void cart_makecart(double *pointx, double *pointy, int npoints,
int xsize, int ysize, double blur)
{
int s,sp;
int step;
double t,h;
double error,dr;
double desiredratio;
/* Calculate the initial density and velocity for snapshot zero */
cart_density(0.0,0,xsize,ysize);
cart_vgrid(0,xsize,ysize);
s = 0;
/* Now integrate the points in the polygons */
step = 0;
t = 0.5*blur*blur;
h = INITH;
do {
/* Do a combined (triple) integration step */
cart_twosteps(pointx,pointy,npoints,t,h,s,xsize,ysize,&error,&dr,&sp);
/* Increase the time by 2h and rotate snapshots */
t += 2.0*h;
step += 2;
s = sp;
/* Adjust the time-step. Factor of 2 arises because the target for
* the two-step process is twice the target for an individual step */
desiredratio = pow(2*TARGETERROR/error,0.2);
if (desiredratio>MAXRATIO) h *= MAXRATIO;
else h *= desiredratio;
/* If no point moved then we are finished */
} while (dr>0.0);
}
void cart_twosteps(double *pointx, double *pointy, int npoints,
double t, double h, int s, int xsize, int ysize,
double *errorp, double *drp, int *spp)
{
int s0,s1,s2,s3,s4;
int p;
double rx1,ry1;
double rx2,ry2;
double rx3,ry3;
double v1x,v1y;
double v2x,v2y;
double v3x,v3y;
double v4x,v4y;
double k1x,k1y;
double k2x,k2y;
double k3x,k3y;
double k4x,k4y;
double dx1,dy1;
double dx2,dy2;
double dx12,dy12;
double dxtotal,dytotal;
double ex,ey;
double esq,esqmax;
double drsq,drsqmax;
s0 = s;
s1 = (s+1)%5;
s2 = (s+2)%5;
s3 = (s+3)%5;
s4 = (s+4)%5;
/* Calculate the density field for the four new time slices */
cart_density(s0, 0.5*h, s1, xsize, ysize);
cart_density(s1, 0.5*h, s2, xsize, ysize);
cart_density(s2, 0.5*h, s3, xsize, ysize);
cart_density(s3, 0.5*h, s4, xsize, ysize);
/* Do all three RK steps for each point in turn */
esqmax = drsqmax = 0.0;
for (p=0; p<npoints; p++) {
rx1 = pointx[p];
ry1 = pointy[p];
/* Do the big combined (2h) RK step */
cart_velocity(rx1,ry1,s0,xsize,ysize,&v1x,&v1y);
k1x = 2*h*v1x;
k1y = 2*h*v1y;
cart_velocity(rx1+0.5*k1x,ry1+0.5*k1y,s2,xsize,ysize,&v2x,&v2y);
k2x = 2*h*v2x;
k2y = 2*h*v2y;
cart_velocity(rx1+0.5*k2x,ry1+0.5*k2y,s2,xsize,ysize,&v3x,&v3y);
k3x = 2*h*v3x;
k3y = 2*h*v3y;
cart_velocity(rx1+k3x,ry1+k3y,s4,xsize,ysize,&v4x,&v4y);
k4x = 2*h*v4x;
k4y = 2*h*v4y;
dx12 = (k1x+k4x+2.0*(k2x+k3x))/6.0;
dy12 = (k1y+k4y+2.0*(k2y+k3y))/6.0;
/* Do the first small RK step. No initial call to cart_velocity() is done
* because it would be the same as the one above, so there's no need
* to do it again */
k1x = h*v1x;
k1y = h*v1y;
cart_velocity(rx1+0.5*k1x,ry1+0.5*k1y,s1,xsize,ysize,&v2x,&v2y);
k2x = h*v2x;
k2y = h*v2y;
cart_velocity(rx1+0.5*k2x,ry1+0.5*k2y,s1,xsize,ysize,&v3x,&v3y);
k3x = h*v3x;
k3y = h*v3y;
cart_velocity(rx1+k3x,ry1+k3y,s2,xsize,ysize,&v4x,&v4y);
k4x = h*v4x;
k4y = h*v4y;
dx1 = (k1x+k4x+2.0*(k2x+k3x))/6.0;
dy1 = (k1y+k4y+2.0*(k2y+k3y))/6.0;
/* Do the second small RK step */
rx2 = rx1 + dx1;
ry2 = ry1 + dy1;
cart_velocity(rx2,ry2,s2,xsize,ysize,&v1x,&v1y);
k1x = h*v1x;
k1y = h*v1y;
cart_velocity(rx2+0.5*k1x,ry2+0.5*k1y,s3,xsize,ysize,&v2x,&v2y);
k2x = h*v2x;
k2y = h*v2y;
cart_velocity(rx2+0.5*k2x,ry2+0.5*k2y,s3,xsize,ysize,&v3x,&v3y);
k3x = h*v3x;
k3y = h*v3y;
cart_velocity(rx2+k3x,ry2+k3y,s4,xsize,ysize,&v4x,&v4y);
k4x = h*v4x;
k4y = h*v4y;
dx2 = (k1x+k4x+2.0*(k2x+k3x))/6.0;
dy2 = (k1y+k4y+2.0*(k2y+k3y))/6.0;
/* Calculate the (squared) error */
ex = (dx1+dx2-dx12)/15;
ey = (dy1+dy2-dy12)/15;
esq = ex*ex + ey*ey;
if (esq>esqmax) esqmax = esq;
/* Update the position of the vertex using the more accurate (two small
* steps) result, and deal with the boundary conditions. This code
* does 5th-order "local extrapolation" (which just means taking
* the estimate of the 5th-order term above and adding it to our
* 4th-order result get a result accurate to the next highest order) */
dxtotal = dx1 + dx2 + ex; // Last term is local extrapolation
dytotal = dy1 + dy2 + ey; // Last term is local extrapolation
drsq = dxtotal*dxtotal + dytotal*dytotal;
if (drsq>drsqmax) drsqmax = drsq;
rx3 = rx1 + dxtotal;
ry3 = ry1 + dytotal;
if (rx3<0) rx3 = 0;
else if (rx3>xsize) rx3 = xsize;
if (ry3<0) ry3 = 0;
else if (ry3>ysize) ry3 = ysize;
pointx[p] = rx3;
pointy[p] = ry3;
}
*errorp = sqrt(esqmax);
*drp = sqrt(drsqmax);
*spp = s4;
}
/* Function to do the transformation of the given set of points
* to the cartogram */
void cart_makecart(double *pointx, double *pointy, int npoints,
int xsize, int ysize, double blur)
{
int i;
int s,sp;
int step;
int done;
double t,h;
double error,dr;
double desiredratio;
/* Calculate the initial density and velocity for snapshot zero */
cart_density(0.0,0,xsize,ysize);
cart_vgrid(0,xsize,ysize);
s = 0;
/* Now integrate the points in the polygons */
step = 0;
t = 0.5*blur*blur;
h = INITH;
do {
/* Do a combined (triple) integration step */
cart_twosteps(pointx,pointy,npoints,t,h,s,xsize,ysize,&error,&dr,&sp);
/* Increase the time by 2h and rotate snapshots */
t += 2.0*h;
step += 2;
s = sp;
/* Adjust the time-step. Factor of 2 arises because the target for
* the two-step process is twice the target for an individual step */
desiredratio = pow(2*TARGETERROR/error,0.2);
if (desiredratio>MAXRATIO) h *= MAXRATIO;
else h *= desiredratio;
done = cart_complete(t);
#ifdef PERCENT
fprintf(stdout,"%i\n",done);
#endif
#ifndef NOPROGRESS
fprintf(stderr," %3i%% |",done);
for (i=0; i<done/2; i++) fprintf(stderr,"=");
for (i=done/2; i<50; i++) fprintf(stderr," ");
fprintf(stderr,"|\r");
#endif
/* If no point moved then we are finished */
} while (dr>0.0);
#ifdef PERCENT
fprintf(stdout,"\n");
#endif
#ifndef NOPROGRESS
fprintf(stderr," 100%% |==================================================|\n");
#endif
}