// Finds fastest trajectory for sliding from (0,y0) to (x,0)
#include <cstdlib>
#include <cmath>
#include <cstdio>
const double PI=4.0*atan(1.0);
using namespace std;

void GetX1(double a,double y,double &x,double &dxda){
	double theta,root;
	root=sqrt(2.0*a*y-y*y);
	theta=acos(1.0-y/a);
	x=-root+a*theta;
	dxda=-(2.0*y/root)+theta;
}

void GetX2(double a,double y,double &x,double &dxda){
	double theta,root;
	root=sqrt(2.0*a*y-y*y);
	theta=acos(1.0-y/a);
	theta=2.0*PI-theta;
	x=-root+a*theta;
	dxda=theta;
}

int main(int argc,char *argv[]){
	double a,y0,x,dxda,xtarget;
	int ntries;
	printf("Enter xtarget and y0: ");
	scanf("%lf %lf",&xtarget,&y0);
	a=3.0;
	ntries=0;
	do{
		ntries+=1;
		if(xtarget<PI*y0*0.5)
			GetX1(a,y0,x,dxda);
		else
			GetX2(a,y0,x,dxda);
		a=a-(x-xtarget)/dxda;
		if(a<0.5*y0)
			a=0.5*y0+0.000001;
		printf("searching: a=%g, Zero=?%g\n",a,x-xtarget);
	}while(fabs(x-xtarget)>1.0E-10 && ntries<10);
	printf("x=%g, a=%g\n",x,a);
	// Print trajectory
	double time=0.0,y;
	printf("  x     y\n");
	printf("%8.5f %8.5f\n",0.0,y0);
	for(y=0.05*a;y<2.0*a;y+=0.1*a){
		GetX1(a,y,x,dxda);
		printf("%8.5f %8.5f\n",x,y0-y);
	}
	for(y=2.0*a-0.05*a;y>0;y-=0.1*a){
		GetX2(a,y,x,dxda);
		printf("%8.5f %8.5f\n",x,y0-y);
	}
}