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lammps/src/math_extra.cpp
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/* ----------------------------------------------------------------------
LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
http://lammps.sandia.gov, Sandia National Laboratories
Steve Plimpton, sjplimp@sandia.gov
Copyright (2003) Sandia Corporation. Under the terms of Contract
DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
certain rights in this software. This software is distributed under
the GNU General Public License.
See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */
/* ----------------------------------------------------------------------
Contributing author: Mike Brown (SNL)
------------------------------------------------------------------------- */
#include <stdio.h>
#include <string.h>
#include "math_extra.h"
#define MAXJACOBI 50
namespace MathExtra {
/* ----------------------------------------------------------------------
output a matrix
------------------------------------------------------------------------- */
void write3(const double mat[3][3])
{
for (unsigned i = 0; i < 3; i++) {
for (unsigned j = 0; j < 3; j++) printf("%g ",mat[i][j]);
printf("\n");
}
}
/* ----------------------------------------------------------------------
solve Ax = b or M ans = v
use gaussian elimination & partial pivoting on matrix
------------------------------------------------------------------------- */
int mldivide3(const double m[3][3], const double *v, double *ans)
{
// create augmented matrix for pivoting
double aug[3][4];
for (unsigned i = 0; i < 3; i++) {
aug[i][3] = v[i];
for (unsigned j = 0; j < 3; j++) aug[i][j] = m[i][j];
}
for (unsigned i = 0; i < 2; i++) {
unsigned p = i;
for (unsigned j = i+1; j < 3; j++) {
if (fabs(aug[j][i]) > fabs(aug[i][i])) {
double tempv[4];
memcpy(tempv,aug[i],4*sizeof(double));
memcpy(aug[i],aug[j],4*sizeof(double));
memcpy(aug[j],tempv,4*sizeof(double));
}
}
while (p < 3 && aug[p][i] == 0.0) p++;
if (p == 3) return 1;
else
if (p != i) {
double tempv[4];
memcpy(tempv,aug[i],4*sizeof(double));
memcpy(aug[i],aug[p],4*sizeof(double));
memcpy(aug[p],tempv,4*sizeof(double));
}
for (unsigned j = i+1; j < 3; j++) {
double m = aug[j][i]/aug[i][i];
for (unsigned k=i+1; k<4; k++) aug[j][k]-=m*aug[i][k];
}
}
if (aug[2][2] == 0.0) return 1;
// back substitution
ans[2] = aug[2][3]/aug[2][2];
for (int i = 1; i >= 0; i--) {
double sumax = 0.0;
for (unsigned j = i+1; j < 3; j++) sumax += aug[i][j]*ans[j];
ans[i] = (aug[i][3]-sumax) / aug[i][i];
}
return 0;
}
/* ----------------------------------------------------------------------
compute evalues and evectors of 3x3 real symmetric matrix
based on Jacobi rotations
adapted from Numerical Recipes jacobi() function
------------------------------------------------------------------------- */
int jacobi(double matrix[3][3], double *evalues, double evectors[3][3])
{
int i,j,k;
double tresh,theta,tau,t,sm,s,h,g,c,b[3],z[3];
for (i = 0; i < 3; i++) {
for (j = 0; j < 3; j++) evectors[i][j] = 0.0;
evectors[i][i] = 1.0;
}
for (i = 0; i < 3; i++) {
b[i] = evalues[i] = matrix[i][i];
z[i] = 0.0;
}
for (int iter = 1; iter <= MAXJACOBI; iter++) {
sm = 0.0;
for (i = 0; i < 2; i++)
for (j = i+1; j < 3; j++)
sm += fabs(matrix[i][j]);
if (sm == 0.0) return 0;
if (iter < 4) tresh = 0.2*sm/(3*3);
else tresh = 0.0;
for (i = 0; i < 2; i++) {
for (j = i+1; j < 3; j++) {
g = 100.0*fabs(matrix[i][j]);
if (iter > 4 && fabs(evalues[i])+g == fabs(evalues[i])
&& fabs(evalues[j])+g == fabs(evalues[j]))
matrix[i][j] = 0.0;
else if (fabs(matrix[i][j]) > tresh) {
h = evalues[j]-evalues[i];
if (fabs(h)+g == fabs(h)) t = (matrix[i][j])/h;
else {
theta = 0.5*h/(matrix[i][j]);
t = 1.0/(fabs(theta)+sqrt(1.0+theta*theta));
if (theta < 0.0) t = -t;
}
c = 1.0/sqrt(1.0+t*t);
s = t*c;
tau = s/(1.0+c);
h = t*matrix[i][j];
z[i] -= h;
z[j] += h;
evalues[i] -= h;
evalues[j] += h;
matrix[i][j] = 0.0;
for (k = 0; k < i; k++) rotate(matrix,k,i,k,j,s,tau);
for (k = i+1; k < j; k++) rotate(matrix,i,k,k,j,s,tau);
for (k = j+1; k < 3; k++) rotate(matrix,i,k,j,k,s,tau);
for (k = 0; k < 3; k++) rotate(evectors,k,i,k,j,s,tau);
}
}
}
for (i = 0; i < 3; i++) {
evalues[i] = b[i] += z[i];
z[i] = 0.0;
}
}
return 1;
}
/* ----------------------------------------------------------------------
perform a single Jacobi rotation
------------------------------------------------------------------------- */
void rotate(double matrix[3][3], int i, int j, int k, int l,
double s, double tau)
{
double g = matrix[i][j];
double h = matrix[k][l];
matrix[i][j] = g-s*(h+g*tau);
matrix[k][l] = h+s*(g-h*tau);
}
/* ----------------------------------------------------------------------
Richardson iteration to update quaternion from angular momentum
return new normalized quaternion q
also returns updated omega at 1/2 step
------------------------------------------------------------------------- */
void richardson(double *q, double *m, double *w, double *moments, double dtq)
{
// full update from dq/dt = 1/2 w q
double wq[4];
MathExtra::vecquat(w,q,wq);
double qfull[4];
qfull[0] = q[0] + dtq * wq[0];
qfull[1] = q[1] + dtq * wq[1];
qfull[2] = q[2] + dtq * wq[2];
qfull[3] = q[3] + dtq * wq[3];
MathExtra::qnormalize(qfull);
// 1st half update from dq/dt = 1/2 w q
double qhalf[4];
qhalf[0] = q[0] + 0.5*dtq * wq[0];
qhalf[1] = q[1] + 0.5*dtq * wq[1];
qhalf[2] = q[2] + 0.5*dtq * wq[2];
qhalf[3] = q[3] + 0.5*dtq * wq[3];
MathExtra::qnormalize(qhalf);
// re-compute omega at 1/2 step from m at 1/2 step and q at 1/2 step
// recompute wq
MathExtra::mq_to_omega(m,qhalf,moments,w);
MathExtra::vecquat(w,qhalf,wq);
// 2nd half update from dq/dt = 1/2 w q
qhalf[0] += 0.5*dtq * wq[0];
qhalf[1] += 0.5*dtq * wq[1];
qhalf[2] += 0.5*dtq * wq[2];
qhalf[3] += 0.5*dtq * wq[3];
MathExtra::qnormalize(qhalf);
// corrected Richardson update
q[0] = 2.0*qhalf[0] - qfull[0];
q[1] = 2.0*qhalf[1] - qfull[1];
q[2] = 2.0*qhalf[2] - qfull[2];
q[3] = 2.0*qhalf[3] - qfull[3];
MathExtra::qnormalize(q);
}
/* ----------------------------------------------------------------------
apply evolution operators to quat, quat momentum
Miller et al., J Chem Phys. 116, 8649-8659 (2002)
------------------------------------------------------------------------- */
void no_squish_rotate(int k, double *p, double *q, double *inertia,
double dt)
{
double phi,c_phi,s_phi,kp[4],kq[4];
// apply permuation operator on p and q, get kp and kq
if (k == 1) {
kq[0] = -q[1]; kp[0] = -p[1];
kq[1] = q[0]; kp[1] = p[0];
kq[2] = q[3]; kp[2] = p[3];
kq[3] = -q[2]; kp[3] = -p[2];
} else if (k == 2) {
kq[0] = -q[2]; kp[0] = -p[2];
kq[1] = -q[3]; kp[1] = -p[3];
kq[2] = q[0]; kp[2] = p[0];
kq[3] = q[1]; kp[3] = p[1];
} else if (k == 3) {
kq[0] = -q[3]; kp[0] = -p[3];
kq[1] = q[2]; kp[1] = p[2];
kq[2] = -q[1]; kp[2] = -p[1];
kq[3] = q[0]; kp[3] = p[0];
}
// obtain phi, cosines and sines
phi = p[0]*kq[0] + p[1]*kq[1] + p[2]*kq[2] + p[3]*kq[3];
if (fabs(inertia[k-1]) < 1e-6) phi *= 0.0;
else phi /= 4.0 * inertia[k-1];
c_phi = cos(dt * phi);
s_phi = sin(dt * phi);
// advance p and q
p[0] = c_phi*p[0] + s_phi*kp[0];
p[1] = c_phi*p[1] + s_phi*kp[1];
p[2] = c_phi*p[2] + s_phi*kp[2];
p[3] = c_phi*p[3] + s_phi*kp[3];
q[0] = c_phi*q[0] + s_phi*kq[0];
q[1] = c_phi*q[1] + s_phi*kq[1];
q[2] = c_phi*q[2] + s_phi*kq[2];
q[3] = c_phi*q[3] + s_phi*kq[3];
}
/* ----------------------------------------------------------------------
compute omega from angular momentum, both in space frame
only know Idiag so need to do M = Iw in body frame
ex,ey,ez are column vectors of rotation matrix P
Mbody = P_transpose Mspace
wbody = Mbody / Idiag
wspace = P wbody
set wbody component to 0.0 if inertia component is 0.0
otherwise body can spin easily around that axis
------------------------------------------------------------------------- */
void angmom_to_omega(double *m, double *ex, double *ey, double *ez,
double *idiag, double *w)
{
double wbody[3];
if (idiag[0] == 0.0) wbody[0] = 0.0;
else wbody[0] = (m[0]*ex[0] + m[1]*ex[1] + m[2]*ex[2]) / idiag[0];
if (idiag[1] == 0.0) wbody[1] = 0.0;
else wbody[1] = (m[0]*ey[0] + m[1]*ey[1] + m[2]*ey[2]) / idiag[1];
if (idiag[2] == 0.0) wbody[2] = 0.0;
else wbody[2] = (m[0]*ez[0] + m[1]*ez[1] + m[2]*ez[2]) / idiag[2];
w[0] = wbody[0]*ex[0] + wbody[1]*ey[0] + wbody[2]*ez[0];
w[1] = wbody[0]*ex[1] + wbody[1]*ey[1] + wbody[2]*ez[1];
w[2] = wbody[0]*ex[2] + wbody[1]*ey[2] + wbody[2]*ez[2];
}
/* ----------------------------------------------------------------------
compute omega from angular momentum
w = omega = angular velocity in space frame
wbody = angular velocity in body frame
project space-frame angular momentum onto body axes
and divide by principal moments
------------------------------------------------------------------------- */
void mq_to_omega(double *m, double *q, double *moments, double *w)
{
double wbody[3];
double rot[3][3];
MathExtra::quat_to_mat(q,rot);
MathExtra::transpose_matvec(rot,m,wbody);
if (moments[0] == 0.0) wbody[0] = 0.0;
else wbody[0] /= moments[0];
if (moments[1] == 0.0) wbody[1] = 0.0;
else wbody[1] /= moments[1];
if (moments[2] == 0.0) wbody[2] = 0.0;
else wbody[2] /= moments[2];
MathExtra::matvec(rot,wbody,w);
}
/* ----------------------------------------------------------------------
compute angular momentum from omega, both in space frame
only know Idiag so need to do M = Iw in body frame
ex,ey,ez are column vectors of rotation matrix P
wbody = P_transpose wspace
Mbody = Idiag wbody
Mspace = P Mbody
------------------------------------------------------------------------- */
void omega_to_angmom(double *w, double *ex, double *ey, double *ez,
double *idiag, double *m)
{
double mbody[3];
mbody[0] = (w[0]*ex[0] + w[1]*ex[1] + w[2]*ex[2]) * idiag[0];
mbody[1] = (w[0]*ey[0] + w[1]*ey[1] + w[2]*ey[2]) * idiag[1];
mbody[2] = (w[0]*ez[0] + w[1]*ez[1] + w[2]*ez[2]) * idiag[2];
m[0] = mbody[0]*ex[0] + mbody[1]*ey[0] + mbody[2]*ez[0];
m[1] = mbody[0]*ex[1] + mbody[1]*ey[1] + mbody[2]*ez[1];
m[2] = mbody[0]*ex[2] + mbody[1]*ey[2] + mbody[2]*ez[2];
}
/* ----------------------------------------------------------------------
create unit quaternion from space-frame ex,ey,ez
ex,ey,ez are columns of a rotation matrix
------------------------------------------------------------------------- */
void exyz_to_q(double *ex, double *ey, double *ez, double *q)
{
// squares of quaternion components
double q0sq = 0.25 * (ex[0] + ey[1] + ez[2] + 1.0);
double q1sq = q0sq - 0.5 * (ey[1] + ez[2]);
double q2sq = q0sq - 0.5 * (ex[0] + ez[2]);
double q3sq = q0sq - 0.5 * (ex[0] + ey[1]);
// some component must be greater than 1/4 since they sum to 1
// compute other components from it
if (q0sq >= 0.25) {
q[0] = sqrt(q0sq);
q[1] = (ey[2] - ez[1]) / (4.0*q[0]);
q[2] = (ez[0] - ex[2]) / (4.0*q[0]);
q[3] = (ex[1] - ey[0]) / (4.0*q[0]);
} else if (q1sq >= 0.25) {
q[1] = sqrt(q1sq);
q[0] = (ey[2] - ez[1]) / (4.0*q[1]);
q[2] = (ey[0] + ex[1]) / (4.0*q[1]);
q[3] = (ex[2] + ez[0]) / (4.0*q[1]);
} else if (q2sq >= 0.25) {
q[2] = sqrt(q2sq);
q[0] = (ez[0] - ex[2]) / (4.0*q[2]);
q[1] = (ey[0] + ex[1]) / (4.0*q[2]);
q[3] = (ez[1] + ey[2]) / (4.0*q[2]);
} else if (q3sq >= 0.25) {
q[3] = sqrt(q3sq);
q[0] = (ex[1] - ey[0]) / (4.0*q[3]);
q[1] = (ez[0] + ex[2]) / (4.0*q[3]);
q[2] = (ez[1] + ey[2]) / (4.0*q[3]);
}
qnormalize(q);
}
/* ----------------------------------------------------------------------
compute space-frame ex,ey,ez from current quaternion q
ex,ey,ez = space-frame coords of 1st,2nd,3rd principal axis
operation is ex = q' d q = Q d, where d is (1,0,0) = 1st axis in body frame
------------------------------------------------------------------------- */
void q_to_exyz(double *q, double *ex, double *ey, double *ez)
{
ex[0] = q[0]*q[0] + q[1]*q[1] - q[2]*q[2] - q[3]*q[3];
ex[1] = 2.0 * (q[1]*q[2] + q[0]*q[3]);
ex[2] = 2.0 * (q[1]*q[3] - q[0]*q[2]);
ey[0] = 2.0 * (q[1]*q[2] - q[0]*q[3]);
ey[1] = q[0]*q[0] - q[1]*q[1] + q[2]*q[2] - q[3]*q[3];
ey[2] = 2.0 * (q[2]*q[3] + q[0]*q[1]);
ez[0] = 2.0 * (q[1]*q[3] + q[0]*q[2]);
ez[1] = 2.0 * (q[2]*q[3] - q[0]*q[1]);
ez[2] = q[0]*q[0] - q[1]*q[1] - q[2]*q[2] + q[3]*q[3];
}
/* ----------------------------------------------------------------------
compute rotation matrix from quaternion
quat = [w i j k]
------------------------------------------------------------------------- */
void quat_to_mat(const double *quat, double mat[3][3])
{
double w2 = quat[0]*quat[0];
double i2 = quat[1]*quat[1];
double j2 = quat[2]*quat[2];
double k2 = quat[3]*quat[3];
double twoij = 2.0*quat[1]*quat[2];
double twoik = 2.0*quat[1]*quat[3];
double twojk = 2.0*quat[2]*quat[3];
double twoiw = 2.0*quat[1]*quat[0];
double twojw = 2.0*quat[2]*quat[0];
double twokw = 2.0*quat[3]*quat[0];
mat[0][0] = w2+i2-j2-k2;
mat[0][1] = twoij-twokw;
mat[0][2] = twojw+twoik;
mat[1][0] = twoij+twokw;
mat[1][1] = w2-i2+j2-k2;
mat[1][2] = twojk-twoiw;
mat[2][0] = twoik-twojw;
mat[2][1] = twojk+twoiw;
mat[2][2] = w2-i2-j2+k2;
}
/* ----------------------------------------------------------------------
compute rotation matrix from quaternion conjugate
quat = [w i j k]
------------------------------------------------------------------------- */
void quat_to_mat_trans(const double *quat, double mat[3][3])
{
double w2 = quat[0]*quat[0];
double i2 = quat[1]*quat[1];
double j2 = quat[2]*quat[2];
double k2 = quat[3]*quat[3];
double twoij = 2.0*quat[1]*quat[2];
double twoik = 2.0*quat[1]*quat[3];
double twojk = 2.0*quat[2]*quat[3];
double twoiw = 2.0*quat[1]*quat[0];
double twojw = 2.0*quat[2]*quat[0];
double twokw = 2.0*quat[3]*quat[0];
mat[0][0] = w2+i2-j2-k2;
mat[1][0] = twoij-twokw;
mat[2][0] = twojw+twoik;
mat[0][1] = twoij+twokw;
mat[1][1] = w2-i2+j2-k2;
mat[2][1] = twojk-twoiw;
mat[0][2] = twoik-twojw;
mat[1][2] = twojk+twoiw;
mat[2][2] = w2-i2-j2+k2;
}
/* ----------------------------------------------------------------------
compute space-frame inertia tensor of an ellipsoid
radii = 3 radii of ellipsoid
quat = orientiation quaternion of ellipsoid
return symmetric inertia tensor as 6-vector in Voigt notation
------------------------------------------------------------------------- */
void inertia_ellipsoid(double *radii, double *quat, double mass,
double *inertia)
{
double p[3][3],ptrans[3][3],itemp[3][3],tensor[3][3];
double idiag[3];
quat_to_mat(quat,p);
quat_to_mat_trans(quat,ptrans);
idiag[0] = 0.2*mass * (radii[1]*radii[1] + radii[2]*radii[2]);
idiag[1] = 0.2*mass * (radii[0]*radii[0] + radii[2]*radii[2]);
idiag[2] = 0.2*mass * (radii[0]*radii[0] + radii[1]*radii[1]);
diag_times3(idiag,ptrans,itemp);
times3(p,itemp,tensor);
inertia[0] = tensor[0][0];
inertia[1] = tensor[1][1];
inertia[2] = tensor[2][2];
inertia[3] = tensor[1][2];
inertia[4] = tensor[0][2];
inertia[5] = tensor[0][1];
}
/* ----------------------------------------------------------------------
compute space-frame inertia tensor of a line segment in 2d
length = length of line
theta = orientiation of line
return symmetric inertia tensor as 6-vector in Voigt notation
------------------------------------------------------------------------- */
void inertia_line(double length, double theta, double mass, double *inertia)
{
double p[3][3],ptrans[3][3],itemp[3][3],tensor[3][3];
double q[4],idiag[3];
q[0] = cos(0.5*theta);
q[1] = q[2] = 0.0;
q[3] = sin(0.5*theta);
MathExtra::quat_to_mat(q,p);
MathExtra::quat_to_mat_trans(q,ptrans);
idiag[0] = 0.0;
idiag[1] = 1.0/12.0 * mass * length*length;
idiag[2] = 1.0/12.0 * mass * length*length;
MathExtra::diag_times3(idiag,ptrans,itemp);
MathExtra::times3(p,itemp,tensor);
inertia[0] = tensor[0][0];
inertia[1] = tensor[1][1];
inertia[2] = tensor[2][2];
inertia[3] = tensor[1][2];
inertia[4] = tensor[0][2];
inertia[5] = tensor[0][1];
}
/* ----------------------------------------------------------------------
compute space-frame inertia tensor of a triangle
v0,v1,v2 = 3 vertices of triangle
from http://en.wikipedia.org/wiki/Inertia_tensor_of_triangle
inertia tensor = a/24 (v0^2 + v1^2 + v2^2 + (v0+v1+v2)^2) I - a Vt S V
a = 2*area of tri = |(v1-v0) x (v2-v0)|
I = 3x3 identity matrix
V = 3x3 matrix with v0,v1,v2 as rows
Vt = 3x3 matrix with v0,v1,v2 as columns
S = 1/24 [2 1 1]
[1 2 1]
[1 1 2]
return symmetric inertia tensor as 6-vector in Voigt notation
------------------------------------------------------------------------- */
void inertia_triangle(double *v0, double *v1, double *v2,
double mass, double *inertia)
{
double s[3][3] = {{2.0, 1.0, 1.0}, {1.0, 2.0, 1.0}, {1.0, 1.0, 2.0}};
double v[3][3],sv[3][3],vtsv[3][3];
double vvv[3],v1mv0[3],v2mv0[3],normal[3];
v[0][0] = v0[0]; v[0][1] = v0[1]; v[0][2] = v0[2];
v[1][0] = v1[0]; v[1][1] = v1[1]; v[1][2] = v1[2];
v[2][0] = v2[0]; v[2][1] = v2[1]; v[2][2] = v2[2];
times3(s,v,sv);
transpose_times3(v,sv,vtsv);
double sum = lensq3(v0) + lensq3(v1) + lensq3(v2);
vvv[0] = v0[0] + v1[0] + v2[0];
vvv[1] = v0[1] + v1[1] + v2[1];
vvv[2] = v0[2] + v1[2] + v2[2];
sum += lensq3(vvv);
sub3(v1,v0,v1mv0);
sub3(v2,v0,v2mv0);
cross3(v1mv0,v2mv0,normal);
double a = len3(normal);
double inv24 = mass/24.0;
inertia[0] = inv24*a*(sum-vtsv[0][0]);
inertia[1] = inv24*a*(sum-vtsv[1][1]);
inertia[2] = inv24*a*(sum-vtsv[2][2]);
inertia[3] = -inv24*a*vtsv[1][2];
inertia[4] = -inv24*a*vtsv[0][2];
inertia[5] = -inv24*a*vtsv[0][1];
}
/* ----------------------------------------------------------------------
compute space-frame inertia tensor of a triangle
idiag = previously computed diagonal inertia tensor
quat = orientiation quaternion of triangle
return symmetric inertia tensor as 6-vector in Voigt notation
------------------------------------------------------------------------- */
void inertia_triangle(double *idiag, double *quat, double mass,
double *inertia)
{
double p[3][3],ptrans[3][3],itemp[3][3],tensor[3][3];
quat_to_mat(quat,p);
quat_to_mat_trans(quat,ptrans);
diag_times3(idiag,ptrans,itemp);
times3(p,itemp,tensor);
inertia[0] = tensor[0][0];
inertia[1] = tensor[1][1];
inertia[2] = tensor[2][2];
inertia[3] = tensor[1][2];
inertia[4] = tensor[0][2];
inertia[5] = tensor[0][1];
}
/* ----------------------------------------------------------------------
Build rotation matrix for a small angle rotation around the X axis
------------------------------------------------------------------------- */
void BuildRxMatrix(double R[3][3], const double angle)
{
const double angleSq = angle * angle;
const double cosAngle = (1.0 - angleSq * 0.25) / (1.0 + angleSq * 0.25);
const double sinAngle = angle / (1.0 + angleSq * 0.25);
R[0][0] = 1.0; R[0][1] = 0.0; R[0][2] = 0.0;
R[1][0] = 0.0; R[1][1] = cosAngle; R[1][2] = -sinAngle;
R[2][0] = 0.0; R[2][1] = sinAngle; R[2][2] = cosAngle;
}
/* ----------------------------------------------------------------------
Build rotation matrix for a small angle rotation around the Y axis
------------------------------------------------------------------------- */
void BuildRyMatrix(double R[3][3], const double angle)
{
const double angleSq = angle * angle;
const double cosAngle = (1.0 - angleSq * 0.25) / (1.0 + angleSq * 0.25);
const double sinAngle = angle / (1.0 + angleSq * 0.25);
R[0][0] = cosAngle; R[0][1] = 0.0; R[0][2] = sinAngle;
R[1][0] = 0.0; R[1][1] = 1.0; R[1][2] = 0.0;
R[2][0] = -sinAngle; R[2][1] = 0.0; R[2][2] = cosAngle;
}
/* ----------------------------------------------------------------------
Build rotation matrix for a small angle rotation around the Y axis
------------------------------------------------------------------------- */
void BuildRzMatrix(double R[3][3], const double angle)
{
const double angleSq = angle * angle;
const double cosAngle = (1.0 - angleSq * 0.25) / (1.0 + angleSq * 0.25);
const double sinAngle = angle / (1.0 + angleSq * 0.25);
R[0][0] = cosAngle; R[0][1] = -sinAngle; R[0][2] = 0.0;
R[1][0] = sinAngle; R[1][1] = cosAngle; R[1][2] = 0.0;
R[2][0] = 0.0; R[2][1] = 0.0; R[2][2] = 1.0;
}
/* ---------------------------------------------------------------------- */
}
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