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| #include "utils.h"
#define PARABOLA_FIT_CENTER_WEIGHT .1
// Define this to use a x^iy^j polynomial with i = 0...NP-1, j = 0...NP-1
// #define NP 3
typedef struct {
coord o;
#if dimension == 2 /* y = a[0]*x^2 + a[1]*x + a[2] */
coord m;
double ** M, rhs[3], a[3];
#else /* 3D z = a[0]*x^2 + a[1]*y^2 + a[2]*x*y + a[3]*x + a[4]*y + a[5] */
double t[3][3];
# ifdef NP
double ** M, rhs[NP*NP], a[NP*NP];
# else
double ** M, rhs[6], a[6];
# endif
#endif /* 3D */
} ParabolaFit;
static void parabola_fit_init (ParabolaFit * p, coord o, coord m)
{
foreach_dimension()
p->o.x = o.x;
#if dimension == 2
foreach_dimension()
p->m.x = m.x;
normalize (&p->m);
int n = 3;
#else /* 3D */
double max;
coord nx = {0., 0., 0.}, ny, nz;
int d = 0;
foreach_dimension()
nz.x = m.x;
normalize (&nz);
max = sq(nz.x);
/* build a vector orthogonal to nz */
if (sq(nz.y) > max) { max = sq(nz.y); d = 1; }
if (sq(nz.z) > max) d = 2;
switch (d) {
case 0: nx.x = - nz.z/nz.x; nx.z = 1.0; break;
case 1: nx.y = - nz.z/nz.y; nx.z = 1.0; break;
case 2: nx.z = - nz.x/nz.z; nx.x = 1.0; break;
}
normalize (&nx);
/* build a second vector orthogonal to nx and nz */
foreach_dimension()
ny.x = nz.y*nx.z - nz.z*nx.y;
/* transformation matrix from (i,j,k) to (nx, ny, nz) */
p->t[0][0] = nx.x; p->t[0][1] = nx.y; p->t[0][2] = nx.z;
p->t[1][0] = ny.x; p->t[1][1] = ny.y; p->t[1][2] = ny.z;
p->t[2][0] = nz.x; p->t[2][1] = nz.y; p->t[2][2] = nz.z;
# ifdef NP
int n = NP*NP;
# else
int n = 6;
# endif
#endif /* 3D */
p->M = (double **) matrix_new (n, n, sizeof(double));
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
p->M[i][j] = 0.;
p->rhs[i] = 0.;
}
}
static void parabola_fit_add (ParabolaFit * p, coord m, double w)
{
#if dimension == 2
double x1 = m.x - p->o.x, y1 = m.y - p->o.y;
double x = p->m.y*x1 - p->m.x*y1;
double y = p->m.x*x1 + p->m.y*y1;
double x2 = w*x*x, x3 = x2*x, x4 = x3*x;
p->M[0][0] += x4;
p->M[1][0] += x3; p->M[1][1] += x2;
p->M[2][1] += w*x; p->M[2][2] += w;
p->rhs[0] += x2*y; p->rhs[1] += w*x*y; p->rhs[2] += w*y;
#else /* 3D */
double x1 = m.x - p->o.x, y1 = m.y - p->o.y, z1 = m.z - p->o.z;
double x = p->t[0][0]*x1 + p->t[0][1]*y1 + p->t[0][2]*z1;
double y = p->t[1][0]*x1 + p->t[1][1]*y1 + p->t[1][2]*z1;
double z = p->t[2][0]*x1 + p->t[2][1]*y1 + p->t[2][2]*z1;
# ifdef NP
for (int i = 0; i < NP; i++)
for (int j = 0; j < NP; j++) {
for (int k = 0; k < NP; k++)
for (int l = 0; l < NP; l++)
p->M[i*NP + j][k*NP + l] += w*pow(x, i + k)*pow(y, j + l);
p->rhs[i*NP + j] += w*z*pow(x, i)*pow(y, j);
}
# else // !NP
double x2 = x*x, x3 = x2*x, x4 = x3*x;
double y2 = y*y, y3 = y2*y, y4 = y3*y;
p->M[0][0] += w*x4; p->M[1][1] += w*y4; p->M[2][2] += w*x2*y2;
p->M[3][3] += w*x2; p->M[4][4] += w*y2; p->M[5][5] += w;
p->M[0][2] += w*x3*y; p->M[0][3] += w*x3; p->M[0][4] += w*x2*y;
p->M[1][2] += w*x*y3; p->M[1][3] += w*x*y2; p->M[1][4] += w*y3;
p->M[2][5] += w*x*y;
p->M[3][5] += w*x;
p->M[4][5] += w*y;
p->rhs[0] += w*x2*z; p->rhs[1] += w*y2*z; p->rhs[2] += w*x*y*z;
p->rhs[3] += w*x*z; p->rhs[4] += w*y*z; p->rhs[5] += w*z;
# endif // !NP
#endif /* 3D */
}
static double parabola_fit_solve (ParabolaFit * p)
{
#if dimension == 2
p->M[0][1] = p->M[1][0];
p->M[0][2] = p->M[2][0] = p->M[1][1];
p->M[1][2] = p->M[2][1];
double pivmin = matrix_inverse (p->M, 3, 1e-10);
if (pivmin) {
p->a[0] = p->M[0][0]*p->rhs[0] + p->M[0][1]*p->rhs[1] + p->M[0][2]*p->rhs[2];
p->a[1] = p->M[1][0]*p->rhs[0] + p->M[1][1]*p->rhs[1] + p->M[1][2]*p->rhs[2];
}
else /* this may be a degenerate/isolated interface fragment */
p->a[0] = p->a[1] = 0.;
#else /* 3D */
# ifdef NP
double pivmin = matrix_inverse (p->M, NP*NP, 1e-10);
if (pivmin)
for (int i = 0; i < NP*NP; i++) {
p->a[i] = 0.;
for (int j = 0; j < NP*NP; j++)
p->a[i] += p->M[i][j]*p->rhs[j];
}
else /* this may be a degenerate/isolated interface fragment */
for (int i = 0; i < NP*NP; i++)
p->a[i] = 0.;
# else // !NP
p->M[0][1] = p->M[2][2]; p->M[0][5] = p->M[3][3];
p->M[1][5] = p->M[4][4];
p->M[2][3] = p->M[0][4]; p->M[2][4] = p->M[1][3];
p->M[3][4] = p->M[2][5];
for (int i = 1; i < 6; i++)
for (int j = 0; j < i; j++)
p->M[i][j] = p->M[j][i];
double pivmin = matrix_inverse (p->M, 6, 1e-10);
if (pivmin)
for (int i = 0; i < 6; i++) {
p->a[i] = 0.;
for (int j = 0; j < 6; j++)
p->a[i] += p->M[i][j]*p->rhs[j];
}
else /* this may be a degenerate/isolated interface fragment */
for (int i = 0; i < 6; i++)
p->a[i] = 0.;
# endif // !NP
#endif /* 3D */
matrix_free (p->M);
return pivmin;
}
static double parabola_fit_curvature (ParabolaFit * p,
double kappamax, double * kmax)
{
double kappa;
#if dimension == 2
double dnm = 1.[0] + sq(p->a[1]);
kappa = - 2.*p->a[0]/pow(dnm, 3/2.);
if (kmax)
*kmax = fabs (kappa);
#else /* 3D */
# ifdef NP
double hxx = 2.*p->a[2*NP], hyy = 2.*p->a[2], hxy = p->a[NP + 1];
double hx = p->a[NP], hy = p->a[1];
# else
double hxx = 2.*p->a[0], hyy = 2.*p->a[1], hxy = p->a[2];
double hx = p->a[3], hy = p->a[4];
# endif
double dnm = 1. + sq(hx) + sq(hy);
kappa = - (hxx*(1. + sq(hy)) + hyy*(1. + sq(hx)) - 2.*hxy*hx*hy)
/sqrt (dnm*dnm*dnm);
if (kmax) {
double kg = (hxx*hyy - hxy*hxy)/(dnm*dnm);
double a = kappa*kappa/4. - kg;
*kmax = fabs (kappa/2.);
if (a >= 0.)
*kmax += sqrt (a);
}
#endif /* 3D */
if (fabs (kappa) > kappamax) {
if (kmax)
*kmax = kappamax;
return kappa > 0. ? kappamax : - kappamax;
}
return kappa;
}
#if AXI
static void parabola_fit_axi_curvature (const ParabolaFit * p,
double r, double h,
double * kappa, double * kmax)
{
double nr = (p->m.x*p->a[1] + p->m.y)/sqrt (1. + sq(p->a[1]));
/* limit the minimum radius to half the grid size */
double kaxi = nr/max(r, h/2.);
*kappa += kaxi;
if (kmax)
*kmax = max (*kmax, fabs (kaxi));
}
#endif /* 2D */
|
