source: XIOS/dev/branch_openmp/extern/remap/src/polyg.cpp @ 1289

/* utilities related to polygons */

#include <cassert>
#include <iostream>
#include "elt.hpp"
#include "errhandle.hpp"

#include "polyg.hpp"
#include <stdio.h>

namespace sphereRemap {

using namespace std;

/* given `N` `vertex`es, `N` `edge`s and `N` `d`s (d for small circles)
   and `g` the barycentre,
   this function reverses the order of arrays of `vertex`es, `edge`s and `d`s 
   but only if it is required!
  (because computing intersection area requires both polygons to have same orientation)
*/
void orient(int N, Coord *vertex, Coord *edge, double *d, const Coord &g)
{
        Coord ga = vertex[0] - g;
        Coord gb = vertex[1] - g;
        Coord vertical = crossprod(ga, gb);
        if (N > 2 && scalarprod(g, vertical) < 0)  // (GAxGB).G
        {
                for (int i = 0; i < N/2; i++)
                        swap(vertex[i], vertex[N-1-i]);

                for (int i = 0; i < (N-1)/2; i++)
                {
                        swap(edge[N-2-i], edge[i]);
                        swap(d[i], d[N-2-i]);
                }
        }
}


void normals(Coord *x, int n, Coord *a)
{
        for (int i = 0; i<n; i++)
                a[i] = crossprod(x[(i+1)%n], x[i]);
}

Coord barycentre(const Coord *x, int n)
{

        if (n == 0) return ORIGIN;
        Coord bc = ORIGIN;
        for (int i = 0; i < n; i++)
        {
                bc = bc + x[i];
        }       
        /* both distances can be equal down to roundoff when norm(bc) < mashineepsilon 
           which can occur when weighted with tiny area */
  
        
  //assert(squaredist(bc, proj(bc)) <= squaredist(bc, proj(bc * (-1.0))));      

        return proj(bc);
}

/** computes the barycentre of the area which is the difference
  of a side ABO of the spherical tetrahedron and of the straight tetrahedron */
static Coord tetrah_side_diff_centre(Coord a, Coord b)
{
        Coord n = crossprod(a,b);
        double sinc2 = n.x*n.x + n.y*n.y + n.z*n.z;
        assert(sinc2 < 1.0 + EPS);

        // exact: u = asin(sinc)/sinc - 1; asin(sinc) = geodesic length of arc ab
        // approx:
        // double u = sinc2/6. + (3./40.)*sinc2*sinc2;

        // exact  
        if (sinc2 > 1.0 - EPS) /* if round-off leads to sinc > 1 asin produces NaN */
                return n * (M_PI_2 - 1);
        double sinc = sqrt(sinc2);
        double u = asin(sinc)/sinc - 1;

        return n*u;
}

/* compute the barycentre as the negative sum of all the barycentres of sides of the tetrahedron */
Coord gc_normalintegral(const Coord *x, int n)
{
        Coord m = barycentre(x, n);
        Coord bc = crossprod(x[n-1]-m, x[0]-m) + tetrah_side_diff_centre(x[n-1], x[0]);
        for (int i = 1; i < n; i++)
                bc = bc + crossprod(x[i-1]-m, x[i]-m) + tetrah_side_diff_centre(x[i-1], x[i]);
        return bc*0.5;
}

Coord exact_barycentre(const Coord *x, int n)
{
        if (n >= 3)
        {
                return  proj(gc_normalintegral(x, n));
        }
        else if (n == 0) return ORIGIN;
        else if (n == 2) return midpoint(x[0], x[1]);
        else if (n == 1) return x[0];
}

Coord sc_gc_moon_normalintegral(Coord a, Coord b, Coord pole)
{
        double hemisphere = (a.z > 0) ? 1: -1;

        double lat = hemisphere * (M_PI_2 - acos(a.z));
        double lon1 = atan2(a.y, a.x);
        double lon2 = atan2(b.y, b.x);
        double lon_diff = lon2 - lon1;

        // wraparound at lon=-pi=pi
        if (lon_diff < -M_PI) lon_diff += 2.0*M_PI;
        else if (lon_diff > M_PI) lon_diff -= 2.0*M_PI;

        // integral of the normal over the surface bound by great arcs a-pole and b-pole and small arc a-b 
        Coord sc_normalintegral = Coord(0.5*(sin(lon2)-sin(lon1))*(M_PI_2 - lat - 0.5*sin(2.0*lat)),
                                        0.5*(cos(lon1)-cos(lon2))*(M_PI_2 - lat - 0.5*sin(2.0*lat)),
                                        hemisphere * lon_diff * 0.25 * (cos(2.0*lat) + 1.0));
        Coord p = Coord(0,0,hemisphere); // TODO assumes north pole is (0,0,1)
        Coord t[] = {a, b, p};
        if (hemisphere < 0) swap(t[0], t[1]);
        return (sc_normalintegral - gc_normalintegral(t, 3)) * hemisphere;
}


double triarea(Coord& A, Coord& B, Coord& C)
{
        double a = ds(B, C);
        double b = ds(C, A);
        double c = ds(A, B);
        double s = 0.5 * (a + b + c);
        double t = tan(0.5*s) * tan(0.5*(s - a)) * tan(0.5*(s - b)) * tan(0.5*(s - c));
//      assert(t >= 0);
  if (t<1e-20) return 0. ;
        return 4 * atan(sqrt(t));
}

/** Computes area of two two-sided polygon
   needs to have one small and one great circle, otherwise zero
   (name origin: lun is moon in french)
*/
double alun(double b, double d)
{
        double a  = acos(d);
        assert(b <= 2 * a);
        double s  = a + 0.5 * b;
        double t = tan(0.5 * s) * tan(0.5 * (s - a)) * tan(0.5 * (s - a)) * tan(0.5 * (s - b));
        double r  = sqrt(1 - d*d);
        double p  = 2 * asin(sin(0.5*b) / r);
        return p*(1 - d) - 4*atan(sqrt(t));
}

/**
  This function returns the area of a spherical element 
that can be composed of great and small circle arcs.
 The caller must ensure this function is not called when `alun` should be called instaed.
  This function also sets `gg` to the barycentre of the element.
  "air" stands for area and  "bar" for barycentre.
*/
double airbar(int N, const Coord *x, const Coord *c, double *d, const Coord& pole, Coord& gg)
{
        if (N < 3)
                return 0; /* polygons with less than three vertices have zero area */
        Coord t[3];
        t[0] = barycentre(x, N);
        Coord *g = new Coord[N];
        double area = 0;
        Coord gg_exact = gc_normalintegral(x, N);
        for (int i = 0; i < N; i++)
        {
                /* for "spherical circle segment" sum triangular part and "small moon" and => account for small circle */
                int ii = (i + 1) % N;
                t[1] = x[i];
                t[2] = x[ii];
    double sc=scalarprod(crossprod(t[1] - t[0], t[2] - t[0]), t[0]) ;
                assert(sc >= -1e-10); // Error: tri a l'env (wrong orientation)
                double area_gc = triarea(t[0], t[1], t[2]);
                //if(area_gc<=0) printf("area_gc = %e\n", area_gc);
                double area_sc_gc_moon = 0;
                if (d[i]) /* handle small circle case */
                {
                        Coord m = midpoint(t[1], t[2]);
                        double mext = scalarprod(m, c[i]) - d[i];
                        char sgl = (mext > 0) ? -1 : 1;
                        area_sc_gc_moon = sgl * alun(arcdist(t[1], t[2]), fabs(scalarprod(t[1], pole)));
                        //if(area_sc_gc_moon<=0) printf("area_sc_gc_moon = %e\n", area_sc_gc_moon);
                        gg_exact = gg_exact + sc_gc_moon_normalintegral(t[1], t[2], pole);
                }
                area += area_gc + area_sc_gc_moon; /* for "spherical circle segment" sum triangular part (at) and "small moon" and => account for small circle */
                g[i] = barycentre(t, 3) * (area_gc + area_sc_gc_moon);
                //printf("g[%d] = (%e,%e,%e) * (%e+%e) = (%e,%e,%e) norm = %e\n", i, barycentre(t, 3).x, barycentre(t, 3).y, barycentre(t, 3).z, area_gc,  area_sc_gc_moon, g[i].x, g[i].y, g[i].z, norm(g[i]));
        }
        gg = barycentre(g, N);
        gg_exact = proj(gg_exact);
        delete[] g;
        gg = gg_exact;
        return area;
}

double polygonarea(Coord *vertices, int N)
{
        assert(N >= 3);

        /* compute polygon area as sum of triangles */
        Coord centre = barycentre(vertices, N);
        double area = 0;
        for (int i = 0; i < N; i++)
                area += triarea(centre, vertices[i], vertices[(i+1)%N]);
        return area;
}

int packedPolygonSize(const Elt& e)
{
        return sizeof(e.id) + sizeof(e.src_id) + sizeof(e.x) + sizeof(e.val) +
               sizeof(e.n) + e.n*(sizeof(double)+sizeof(Coord));
}

void packPolygon(const Elt& e, char *buffer, int& pos) 
{
        *((GloId *) &(buffer[pos])) = e.id;
        pos += sizeof(e.id);
        *((GloId *) &(buffer[pos])) = e.src_id;
        pos += sizeof(e.src_id);

        *((Coord *) &(buffer[pos])) = e.x;
        pos += sizeof(e.x);

        *((double*) &(buffer[pos])) = e.val;
        pos += sizeof(e.val);

        *((int *) & (buffer[pos])) = e.n;
        pos += sizeof(e.n);

        for (int i = 0; i < e.n; i++)
        {
                *((double *) & (buffer[pos])) = e.d[i];
                pos += sizeof(e.d[i]);

                *((Coord *) &(buffer[pos])) = e.vertex[i];
                pos += sizeof(e.vertex[i]);
        } 

}

void unpackPolygon(Elt& e, const char *buffer, int& pos) 
{
        e.id = *((GloId *) &(buffer[pos]));
        pos += sizeof(e.id);
        e.src_id = *((GloId *) &(buffer[pos]));
        pos += sizeof(e.src_id);

        e.x = *((Coord *) & (buffer[pos]) );
        pos += sizeof(e.x);

        e.val = *((double *) & (buffer[pos]));
        pos += sizeof(double);

        e.n = *((int *) & (buffer[pos]));
        pos += sizeof(int);

        for (int i = 0; i < e.n; i++)
        {
                e.d[i] = *((double *) & (buffer[pos]));
                pos += sizeof(double);

                e.vertex[i] = *((Coord *) & (buffer[pos]));
                pos += sizeof(Coord);
        }
}

int packIntersectionSize(const Elt& elt) 
{
        return elt.is.size() * (2*sizeof(int)+ sizeof(GloId) + 5*sizeof(double));
}

void packIntersection(const Elt& e, char* buffer,int& pos) 
{
  for (list<Polyg *>::const_iterator it = e.is.begin(); it != e.is.end(); ++it)
        {
                *((int *) &(buffer[0])) += 1;

                *((int *) &(buffer[pos])) = e.id.ind;
                pos += sizeof(int);

    *((double *) &(buffer[pos])) = e.area;
    pos += sizeof(double);

                *((GloId *) &(buffer[pos])) = (*it)->id;
                pos += sizeof(GloId);
  
                *((int *) &(buffer[pos])) = (*it)->n;
                pos += sizeof(int);
                *((double *) &(buffer[pos])) = (*it)->area;
                pos += sizeof(double);

                *((Coord *) &(buffer[pos])) = (*it)->x;
                pos += sizeof(Coord);
        }
}

void unpackIntersection(Elt* e, const char* buffer) 
{
        int ind;
        int pos = 0;
  
        int n = *((int *) & (buffer[pos]));
        pos += sizeof(int);
        for (int i = 0; i < n; i++)
        {
                ind = *((int*) &(buffer[pos]));
                pos+=sizeof(int);

                Elt& elt= e[ind];

    elt.area=*((double *) & (buffer[pos]));
                pos += sizeof(double);

                Polyg *polygon = new Polyg;

                polygon->id =  *((GloId *) & (buffer[pos]));
                pos += sizeof(GloId);

                polygon->n =  *((int *) & (buffer[pos]));
                pos += sizeof(int);

                polygon->area =  *((double *) & (buffer[pos]));
                pos += sizeof(double);

                polygon->x = *((Coord *) & (buffer[pos]));
                pos += sizeof(Coord);

                elt.is.push_back(polygon);
        }
}

}