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Kabsch.h
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Kabsch.h
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/**************************************************************************
Implemetation of Kabsch algoritm for finding the best rotation matrix
---------------------------------------------------------------------------
x - x(i,m) are coordinates of atom m in set x (input)
y - y(i,m) are coordinates of atom m in set y (input)
n - n is number of atom pairs (input)
mode - 0:calculate rms only (input)
1:calculate u,t only (takes medium)
2:calculate rms,u,t (takes longer)
rms - sum of w*(ux+t-y)**2 over all atom pairs (output)
u - u(i,j) is rotation matrix for best superposition (output)
t - t(i) is translation vector for best superposition (output)
**************************************************************************/
bool Kabsch(double **x, double **y, int n, int mode, double *rms,
double t[3], double u[3][3])
{
int i, j, m, m1, l, k;
double e0, rms1, d, h, g;
double cth, sth, sqrth, p, det, sigma;
double xc[3], yc[3];
double a[3][3], b[3][3], r[3][3], e[3], rr[6], ss[6];
double sqrt3 = 1.73205080756888, tol = 0.01;
int ip[] = { 0, 1, 3, 1, 2, 4, 3, 4, 5 };
int ip2312[] = { 1, 2, 0, 1 };
int a_failed = 0, b_failed = 0;
double epsilon = 0.00000001;
//initialization
*rms = 0;
rms1 = 0;
e0 = 0;
double c1[3], c2[3];
double s1[3], s2[3];
double sx[3], sy[3], sz[3];
for (i = 0; i < 3; i++)
{
s1[i] = 0.0;
s2[i] = 0.0;
sx[i] = 0.0;
sy[i] = 0.0;
sz[i] = 0.0;
}
for (i = 0; i<3; i++)
{
xc[i] = 0.0;
yc[i] = 0.0;
t[i] = 0.0;
for (j = 0; j<3; j++)
{
u[i][j] = 0.0;
r[i][j] = 0.0;
a[i][j] = 0.0;
if (i == j)
{
u[i][j] = 1.0;
a[i][j] = 1.0;
}
}
}
if (n<1) return false;
//compute centers for vector sets x, y
for (i = 0; i<n; i++)
{
for (j = 0; j < 3; j++)
{
c1[j] = x[i][j];
c2[j] = y[i][j];
s1[j] += c1[j];
s2[j] += c2[j];
}
for (j = 0; j < 3; j++)
{
sx[j] += c1[0] * c2[j];
sy[j] += c1[1] * c2[j];
sz[j] += c1[2] * c2[j];
}
}
for (i = 0; i < 3; i++)
{
xc[i] = s1[i] / n;
yc[i] = s2[i] / n;
}
if (mode == 2 || mode == 0)
for (int mm = 0; mm < n; mm++)
for (int nn = 0; nn < 3; nn++)
e0 += (x[mm][nn] - xc[nn]) * (x[mm][nn] - xc[nn]) +
(y[mm][nn] - yc[nn]) * (y[mm][nn] - yc[nn]);
for (j = 0; j < 3; j++)
{
r[j][0] = sx[j] - s1[0] * s2[j] / n;
r[j][1] = sy[j] - s1[1] * s2[j] / n;
r[j][2] = sz[j] - s1[2] * s2[j] / n;
}
//compute determinant of matrix r
det = r[0][0] * (r[1][1] * r[2][2] - r[1][2] * r[2][1])\
- r[0][1] * (r[1][0] * r[2][2] - r[1][2] * r[2][0])\
+ r[0][2] * (r[1][0] * r[2][1] - r[1][1] * r[2][0]);
sigma = det;
//compute tras(r)*r
m = 0;
for (j = 0; j<3; j++)
{
for (i = 0; i <= j; i++)
{
rr[m] = r[0][i] * r[0][j] + r[1][i] * r[1][j] + r[2][i] * r[2][j];
m++;
}
}
double spur = (rr[0] + rr[2] + rr[5]) / 3.0;
double cof = (((((rr[2] * rr[5] - rr[4] * rr[4]) + rr[0] * rr[5])\
- rr[3] * rr[3]) + rr[0] * rr[2]) - rr[1] * rr[1]) / 3.0;
det = det*det;
for (i = 0; i<3; i++) e[i] = spur;
if (spur>0)
{
d = spur*spur;
h = d - cof;
g = (spur*cof - det) / 2.0 - spur*h;
if (h>0)
{
sqrth = sqrt(h);
d = h*h*h - g*g;
if (d<0.0) d = 0.0;
d = atan2(sqrt(d), -g) / 3.0;
cth = sqrth * cos(d);
sth = sqrth*sqrt3*sin(d);
e[0] = (spur + cth) + cth;
e[1] = (spur - cth) + sth;
e[2] = (spur - cth) - sth;
if (mode != 0)
{//compute a
for (l = 0; l<3; l = l + 2)
{
d = e[l];
ss[0] = (d - rr[2]) * (d - rr[5]) - rr[4] * rr[4];
ss[1] = (d - rr[5]) * rr[1] + rr[3] * rr[4];
ss[2] = (d - rr[0]) * (d - rr[5]) - rr[3] * rr[3];
ss[3] = (d - rr[2]) * rr[3] + rr[1] * rr[4];
ss[4] = (d - rr[0]) * rr[4] + rr[1] * rr[3];
ss[5] = (d - rr[0]) * (d - rr[2]) - rr[1] * rr[1];
if (fabs(ss[0]) <= epsilon) ss[0] = 0.0;
if (fabs(ss[1]) <= epsilon) ss[1] = 0.0;
if (fabs(ss[2]) <= epsilon) ss[2] = 0.0;
if (fabs(ss[3]) <= epsilon) ss[3] = 0.0;
if (fabs(ss[4]) <= epsilon) ss[4] = 0.0;
if (fabs(ss[5]) <= epsilon) ss[5] = 0.0;
if (fabs(ss[0]) >= fabs(ss[2]))
{
j = 0;
if (fabs(ss[0]) < fabs(ss[5])) j = 2;
}
else if (fabs(ss[2]) >= fabs(ss[5])) j = 1;
else j = 2;
d = 0.0;
j = 3 * j;
for (i = 0; i<3; i++)
{
k = ip[i + j];
a[i][l] = ss[k];
d = d + ss[k] * ss[k];
}
//if( d > 0.0 ) d = 1.0 / sqrt(d);
if (d > epsilon) d = 1.0 / sqrt(d);
else d = 0.0;
for (i = 0; i<3; i++) a[i][l] = a[i][l] * d;
}//for l
d = a[0][0] * a[0][2] + a[1][0] * a[1][2] + a[2][0] * a[2][2];
if ((e[0] - e[1]) >(e[1] - e[2]))
{
m1 = 2;
m = 0;
}
else
{
m1 = 0;
m = 2;
}
p = 0;
for (i = 0; i<3; i++)
{
a[i][m1] = a[i][m1] - d*a[i][m];
p = p + a[i][m1] * a[i][m1];
}
if (p <= tol)
{
p = 1.0;
for (i = 0; i<3; i++)
{
if (p < fabs(a[i][m])) continue;
p = fabs(a[i][m]);
j = i;
}
k = ip2312[j];
l = ip2312[j + 1];
p = sqrt(a[k][m] * a[k][m] + a[l][m] * a[l][m]);
if (p > tol)
{
a[j][m1] = 0.0;
a[k][m1] = -a[l][m] / p;
a[l][m1] = a[k][m] / p;
}
else a_failed = 1;
}//if p<=tol
else
{
p = 1.0 / sqrt(p);
for (i = 0; i<3; i++) a[i][m1] = a[i][m1] * p;
}//else p<=tol
if (a_failed != 1)
{
a[0][1] = a[1][2] * a[2][0] - a[1][0] * a[2][2];
a[1][1] = a[2][2] * a[0][0] - a[2][0] * a[0][2];
a[2][1] = a[0][2] * a[1][0] - a[0][0] * a[1][2];
}
}//if(mode!=0)
}//h>0
//compute b anyway
if (mode != 0 && a_failed != 1)//a is computed correctly
{
//compute b
for (l = 0; l<2; l++)
{
d = 0.0;
for (i = 0; i<3; i++)
{
b[i][l] = r[i][0] * a[0][l] +
r[i][1] * a[1][l] + r[i][2] * a[2][l];
d = d + b[i][l] * b[i][l];
}
//if( d > 0 ) d = 1.0 / sqrt(d);
if (d > epsilon) d = 1.0 / sqrt(d);
else d = 0.0;
for (i = 0; i<3; i++) b[i][l] = b[i][l] * d;
}
d = b[0][0] * b[0][1] + b[1][0] * b[1][1] + b[2][0] * b[2][1];
p = 0.0;
for (i = 0; i<3; i++)
{
b[i][1] = b[i][1] - d*b[i][0];
p += b[i][1] * b[i][1];
}
if (p <= tol)
{
p = 1.0;
for (i = 0; i<3; i++)
{
if (p<fabs(b[i][0])) continue;
p = fabs(b[i][0]);
j = i;
}
k = ip2312[j];
l = ip2312[j + 1];
p = sqrt(b[k][0] * b[k][0] + b[l][0] * b[l][0]);
if (p > tol)
{
b[j][1] = 0.0;
b[k][1] = -b[l][0] / p;
b[l][1] = b[k][0] / p;
}
else b_failed = 1;
}//if( p <= tol )
else
{
p = 1.0 / sqrt(p);
for (i = 0; i<3; i++) b[i][1] = b[i][1] * p;
}
if (b_failed != 1)
{
b[0][2] = b[1][0] * b[2][1] - b[1][1] * b[2][0];
b[1][2] = b[2][0] * b[0][1] - b[2][1] * b[0][0];
b[2][2] = b[0][0] * b[1][1] - b[0][1] * b[1][0];
//compute u
for (i = 0; i<3; i++)
for (j = 0; j<3; j++)
u[i][j] = b[i][0] * a[j][0] +
b[i][1] * a[j][1] + b[i][2] * a[j][2];
}
//compute t
for (i = 0; i<3; i++)
t[i] = ((yc[i] - u[i][0] * xc[0]) - u[i][1] * xc[1]) -
u[i][2] * xc[2];
}//if(mode!=0 && a_failed!=1)
}//spur>0
else //just compute t and errors
{
//compute t
for (i = 0; i<3; i++)
t[i] = ((yc[i] - u[i][0] * xc[0]) - u[i][1] * xc[1]) -
u[i][2] * xc[2];
}//else spur>0
//compute rms
for (i = 0; i<3; i++)
{
if (e[i] < 0) e[i] = 0;
e[i] = sqrt(e[i]);
}
d = e[2];
if (sigma < 0.0) d = -d;
d = (d + e[1]) + e[0];
if (mode == 2 || mode == 0)
{
rms1 = (e0 - d) - d;
if (rms1 < 0.0) rms1 = 0.0;
}
*rms = rms1;
return true;
}