# src/parabola.h

 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 ``` ``````#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 normalize (coord * n) { double norm = 0.; foreach_dimension() norm += sq(n->x); norm = sqrt(norm); foreach_dimension() n->x /= norm; } 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 = 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. + 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 */``````