9#define mat_pad(A) (A[W][X]=A[X][W]=A[W][Y]=A[Y][W]=A[W][Z]=A[Z][W]=0,A[W][W]=1)
12#define mat_copy(C,gets,A,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
13 C[i][j] gets (A[i][j]);}
16#define mat_tpose(AT,gets,A,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
17 AT[i][j] gets (A[j][i]);}
20#define mat_binop(C,gets,A,op,B,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
21 C[i][j] gets (A[i][j]) op (B[i][j]);}
24void mat_mult(HMatrix A, HMatrix B, HMatrix AB)
27 for (i=0; i<3; i++)
for (j=0; j<3; j++)
28 AB[i][j] = A[i][0]*B[0][j] + A[i][1]*B[1][j] + A[i][2]*B[2][j];
32float vdot(
float *va,
float *vb)
34 return (va[0]*vb[0] + va[1]*vb[1] + va[2]*vb[2]);
38void vcross(
float *va,
float *vb,
float *v)
40 v[0] = va[1]*vb[2] - va[2]*vb[1];
41 v[1] = va[2]*vb[0] - va[0]*vb[2];
42 v[2] = va[0]*vb[1] - va[1]*vb[0];
46void adjoint_transpose(HMatrix M, HMatrix MadjT)
48 vcross(M[1], M[2], MadjT[0]);
49 vcross(M[2], M[0], MadjT[1]);
50 vcross(M[0], M[1], MadjT[2]);
56Quat Qt_(
float x,
float y,
float z,
float w)
59 qq.x = x; qq.y = y; qq.z = z; qq.w = w;
67 qq.x = -q.x; qq.y = -q.y; qq.z = -q.z; qq.w = q.w;
77 qq.w = qL.w*qR.w - qL.x*qR.x - qL.y*qR.y - qL.z*qR.z;
78 qq.x = qL.w*qR.x + qL.x*qR.w + qL.y*qR.z - qL.z*qR.y;
79 qq.y = qL.w*qR.y + qL.y*qR.w + qL.z*qR.x - qL.x*qR.z;
80 qq.z = qL.w*qR.z + qL.z*qR.w + qL.x*qR.y - qL.y*qR.x;
88 qq.w = q.w*w; qq.x = q.x*w; qq.y = q.y*w; qq.z = q.z*w;
96Quat Qt_FromMatrix(HMatrix mat)
104 register double tr, s;
106 tr = mat[X][X] + mat[Y][Y]+ mat[Z][Z];
108 s = sqrt(tr + mat[W][W]);
111 qu.x = (mat[Z][Y] - mat[Y][Z]) * s;
112 qu.y = (mat[X][Z] - mat[Z][X]) * s;
113 qu.z = (mat[Y][X] - mat[X][Y]) * s;
116 if (mat[Y][Y] > mat[X][X]) h = Y;
117 if (mat[Z][Z] > mat[h][h]) h = Z;
119#define caseMacro(i,j,k,I,J,K) \
121 s = sqrt( (mat[I][I] - (mat[J][J]+mat[K][K])) + mat[W][W] );\
124 qu.j = (mat[I][J] + mat[J][I]) * s;\
125 qu.k = (mat[K][I] + mat[I][K]) * s;\
126 qu.w = (mat[K][J] - mat[J][K]) * s;\
128 caseMacro(x,y,z,X,Y,Z);
129 caseMacro(y,z,x,Y,Z,X);
130 caseMacro(z,x,y,Z,X,Y);
133 if (mat[W][W] != 1.0) qu = Qt_Scale(qu, 1/sqrt(mat[W][W]));
138static HMatrix mat_id = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}};
141float mat_norm(HMatrix M,
int tpose)
146 for (i=0; i<3; i++) {
147 if (tpose) sum = fabs(M[0][i])+fabs(M[1][i])+fabs(M[2][i]);
148 else sum = fabs(M[i][0])+fabs(M[i][1])+fabs(M[i][2]);
149 if (max<sum) max = sum;
154float norm_inf(HMatrix M) {
return mat_norm(M, 0);}
155float norm_one(HMatrix M) {
return mat_norm(M, 1);}
158int find_max_col(HMatrix M)
163 for (i=0; i<3; i++)
for (j=0; j<3; j++) {
164 abs = M[i][j];
if (abs<0.0) abs = -abs;
165 if (abs>max) {max = abs; col = j;}
171void make_reflector(
float *v,
float *u)
173 float s = sqrt(vdot(v, v));
174 u[0] = v[0]; u[1] = v[1];
175 u[2] = v[2] + ((v[2]<0.0) ? -s : s);
176 s = sqrt(2.0/vdot(u, u));
177 u[0] = u[0]*s; u[1] = u[1]*s; u[2] = u[2]*s;
181void reflect_cols(HMatrix M,
float *u)
184 for (i=0; i<3; i++) {
185 float s = u[0]*M[0][i] + u[1]*M[1][i] + u[2]*M[2][i];
186 for (j=0; j<3; j++) M[j][i] -= u[j]*s;
190void reflect_rows(HMatrix M,
float *u)
193 for (i=0; i<3; i++) {
194 float s = vdot(u, M[i]);
195 for (j=0; j<3; j++) M[i][j] -= u[j]*s;
200void do_rank1(HMatrix M, HMatrix Q)
202 float v1[3], v2[3], s;
204 mat_copy(Q,=,mat_id,4);
206 col = find_max_col(M);
208 v1[0] = M[0][col]; v1[1] = M[1][col]; v1[2] = M[2][col];
209 make_reflector(v1, v1); reflect_cols(M, v1);
210 v2[0] = M[2][0]; v2[1] = M[2][1]; v2[2] = M[2][2];
211 make_reflector(v2, v2); reflect_rows(M, v2);
213 if (s<0.0) Q[2][2] = -1.0;
214 reflect_cols(Q, v1); reflect_rows(Q, v2);
218void do_rank2(HMatrix M, HMatrix MadjT, HMatrix Q)
221 float w, x, y, z, c, s, d;
224 col = find_max_col(MadjT);
225 if (col<0) {do_rank1(M, Q);
return;}
226 v1[0] = MadjT[0][col]; v1[1] = MadjT[1][col]; v1[2] = MadjT[2][col];
227 make_reflector(v1, v1); reflect_cols(M, v1);
228 vcross(M[0], M[1], v2);
229 make_reflector(v2, v2); reflect_rows(M, v2);
230 w = M[0][0]; x = M[0][1]; y = M[1][0]; z = M[1][1];
232 c = z+w; s = y-x; d = sqrt(c*c+s*s); c = c/d; s = s/d;
233 Q[0][0] = Q[1][1] = c; Q[0][1] = -(Q[1][0] = s);
235 c = z-w; s = y+x; d = sqrt(c*c+s*s); c = c/d; s = s/d;
236 Q[0][0] = -(Q[1][1] = c); Q[0][1] = Q[1][0] = s;
238 Q[0][2] = Q[2][0] = Q[1][2] = Q[2][1] = 0.0; Q[2][2] = 1.0;
239 reflect_cols(Q, v1); reflect_rows(Q, v2);
251float polar_decomp(HMatrix M, HMatrix Q, HMatrix S)
254 HMatrix Mk, MadjTk, Ek;
255 float det, M_one, M_inf, MadjT_one, MadjT_inf, E_one, gamma, g1, g2;
258 M_one = norm_one(Mk); M_inf = norm_inf(Mk);
260 adjoint_transpose(Mk, MadjTk);
261 det = vdot(Mk[0], MadjTk[0]);
262 if (det==0.0) {do_rank2(Mk, MadjTk, Mk);
break;}
263 MadjT_one = norm_one(MadjTk); MadjT_inf = norm_inf(MadjTk);
264 gamma = sqrt(sqrt((MadjT_one*MadjT_inf)/(M_one*M_inf))/fabs(det));
266 g2 = 0.5/(gamma*det);
268 mat_binop(Mk,=,g1*Mk,+,g2*MadjTk,3);
269 mat_copy(Ek,-=,Mk,3);
270 E_one = norm_one(Ek);
271 M_one = norm_one(Mk); M_inf = norm_inf(Mk);
272 }
while (E_one>(M_one*TOL));
273 mat_tpose(Q,=,Mk,3); mat_pad(Q);
274 mat_mult(Mk, M, S); mat_pad(S);
275 for (i=0; i<3; i++)
for (j=i; j<3; j++)
276 S[i][j] = S[j][i] = 0.5*(S[i][j]+S[j][i]);
303HVect spect_decomp(HMatrix S, HMatrix U)
306 double Diag[3],OffD[3];
307 double g,h,fabsh,fabsOffDi,t,theta,c,s,tau,ta,OffDq,a,b;
308 static char nxt[] = {Y,Z,X};
310 mat_copy(U,=,mat_id,4);
311 Diag[X] = S[X][X]; Diag[Y] = S[Y][Y]; Diag[Z] = S[Z][Z];
312 OffD[X] = S[Y][Z]; OffD[Y] = S[Z][X]; OffD[Z] = S[X][Y];
313 for (sweep=20; sweep>0; sweep--) {
314 float sm = fabs(OffD[X])+fabs(OffD[Y])+fabs(OffD[Z]);
316 for (i=Z; i>=X; i--) {
317 int p = nxt[i];
int q = nxt[p];
318 fabsOffDi = fabs(OffD[i]);
321 h = Diag[q] - Diag[p];
323 if (fabsh+g==fabsh) {
326 theta = 0.5*h/OffD[i];
327 t = 1.0/(fabs(theta)+sqrt(theta*theta+1.0));
328 if (theta<0.0) t = -t;
330 c = 1.0/sqrt(t*t+1.0); s = t*c;
332 ta = t*OffD[i]; OffD[i] = 0.0;
333 Diag[p] -= ta; Diag[q] += ta;
335 OffD[q] -= s*(OffD[p] + tau*OffD[q]);
336 OffD[p] += s*(OffDq - tau*OffD[p]);
337 for (j=Z; j>=X; j--) {
338 a = U[j][p]; b = U[j][q];
339 U[j][p] -= s*(b + tau*a);
340 U[j][q] += s*(a - tau*b);
345 kv.x = Diag[X]; kv.y = Diag[Y]; kv.z = Diag[Z]; kv.w = 1.0;
360#define SQRTHALF (0.7071067811865475244)
361#define sgn(n,v) ((n)?-(v):(v))
362#define swap(a,i,j) {a[3]=a[i]; a[i]=a[j]; a[j]=a[3];}
363#define cycle(a,p) if (p) {a[3]=a[0]; a[0]=a[1]; a[1]=a[2]; a[2]=a[3];}\
364 else {a[3]=a[2]; a[2]=a[1]; a[1]=a[0]; a[0]=a[3];}
368 ka[X] = k->x; ka[Y] = k->y; ka[Z] = k->z;
369 if (ka[X]==ka[Y]) {
if (ka[X]==ka[Z]) turn = W;
else turn = Z;}
370 else {
if (ka[X]==ka[Z]) turn = Y;
else if (ka[Y]==ka[Z]) turn = X;}
373 unsigned neg[3], win;
375 static Quat qxtoz = {0,SQRTHALF,0,SQRTHALF};
376 static Quat qytoz = {SQRTHALF,0,0,SQRTHALF};
377 static Quat qppmm = { 0.5, 0.5,-0.5,-0.5};
378 static Quat qpppp = { 0.5, 0.5, 0.5, 0.5};
379 static Quat qmpmm = {-0.5, 0.5,-0.5,-0.5};
380 static Quat qpppm = { 0.5, 0.5, 0.5,-0.5};
381 static Quat q0001 = { 0.0, 0.0, 0.0, 1.0};
382 static Quat q1000 = { 1.0, 0.0, 0.0, 0.0};
384 default:
return (Qt_Conj(q));
385 case X: q = Qt_Mul(q, qtoz = qxtoz); swap(ka,X,Z)
break;
386 case Y: q = Qt_Mul(q, qtoz = qytoz); swap(ka,Y,Z)
break;
387 case Z: qtoz = q0001;
break;
390 mag[0] = (double)q.z*q.z+(
double)q.w*q.w-0.5;
391 mag[1] = (double)q.x*q.z-(
double)q.y*q.w;
392 mag[2] = (double)q.y*q.z+(
double)q.x*q.w;
393 for (i=0; i<3; i++)
if (neg[i] = (mag[i]<0.0)) mag[i] = -mag[i];
394 if (mag[0]>mag[1]) {
if (mag[0]>mag[2]) win = 0;
else win = 2;}
395 else {
if (mag[1]>mag[2]) win = 1;
else win = 2;}
397 case 0:
if (neg[0]) p = q1000;
else p = q0001;
break;
398 case 1:
if (neg[1]) p = qppmm;
else p = qpppp; cycle(ka,0) break;
399 case 2: if (neg[2]) p = qmpmm; else p = qpppm; cycle(ka,1) break;
402 t = sqrt(mag[win]+0.5);
403 p = Qt_Mul(p, Qt_(0.0,0.0,-qp.z/t,qp.w/t));
404 p = Qt_Mul(qtoz, Qt_Conj(p));
407 unsigned lo, hi, neg[4], par = 0;
408 double all, big, two;
409 qa[0] = q.x; qa[1] = q.y; qa[2] = q.z; qa[3] = q.w;
410 for (i=0; i<4; i++) {
412 if (neg[i] = (qa[i]<0.0)) qa[i] = -qa[i];
416 if (qa[0]>qa[1]) lo = 0;
else lo = 1;
417 if (qa[2]>qa[3]) hi = 2;
else hi = 3;
419 if (qa[lo^1]>qa[hi]) {hi = lo; lo ^= 1;}
420 else {hi ^= lo; lo ^= hi; hi ^= lo;}
421 }
else {
if (qa[hi^1]>qa[lo]) lo = hi^1;}
422 all = (qa[0]+qa[1]+qa[2]+qa[3])*0.5;
423 two = (qa[hi]+qa[lo])*SQRTHALF;
427 {
int i;
for (i=0; i<4; i++) pa[i] = sgn(neg[i], 0.5);}
429 }
else { pa[hi] = sgn(neg[hi],1.0);}
432 pa[hi] = sgn(neg[hi],SQRTHALF); pa[lo] = sgn(neg[lo], SQRTHALF);
433 if (lo>hi) {hi ^= lo; lo ^= hi; hi ^= lo;}
434 if (hi==W) {hi =
"\001\002\000"[lo]; lo = 3-hi-lo;}
436 }
else { pa[hi] = sgn(neg[hi],1.0);}
438 p.x = -pa[0]; p.y = -pa[1]; p.z = -pa[2]; p.w = pa[3];
440 k->x = ka[X]; k->y = ka[Y]; k->z = ka[Z];
468 parts->t = Qt_(A[X][W], A[Y][W], A[Z][W], 0);
469 det = polar_decomp(A, Q, S);
474 parts->q = Qt_FromMatrix(Q);
475 parts->k = spect_decomp(S, U);
476 parts->u = Qt_FromMatrix(U);
477 p = snuggle(parts->u, &parts->k);
478 parts->u = Qt_Mul(parts->u, p);
488 inverse->f = parts->f;
489 inverse->q = Qt_Conj(parts->q);
490 inverse->u = Qt_Mul(parts->q, parts->u);
491 inverse->k.x = (parts->k.x==0.0) ? 0.0 : 1.0/parts->k.x;
492 inverse->k.y = (parts->k.y==0.0) ? 0.0 : 1.0/parts->k.y;
493 inverse->k.z = (parts->k.z==0.0) ? 0.0 : 1.0/parts->k.z;
494 inverse->k.w = parts->k.w;
495 t = Qt_(-parts->t.x, -parts->t.y, -parts->t.z, 0);
496 t = Qt_Mul(Qt_Conj(inverse->u), Qt_Mul(t, inverse->u));
497 t = Qt_(inverse->k.x*t.x, inverse->k.y*t.y, inverse->k.z*t.z, 0);
498 p = Qt_Mul(inverse->q, inverse->u);
499 t = Qt_Mul(p, Qt_Mul(t, Qt_Conj(p)));
500 inverse->t = (inverse->f>0.0) ? t : Qt_(-t.x, -t.y, -t.z, 0);