| 1 |
#include <math.h> |
| 2 |
#include "RigidBody.hpp" |
| 3 |
#include "VDWAtom.hpp" |
| 4 |
#include "MatVec3.h" |
| 5 |
|
| 6 |
RigidBody::RigidBody() { |
| 7 |
is_linear = false; |
| 8 |
linear_axis = -1; |
| 9 |
momIntTol = 1e-6; |
| 10 |
} |
| 11 |
|
| 12 |
RigidBody::~RigidBody() { |
| 13 |
} |
| 14 |
|
| 15 |
void RigidBody::addAtom(VDWAtom* at) { |
| 16 |
|
| 17 |
vec3 coords; |
| 18 |
|
| 19 |
myAtoms.push_back(at); |
| 20 |
|
| 21 |
at->getPos(coords.vec); |
| 22 |
refCoords.push_back(coords); |
| 23 |
} |
| 24 |
|
| 25 |
void RigidBody::getPos(double theP[3]){ |
| 26 |
for (int i = 0; i < 3 ; i++) |
| 27 |
theP[i] = pos[i]; |
| 28 |
} |
| 29 |
|
| 30 |
void RigidBody::setPos(double theP[3]){ |
| 31 |
for (int i = 0; i < 3 ; i++) |
| 32 |
pos[i] = theP[i]; |
| 33 |
} |
| 34 |
|
| 35 |
|
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void RigidBody::setEuler( double phi, double theta, double psi ){ |
| 37 |
|
| 38 |
A[0][0] = (cos(phi) * cos(psi)) - (sin(phi) * cos(theta) * sin(psi)); |
| 39 |
A[0][1] = (sin(phi) * cos(psi)) + (cos(phi) * cos(theta) * sin(psi)); |
| 40 |
A[0][2] = sin(theta) * sin(psi); |
| 41 |
|
| 42 |
A[1][0] = -(cos(phi) * sin(psi)) - (sin(phi) * cos(theta) * cos(psi)); |
| 43 |
A[1][1] = -(sin(phi) * sin(psi)) + (cos(phi) * cos(theta) * cos(psi)); |
| 44 |
A[1][2] = sin(theta) * cos(psi); |
| 45 |
|
| 46 |
A[2][0] = sin(phi) * sin(theta); |
| 47 |
A[2][1] = -cos(phi) * sin(theta); |
| 48 |
A[2][2] = cos(theta); |
| 49 |
|
| 50 |
} |
| 51 |
|
| 52 |
void RigidBody::getQ( double q[4] ){ |
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|
| 54 |
double t, s; |
| 55 |
double ad1, ad2, ad3; |
| 56 |
|
| 57 |
t = A[0][0] + A[1][1] + A[2][2] + 1.0; |
| 58 |
if( t > 0.0 ){ |
| 59 |
|
| 60 |
s = 0.5 / sqrt( t ); |
| 61 |
q[0] = 0.25 / s; |
| 62 |
q[1] = (A[1][2] - A[2][1]) * s; |
| 63 |
q[2] = (A[2][0] - A[0][2]) * s; |
| 64 |
q[3] = (A[0][1] - A[1][0]) * s; |
| 65 |
} |
| 66 |
else{ |
| 67 |
|
| 68 |
ad1 = fabs( A[0][0] ); |
| 69 |
ad2 = fabs( A[1][1] ); |
| 70 |
ad3 = fabs( A[2][2] ); |
| 71 |
|
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if( ad1 >= ad2 && ad1 >= ad3 ){ |
| 73 |
|
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s = 2.0 * sqrt( 1.0 + A[0][0] - A[1][1] - A[2][2] ); |
| 75 |
q[0] = (A[1][2] + A[2][1]) / s; |
| 76 |
q[1] = 0.5 / s; |
| 77 |
q[2] = (A[0][1] + A[1][0]) / s; |
| 78 |
q[3] = (A[0][2] + A[2][0]) / s; |
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} |
| 80 |
else if( ad2 >= ad1 && ad2 >= ad3 ){ |
| 81 |
|
| 82 |
s = sqrt( 1.0 + A[1][1] - A[0][0] - A[2][2] ) * 2.0; |
| 83 |
q[0] = (A[0][2] + A[2][0]) / s; |
| 84 |
q[1] = (A[0][1] + A[1][0]) / s; |
| 85 |
q[2] = 0.5 / s; |
| 86 |
q[3] = (A[1][2] + A[2][1]) / s; |
| 87 |
} |
| 88 |
else{ |
| 89 |
|
| 90 |
s = sqrt( 1.0 + A[2][2] - A[0][0] - A[1][1] ) * 2.0; |
| 91 |
q[0] = (A[0][1] + A[1][0]) / s; |
| 92 |
q[1] = (A[0][2] + A[2][0]) / s; |
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q[2] = (A[1][2] + A[2][1]) / s; |
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q[3] = 0.5 / s; |
| 95 |
} |
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} |
| 97 |
} |
| 98 |
|
| 99 |
void RigidBody::setQ( double the_q[4] ){ |
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|
| 101 |
double q0Sqr, q1Sqr, q2Sqr, q3Sqr; |
| 102 |
|
| 103 |
q0Sqr = the_q[0] * the_q[0]; |
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q1Sqr = the_q[1] * the_q[1]; |
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q2Sqr = the_q[2] * the_q[2]; |
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q3Sqr = the_q[3] * the_q[3]; |
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|
| 108 |
A[0][0] = q0Sqr + q1Sqr - q2Sqr - q3Sqr; |
| 109 |
A[0][1] = 2.0 * ( the_q[1] * the_q[2] + the_q[0] * the_q[3] ); |
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A[0][2] = 2.0 * ( the_q[1] * the_q[3] - the_q[0] * the_q[2] ); |
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|
| 112 |
A[1][0] = 2.0 * ( the_q[1] * the_q[2] - the_q[0] * the_q[3] ); |
| 113 |
A[1][1] = q0Sqr - q1Sqr + q2Sqr - q3Sqr; |
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A[1][2] = 2.0 * ( the_q[2] * the_q[3] + the_q[0] * the_q[1] ); |
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|
| 116 |
A[2][0] = 2.0 * ( the_q[1] * the_q[3] + the_q[0] * the_q[2] ); |
| 117 |
A[2][1] = 2.0 * ( the_q[2] * the_q[3] - the_q[0] * the_q[1] ); |
| 118 |
A[2][2] = q0Sqr - q1Sqr -q2Sqr +q3Sqr; |
| 119 |
|
| 120 |
} |
| 121 |
|
| 122 |
void RigidBody::getA( double the_A[3][3] ){ |
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|
| 124 |
for (int i = 0; i < 3; i++) |
| 125 |
for (int j = 0; j < 3; j++) |
| 126 |
the_A[i][j] = A[i][j]; |
| 127 |
|
| 128 |
} |
| 129 |
|
| 130 |
void RigidBody::setA( double the_A[3][3] ){ |
| 131 |
|
| 132 |
for (int i = 0; i < 3; i++) |
| 133 |
for (int j = 0; j < 3; j++) |
| 134 |
A[i][j] = the_A[i][j]; |
| 135 |
|
| 136 |
} |
| 137 |
|
| 138 |
void RigidBody::getI( double the_I[3][3] ){ |
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|
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for (int i = 0; i < 3; i++) |
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for (int j = 0; j < 3; j++) |
| 142 |
the_I[i][j] = I[i][j]; |
| 143 |
|
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} |
| 145 |
|
| 146 |
void RigidBody::lab2Body( double r[3] ){ |
| 147 |
|
| 148 |
double rl[3]; // the lab frame vector |
| 149 |
|
| 150 |
rl[0] = r[0]; |
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rl[1] = r[1]; |
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rl[2] = r[2]; |
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|
| 154 |
r[0] = (A[0][0] * rl[0]) + (A[0][1] * rl[1]) + (A[0][2] * rl[2]); |
| 155 |
r[1] = (A[1][0] * rl[0]) + (A[1][1] * rl[1]) + (A[1][2] * rl[2]); |
| 156 |
r[2] = (A[2][0] * rl[0]) + (A[2][1] * rl[1]) + (A[2][2] * rl[2]); |
| 157 |
|
| 158 |
} |
| 159 |
|
| 160 |
void RigidBody::body2Lab( double r[3] ){ |
| 161 |
|
| 162 |
double rb[3]; // the body frame vector |
| 163 |
|
| 164 |
rb[0] = r[0]; |
| 165 |
rb[1] = r[1]; |
| 166 |
rb[2] = r[2]; |
| 167 |
|
| 168 |
r[0] = (A[0][0] * rb[0]) + (A[1][0] * rb[1]) + (A[2][0] * rb[2]); |
| 169 |
r[1] = (A[0][1] * rb[0]) + (A[1][1] * rb[1]) + (A[2][1] * rb[2]); |
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r[2] = (A[0][2] * rb[0]) + (A[1][2] * rb[1]) + (A[2][2] * rb[2]); |
| 171 |
|
| 172 |
} |
| 173 |
|
| 174 |
void RigidBody::calcRefCoords( ) { |
| 175 |
|
| 176 |
int i, j, it, n_linear_coords, pAxis, maxAxis, midAxis; |
| 177 |
double mtmp; |
| 178 |
vec3 apos; |
| 179 |
double refCOM[3]; |
| 180 |
vec3 ptmp; |
| 181 |
double Itmp[3][3]; |
| 182 |
double pAxisMat[3][3], pAxisRotMat[3][3]; |
| 183 |
double evals[3]; |
| 184 |
double prePos[3], rotPos[3]; |
| 185 |
double r, r2, len; |
| 186 |
double iMat[3][3]; |
| 187 |
|
| 188 |
// First, find the center of mass: |
| 189 |
|
| 190 |
mass = 0.0; |
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for (j=0; j<3; j++) |
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refCOM[j] = 0.0; |
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|
| 194 |
for (i = 0; i < myAtoms.size(); i++) { |
| 195 |
mtmp = myAtoms[i]->getMass(); |
| 196 |
mass += mtmp; |
| 197 |
|
| 198 |
apos = refCoords[i]; |
| 199 |
|
| 200 |
for(j = 0; j < 3; j++) { |
| 201 |
refCOM[j] += apos[j]*mtmp; |
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} |
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} |
| 204 |
|
| 205 |
for(j = 0; j < 3; j++) |
| 206 |
refCOM[j] /= mass; |
| 207 |
|
| 208 |
// Next, move the origin of the reference coordinate system to the COM: |
| 209 |
|
| 210 |
for (i = 0; i < myAtoms.size(); i++) { |
| 211 |
apos = refCoords[i]; |
| 212 |
for (j=0; j < 3; j++) { |
| 213 |
apos[j] = apos[j] - refCOM[j]; |
| 214 |
} |
| 215 |
refCoords[i] = apos; |
| 216 |
} |
| 217 |
|
| 218 |
// Moment of Inertia calculation |
| 219 |
|
| 220 |
for (i = 0; i < 3; i++) |
| 221 |
for (j = 0; j < 3; j++) |
| 222 |
Itmp[i][j] = 0.0; |
| 223 |
|
| 224 |
for (it = 0; it < myAtoms.size(); it++) { |
| 225 |
|
| 226 |
mtmp = myAtoms[it]->getMass(); |
| 227 |
ptmp = refCoords[it]; |
| 228 |
r= norm3(ptmp.vec); |
| 229 |
r2 = r*r; |
| 230 |
|
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for (i = 0; i < 3; i++) { |
| 232 |
for (j = 0; j < 3; j++) { |
| 233 |
|
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if (i==j) Itmp[i][j] += mtmp * r2; |
| 235 |
|
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Itmp[i][j] -= mtmp * ptmp.vec[i]*ptmp.vec[j]; |
| 237 |
} |
| 238 |
} |
| 239 |
} |
| 240 |
|
| 241 |
diagonalize3x3(Itmp, evals, sU); |
| 242 |
|
| 243 |
// zero out I and then fill the diagonals with the moments of inertia: |
| 244 |
|
| 245 |
n_linear_coords = 0; |
| 246 |
|
| 247 |
for (i = 0; i < 3; i++) { |
| 248 |
for (j = 0; j < 3; j++) { |
| 249 |
I[i][j] = 0.0; |
| 250 |
} |
| 251 |
I[i][i] = evals[i]; |
| 252 |
|
| 253 |
if (fabs(evals[i]) < momIntTol) { |
| 254 |
is_linear = true; |
| 255 |
n_linear_coords++; |
| 256 |
linear_axis = i; |
| 257 |
} |
| 258 |
} |
| 259 |
|
| 260 |
if (n_linear_coords > 1) { |
| 261 |
printf( |
| 262 |
"RigidBody error.\n" |
| 263 |
"\tOOPSE found more than one axis in this rigid body with a vanishing \n" |
| 264 |
"\tmoment of inertia. This can happen in one of three ways:\n" |
| 265 |
"\t 1) Only one atom was specified, or \n" |
| 266 |
"\t 2) All atoms were specified at the same location, or\n" |
| 267 |
"\t 3) The programmers did something stupid.\n" |
| 268 |
"\tIt is silly to use a rigid body to describe this situation. Be smarter.\n" |
| 269 |
); |
| 270 |
exit(-1); |
| 271 |
} |
| 272 |
|
| 273 |
//sort and reorder the moment axes |
| 274 |
if (evals[0] < evals[1] && evals[0] < evals[2]) |
| 275 |
pAxis = 0; |
| 276 |
else if (evals[1] < evals[2]) |
| 277 |
pAxis = 1; |
| 278 |
else |
| 279 |
pAxis = 2; |
| 280 |
|
| 281 |
if (evals[0] > evals[1] && evals[0] > evals[2]) |
| 282 |
maxAxis = 0; |
| 283 |
else if (evals[1] > evals[2]) |
| 284 |
maxAxis = 1; |
| 285 |
else |
| 286 |
maxAxis = 2; |
| 287 |
|
| 288 |
midAxis = 0; |
| 289 |
if (midAxis == pAxis || midAxis == pAxis) |
| 290 |
midAxis = 1; |
| 291 |
if (midAxis == pAxis || midAxis == pAxis) |
| 292 |
midAxis = 2; |
| 293 |
|
| 294 |
if (pAxis != maxAxis){ |
| 295 |
//zero out our matrices |
| 296 |
for (i=0; i<3; i++){ |
| 297 |
for (j=0; j<3; j++) { |
| 298 |
pAxisMat[i][j] = 0.0; |
| 299 |
pAxisRotMat[i][j] = 0.0; |
| 300 |
} |
| 301 |
} |
| 302 |
|
| 303 |
//let z be the smallest and x be the largest eigenvalue axes |
| 304 |
for (i=0; i<3; i++){ |
| 305 |
pAxisMat[i][2] = I[i][pAxis]; |
| 306 |
pAxisMat[i][1] = I[i][midAxis]; |
| 307 |
pAxisMat[i][0] = I[i][maxAxis]; |
| 308 |
} |
| 309 |
|
| 310 |
//calculate the proper rotation matrix |
| 311 |
transposeMat3(pAxisMat, pAxisRotMat); |
| 312 |
|
| 313 |
//rotate the rigid body to the principle axis frame |
| 314 |
for (i = 0; i < myAtoms.size(); i++) { |
| 315 |
apos = refCoords[i]; |
| 316 |
for (j=0; j<3; j++) |
| 317 |
prePos[j] = apos[j]; |
| 318 |
|
| 319 |
matVecMul3(pAxisRotMat, prePos, rotPos); |
| 320 |
|
| 321 |
for (j=0; j < 3; j++) |
| 322 |
apos[j] = rotPos[j]; |
| 323 |
|
| 324 |
refCoords[i] = apos; |
| 325 |
} |
| 326 |
|
| 327 |
//the lab and the body frame match up at this point, so A = Identity Matrix |
| 328 |
for (i=0; i<3; i++){ |
| 329 |
for (j=0; j<3; j++){ |
| 330 |
if (i == j) |
| 331 |
iMat[i][j] = 1.0; |
| 332 |
else |
| 333 |
iMat[i][j] = 0.0; |
| 334 |
} |
| 335 |
} |
| 336 |
setA(iMat); |
| 337 |
} |
| 338 |
|
| 339 |
// renormalize column vectors: |
| 340 |
|
| 341 |
for (i=0; i < 3; i++) { |
| 342 |
len = 0.0; |
| 343 |
for (j = 0; j < 3; j++) { |
| 344 |
len += sU[i][j]*sU[i][j]; |
| 345 |
} |
| 346 |
len = sqrt(len); |
| 347 |
for (j = 0; j < 3; j++) { |
| 348 |
sU[i][j] /= len; |
| 349 |
} |
| 350 |
} |
| 351 |
} |
| 352 |
|
| 353 |
void RigidBody::doEulerToRotMat(double euler[3], double myA[3][3] ){ |
| 354 |
|
| 355 |
double phi, theta, psi; |
| 356 |
|
| 357 |
phi = euler[0]; |
| 358 |
theta = euler[1]; |
| 359 |
psi = euler[2]; |
| 360 |
|
| 361 |
myA[0][0] = (cos(phi) * cos(psi)) - (sin(phi) * cos(theta) * sin(psi)); |
| 362 |
myA[0][1] = (sin(phi) * cos(psi)) + (cos(phi) * cos(theta) * sin(psi)); |
| 363 |
myA[0][2] = sin(theta) * sin(psi); |
| 364 |
|
| 365 |
myA[1][0] = -(cos(phi) * sin(psi)) - (sin(phi) * cos(theta) * cos(psi)); |
| 366 |
myA[1][1] = -(sin(phi) * sin(psi)) + (cos(phi) * cos(theta) * cos(psi)); |
| 367 |
myA[1][2] = sin(theta) * cos(psi); |
| 368 |
|
| 369 |
myA[2][0] = sin(phi) * sin(theta); |
| 370 |
myA[2][1] = -cos(phi) * sin(theta); |
| 371 |
myA[2][2] = cos(theta); |
| 372 |
|
| 373 |
} |
| 374 |
|
| 375 |
void RigidBody::updateAtoms() { |
| 376 |
int i, j; |
| 377 |
vec3 ref; |
| 378 |
double apos[3]; |
| 379 |
|
| 380 |
for (i = 0; i < myAtoms.size(); i++) { |
| 381 |
|
| 382 |
ref = refCoords[i]; |
| 383 |
|
| 384 |
body2Lab(ref.vec); |
| 385 |
|
| 386 |
for (j = 0; j<3; j++) |
| 387 |
apos[j] = pos[j] + ref.vec[j]; |
| 388 |
|
| 389 |
myAtoms[i]->setPos(apos); |
| 390 |
|
| 391 |
} |
| 392 |
} |
| 393 |
|
| 394 |
/** |
| 395 |
* getEulerAngles computes a set of Euler angle values consistent |
| 396 |
* with an input rotation matrix. They are returned in the following |
| 397 |
* order: |
| 398 |
* myEuler[0] = phi; |
| 399 |
* myEuler[1] = theta; |
| 400 |
* myEuler[2] = psi; |
| 401 |
*/ |
| 402 |
void RigidBody::getEulerAngles(double myEuler[3]) { |
| 403 |
|
| 404 |
// We use so-called "x-convention", which is the most common |
| 405 |
// definition. In this convention, the rotation given by Euler |
| 406 |
// angles (phi, theta, psi), where the first rotation is by an angle |
| 407 |
// phi about the z-axis, the second is by an angle theta (0 <= theta |
| 408 |
// <= 180) about the x-axis, and the third is by an angle psi about |
| 409 |
// the z-axis (again). |
| 410 |
|
| 411 |
|
| 412 |
double phi,theta,psi,eps; |
| 413 |
double ctheta; |
| 414 |
double stheta; |
| 415 |
|
| 416 |
// set the tolerance for Euler angles and rotation elements |
| 417 |
|
| 418 |
eps = 1.0e-8; |
| 419 |
|
| 420 |
theta = acos(min(1.0,max(-1.0,A[2][2]))); |
| 421 |
ctheta = A[2][2]; |
| 422 |
stheta = sqrt(1.0 - ctheta * ctheta); |
| 423 |
|
| 424 |
// when sin(theta) is close to 0, we need to consider the |
| 425 |
// possibility of a singularity. In this case, we can assign an |
| 426 |
// arbitary value to phi (or psi), and then determine the psi (or |
| 427 |
// phi) or vice-versa. We'll assume that phi always gets the |
| 428 |
// rotation, and psi is 0 in cases of singularity. we use atan2 |
| 429 |
// instead of atan, since atan2 will give us -Pi to Pi. Since 0 <= |
| 430 |
// theta <= 180, sin(theta) will be always non-negative. Therefore, |
| 431 |
// it never changes the sign of both of the parameters passed to |
| 432 |
// atan2. |
| 433 |
|
| 434 |
if (fabs(stheta) <= eps){ |
| 435 |
psi = 0.0; |
| 436 |
phi = atan2(-A[1][0], A[0][0]); |
| 437 |
} |
| 438 |
// we only have one unique solution |
| 439 |
else{ |
| 440 |
phi = atan2(A[2][0], -A[2][1]); |
| 441 |
psi = atan2(A[0][2], A[1][2]); |
| 442 |
} |
| 443 |
|
| 444 |
//wrap phi and psi, make sure they are in the range from 0 to 2*Pi |
| 445 |
//if (phi < 0) |
| 446 |
// phi += M_PI; |
| 447 |
|
| 448 |
//if (psi < 0) |
| 449 |
// psi += M_PI; |
| 450 |
|
| 451 |
myEuler[0] = phi; |
| 452 |
myEuler[1] = theta; |
| 453 |
myEuler[2] = psi; |
| 454 |
|
| 455 |
return; |
| 456 |
} |
| 457 |
|
| 458 |
double RigidBody::max(double x, double y) { |
| 459 |
return (x > y) ? x : y; |
| 460 |
} |
| 461 |
|
| 462 |
double RigidBody::min(double x, double y) { |
| 463 |
return (x > y) ? y : x; |
| 464 |
} |
| 465 |
|
| 466 |
double RigidBody::findMaxExtent(){ |
| 467 |
int i; |
| 468 |
double refAtomPos[3]; |
| 469 |
double maxExtent; |
| 470 |
double tempExtent; |
| 471 |
|
| 472 |
//zero the extent variables |
| 473 |
maxExtent = 0.0; |
| 474 |
tempExtent = 0.0; |
| 475 |
for (i=0; i<3; i++) |
| 476 |
refAtomPos[i] = 0.0; |
| 477 |
|
| 478 |
//loop over all atoms |
| 479 |
for (i=0; i<myAtoms.size(); i++){ |
| 480 |
getAtomRefCoor(refAtomPos, i); |
| 481 |
tempExtent = sqrt(refAtomPos[0]*refAtomPos[0] + refAtomPos[1]*refAtomPos[1] |
| 482 |
+ refAtomPos[2]*refAtomPos[2]); |
| 483 |
if (tempExtent > maxExtent) |
| 484 |
maxExtent = tempExtent; |
| 485 |
} |
| 486 |
return maxExtent; |
| 487 |
} |
| 488 |
|
| 489 |
void RigidBody::findCOM() { |
| 490 |
|
| 491 |
size_t i; |
| 492 |
int j; |
| 493 |
double mtmp; |
| 494 |
double ptmp[3]; |
| 495 |
|
| 496 |
for(j = 0; j < 3; j++) { |
| 497 |
pos[j] = 0.0; |
| 498 |
} |
| 499 |
mass = 0.0; |
| 500 |
|
| 501 |
for (i = 0; i < myAtoms.size(); i++) { |
| 502 |
|
| 503 |
mtmp = myAtoms[i]->getMass(); |
| 504 |
myAtoms[i]->getPos(ptmp); |
| 505 |
|
| 506 |
mass += mtmp; |
| 507 |
|
| 508 |
for(j = 0; j < 3; j++) { |
| 509 |
pos[j] += ptmp[j]*mtmp; |
| 510 |
} |
| 511 |
|
| 512 |
} |
| 513 |
|
| 514 |
for(j = 0; j < 3; j++) { |
| 515 |
pos[j] /= mass; |
| 516 |
} |
| 517 |
|
| 518 |
} |
| 519 |
|
| 520 |
void RigidBody::getAtomPos(double theP[3], int index){ |
| 521 |
vec3 ref; |
| 522 |
|
| 523 |
if (index >= myAtoms.size()) |
| 524 |
printf( "%d is an invalid index, current rigid body contains " |
| 525 |
"%d atoms\n", index, myAtoms.size()); |
| 526 |
|
| 527 |
ref = refCoords[index]; |
| 528 |
body2Lab(ref.vec); |
| 529 |
|
| 530 |
theP[0] = pos[0] + ref[0]; |
| 531 |
theP[1] = pos[1] + ref[1]; |
| 532 |
theP[2] = pos[2] + ref[2]; |
| 533 |
} |
| 534 |
|
| 535 |
|
| 536 |
void RigidBody::getAtomRefCoor(double pos[3], int index){ |
| 537 |
vec3 ref; |
| 538 |
|
| 539 |
ref = refCoords[index]; |
| 540 |
pos[0] = ref[0]; |
| 541 |
pos[1] = ref[1]; |
| 542 |
pos[2] = ref[2]; |
| 543 |
|
| 544 |
} |
