double getResolution(double ctheta0, int N){
    double s=0;
    for (int i=0;i<N;i++){
        double ctheta=1-(i+0.5)/(N-1.0)*(1-ctheta0);
        double stheta2=1-ctheta*ctheta;
        s+=stheta2;
    }
    return s/N;
}

double getResolutionT(double ctheta0, int N){
    double s=0;
    for (int i=0;i<N;i++){
        double ctheta=1-(i+0.5)/(N-1.0)*(1-ctheta0);
        s+=ctheta*ctheta;
    }
    return s/N;