, &VN) && point_in_cone(B1, B2, &VN) )
                {
                        return PIR_INTERSECTS;
                }

                return PIR_NO_INTERACT;
        }

        /* The rest are all intersects variants... */
        rv |= PIR_INTERSECTS;

        /* A touches B */
        if ( a1_side == 0 )
        {
                /* Touches at A1, A2 is on what side? */
                rv |= (a2_side < 0 ? PIR_A_TOUCH_RIGHT : PIR_A_TOUCH_LEFT);
        }
        else if ( a2_side == 0 )
        {
                /* Touches at A2, A1 is on what side? */
                rv |= (a1_side < 0 ? PIR_A_TOUCH_RIGHT : PIR_A_TOUCH_LEFT);
        }

        /* B touches A */
        if ( b1_side == 0 )
        {
                /* Touches at B1, B2 is on what side? */
                rv |= (b2_side < 0 ? PIR_B_TOUCH_RIGHT : PIR_B_TOUCH_LEFT);
        }
        else if ( b2_side == 0 )
        {
                /* Touches at B2, B1 is on what side? */
                rv |= (b1_side < 0 ? PIR_B_TOUCH_RIGHT : PIR_B_TOUCH_LEFT);
        }

        return rv;
}

/**
* This routine returns LW_TRUE if the stabline joining the pt_outside and pt_to_test
* crosses the ring an odd number of times, or if the pt_to_test is on the ring boundary itself,
* returning LW_FALSE otherwise.
* The pt_outside *must* be guaranteed to be outside the ring (use the geography_pt_outside() function
* to derive one in postgis, or the gbox_pt_outside() function if you don't mind burning CPU cycles
* building a gbox first).
*/
int ptarray_contains_point_sphere(const POINTARRAY *pa, const POINT2D *pt_outside, const POINT2D *pt_to_test)
{
        POINT3D S1, S2; /* Stab line end points */
        POINT3D E1, E2; /* Edge end points (3-space) */
        POINT2D p; /* Edge end points (lon/lat) */
        uint32_t count = 0, i, inter;

        /* Null input, not enough points for a ring? You ain't closed! */
        if ( ! pa || pa->npoints < 4 )
                return LW_FALSE;

        /* Set up our stab line */
        ll2cart(pt_to_test, &S1);
        ll2cart(pt_outside, &S2);

        /* Initialize first point */
        getPoint2d_p(pa, 0, &p);
        ll2cart(&p, &E1);

        /* Walk every edge and see if the stab line hits it */
        for ( i = 1; i < pa->npoints; i++ )
        {
                LWDEBUGF(4, "testing edge (%d)", i);
                LWDEBUGF(4, "  start point == POINT(%.12g %.12g)", p.x, p.y);

                /* Read next point. */
                getPoint2d_p(pa, i, &p);
                ll2cart(&p, &E2);

                /* Skip over too-short edges. */
                if ( point3d_equals(&E1, &E2) )
                {
                        continue;
                }

                /* Our test point is on an edge end! Point is "in ring" by our definition */
                if ( point3d_equals(&S1, &E1) )
                {
                        return LW_TRUE;
                }

                /* Calculate relationship between stab line and edge */
                inter = edge_intersects(&S1, &S2, &E1, &E2);

                /* We have some kind of interaction... */
                if ( inter & PIR_INTERSECTS )
                {
                        /* If the stabline is touching the edge, that implies the test point */
                        /* is on the edge, so we're done, the point is in (on) the ring. */
                        if ( (inter & PIR_A_TOUCH_RIGHT) || (inter & PIR_A_TOUCH_LEFT) )
                        {
                                return LW_TRUE;
                        }

                        /* It's a touching interaction, disregard all the left-side ones. */
                        /* It's a co-linear intersection, ignore those. */
                        if ( inter & PIR_B_TOUCH_RIGHT || inter & PIR_COLINEAR )
                        {
                                /* Do nothing, to avoid double counts. */
                                LWDEBUGF(4,"    edge (%d) crossed, disregarding to avoid double count", i, count);
                        }
                        else
                        {
                                /* Increment crossingn count. */
                                count++;
                                LWDEBUGF(4,"    edge (%d) crossed, count == %d", i, count);
                        }
                }
                else
                {
                        LWDEBUGF(4,"    edge (%d) did not cross", i);
                }

                /* Increment to next edge */
                E1 = E2;
        }

        LWDEBUGF(4,"final count == %d", count);

        /* An odd number of crossings implies containment! */
        if ( count % 2 )
        {
                return LW_TRUE;
        }

        return LW_FALSE;
}