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#include "path.h"
#include <math.h>
#include <gsl/gsl_spline.h>
#include <stddef.h> /* for size_t */
#include <gsl/gsl_integration.h>
#include "pdg.h"
#include "vector.h"
#include <stdlib.h> /* for malloc(), calloc(), etc. */
#include "misc.h"
#include <gsl/gsl_sf_psi.h> /* for gsl_sf_psi_n() */
#include "scattering.h"
#define N 1000
static double foo(double s, double range, double theta0, int k)
{
return sqrt(2*pow(theta0,2)*range)*sin(M_PI*s*(k-0.5)/range)/(M_PI*(k-0.5));
}
double path_get_coefficient(unsigned int k, double *s, double *x, double theta0, size_t n)
{
size_t i;
double sum, range;
range = s[n-1];
sum = 0.0;
for (i = 1; i < n; i++) {
sum += (foo(s[i],range,theta0,k)*x[i] + foo(s[i-1],range,theta0,k)*x[i-1])*(s[i]-s[i-1])/2.0;
}
return sum*pow(k-0.5,2)*pow(M_PI,2)/pow(theta0*range,2);
}
path *path_init(double *pos, double *dir, double T0, double range, double theta0, getKineticEnergyFunc *fun, void *params, double *z1, double *z2, size_t n, double m)
{
size_t i, j;
double E, mom, beta, theta, phi;
double normal[3], k[3], tmp[3], tmp2[3];
path *p = malloc(sizeof(path));
p->pos[0] = pos[0];
p->pos[1] = pos[1];
p->pos[2] = pos[2];
p->dir[0] = dir[0];
p->dir[1] = dir[1];
p->dir[2] = dir[2];
p->n = n;
double *s = malloc(sizeof(double)*N);
double *theta1 = calloc(N,sizeof(double));
double *theta2 = calloc(N,sizeof(double));
double *x = calloc(N,sizeof(double));
double *y = calloc(N,sizeof(double));
double *z = calloc(N,sizeof(double));
double *T = calloc(N,sizeof(double));
double *t = calloc(N,sizeof(double));
double *dx = calloc(N,sizeof(double));
double *dy = calloc(N,sizeof(double));
double *dz = calloc(N,sizeof(double));
dz[0] = 1.0;
for (i = 0; i < N; i++) {
s[i] = range*i/(N-1);
for (j = 0; j < n; j++) {
theta1[i] += z1[j]*foo(s[i],range,theta0,j+1);
theta2[i] += z2[j]*foo(s[i],range,theta0,j+1);
}
T[i] = fun(s[i],params);
if (i > 0) {
theta = sqrt(theta1[i]*theta1[i] + theta2[i]*theta2[i]);
phi = atan2(theta2[i],theta1[i]);
dx[i] = (s[i]-s[i-1])*sin(theta)*cos(phi);
dy[i] = (s[i]-s[i-1])*sin(theta)*sin(phi);
dz[i] = (s[i]-s[i-1])*cos(theta);
/* Calculate total energy */
E = T[i] + m;
mom = sqrt(E*E - m*m);
beta = mom/E;
t[i] = t[i-1] + (s[i]-s[i-1])/(beta*SPEED_OF_LIGHT);
x[i] = x[i-1] + dx[i];
y[i] = y[i-1] + dy[i];
z[i] = z[i-1] + dz[i];
}
}
/* Now, we rotate and translate the path so that it starts at `pos` and
* points in the direction `dir`. */
/* We need to compute an arbitrary vector which is orthogonal to the
* direction vector. To do this, all we need is another vector not parallel
* to the direction. */
k[0] = 0.0;
k[1] = 0.0;
k[2] = 1.0;
/* If k is approximately equal to the unit vector in the z direction, we
* switch to the unit vector in the x direction. */
if (allclose(k,dir,3,1e-9,1e-9)) {
k[0] = 1.0;
k[1] = 0.0;
k[2] = 0.0;
}
/* Take the cross product between `k` and `dir` to get a vector orthogonal
* to `dir`. */
CROSS(normal,k,dir);
/* Make sure it's normalized. */
normalize(normal);
/* Compute the angle required to rotate the unit vector to `dir`. */
phi = acos(DOT(k,dir));
/* Now we rotate and translate all the positions and rotate all the
* directions. */
for (i = 0; i < N; i++) {
tmp[0] = x[i];
tmp[1] = y[i];
tmp[2] = z[i];
rotate(tmp2,tmp,normal,phi);
ADD(tmp2,tmp2,pos);
x[i] = tmp2[0];
y[i] = tmp2[1];
z[i] = tmp2[2];
tmp[0] = dx[i];
tmp[1] = dy[i];
tmp[2] = dz[i];
normalize(tmp);
rotate(tmp2,tmp,normal,phi);
dx[i] = tmp2[0];
dy[i] = tmp2[1];
dz[i] = tmp2[2];
}
/* Calculate the approximate residual scattering RMS.
*
* For a given truncated KL expansion the residual error in the
* approximation is given by:
*
* \sum_{n+1}^\infty sqrt(lambda_k)*sqrt(2/range)*sin(pi*s*(k-0.5)/range)
*
* Since there is no nice closed form solution for this, we estimate it by
* evaluating the result at the middle of the track. */
if (p->n > 0) {
p->theta0 = theta0*sqrt(range)*sqrt(gsl_sf_psi_n(1,p->n+0.5))/M_PI;
p->theta0 = fmax(MIN_THETA0, p->theta0);
p->theta0 = fmin(MAX_THETA0, p->theta0);
} else {
p->theta0 = theta0;
}
p->s = s;
p->x = x;
p->y = y;
p->z = z;
p->T = T;
p->t = t;
p->dx = dx;
p->dy = dy;
p->dz = dz;
free(theta1);
free(theta2);
return p;
}
void path_eval(path *p, double s, double *pos, double *dir, double *T, double *t, double *theta0)
{
pos[0] = interp1d(s,p->s,p->x,N);
pos[1] = interp1d(s,p->s,p->y,N);
pos[2] = interp1d(s,p->s,p->z,N);
*T = interp1d(s,p->s,p->T,N);
*t = interp1d(s,p->s,p->t,N);
/* Technically we should be interpolating between the direction vectors
* using an algorithm like Slerp (see https://en.wikipedia.org/wiki/Slerp),
* but since we expect the direction vectors to be pretty close to each
* other, we just linearly interpolate and then normalize it. */
dir[0] = interp1d(s,p->s,p->dx,N);
dir[1] = interp1d(s,p->s,p->dy,N);
dir[2] = interp1d(s,p->s,p->dz,N);
normalize(dir);
if (p->n > 0) {
*theta0 = p->theta0;
} else {
*theta0 = fmax(MIN_THETA0,p->theta0*sqrt(s));
*theta0 = fmin(MAX_THETA0,*theta0);
}
}
void path_free(path *p)
{
free(p->s);
free(p->x);
free(p->y);
free(p->z);
free(p->T);
free(p->t);
free(p->dx);
free(p->dy);
free(p->dz);
free(p);
}
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