/**** Decompose.c ****/
/* Ken Shoemake, 1993 */
#include <math.h>
#include "Decompose.h"
namespace GraphicGems
{
/******* Matrix Preliminaries *******/
/** Fill out 3x3 matrix to 4x4 **/
#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)
/** Copy nxn matrix A to C using "gets" for assignment **/
#define mat_copy(C,gets,A,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
C[i][j] gets (A[i][j]);}
/** Copy transpose of nxn matrix A to C using "gets" for assignment **/
#define mat_tpose(AT,gets,A,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
AT[i][j] gets (A[j][i]);}
/** Assign nxn matrix C the element-wise combination of A and B using "op" **/
#define mat_binop(C,gets,A,op,B,n) {int i,j; for(i=0;i<n;i++) for(j=0;j<n;j++)\
C[i][j] gets (A[i][j]) op (B[i][j]);}
/** Multiply the upper left 3x3 parts of A and B to get AB **/
void mat_mult(HMatrix A, HMatrix B, HMatrix AB)
{
int i, j;
for (i=0; i<3; i++) for (j=0; j<3; j++)
AB[i][j] = A[i][0]*B[0][j] + A[i][1]*B[1][j] + A[i][2]*B[2][j];
}
/** Return dot product of length 3 vectors va and vb **/
double vdot(double *va, double *vb)
{
return (va[0]*vb[0] + va[1]*vb[1] + va[2]*vb[2]);
}
/** Set v to cross product of length 3 vectors va and vb **/
void vcross(double *va, double *vb, double *v)
{
v[0] = va[1]*vb[2] - va[2]*vb[1];
v[1] = va[2]*vb[0] - va[0]*vb[2];
v[2] = va[0]*vb[1] - va[1]*vb[0];
}
/** Set MadjT to transpose of inverse of M times determinant of M **/
void adjoint_transpose(HMatrix M, HMatrix MadjT)
{
vcross(M[1], M[2], MadjT[0]);
vcross(M[2], M[0], MadjT[1]);
vcross(M[0], M[1], MadjT[2]);
}
/******* Quaternion Preliminaries *******/
/* Construct a (possibly non-unit) quaternion from real components. */
Quat Qt_(double x, double y, double z, double w)
{
Quat qq;
qq.x = x;
qq.y = y;
qq.z = z;
qq.w = w;
return (qq);
}
/* Return conjugate of quaternion. */
Quat Qt_Conj(Quat q)
{
Quat qq;
qq.x = -q.x;
qq.y = -q.y;
qq.z = -q.z;
qq.w = q.w;
return (qq);
}
/* Return quaternion product qL * qR. Note: order is important!
* To combine rotations, use the product Mul(qSecond, qFirst),
* which gives the effect of rotating by qFirst then qSecond. */
Quat Qt_Mul(Quat qL, Quat qR)
{
Quat qq;
qq.w = qL.w*qR.w - qL.x*qR.x - qL.y*qR.y - qL.z*qR.z;
qq.x = qL.w*qR.x + qL.x*qR.w + qL.y*qR.z - qL.z*qR.y;
qq.y = qL.w*qR.y + qL.y*qR.w + qL.z*qR.x - qL.x*qR.z;
qq.z = qL.w*qR.z + qL.z*qR.w + qL.x*qR.y - qL.y*qR.x;
return (qq);
}
/* Return product of quaternion q by scalar w. */
Quat Qt_Scale(Quat q, double w)
{
Quat qq;
qq.w = q.w*w;
qq.x = q.x*w;
qq.y = q.y*w;
qq.z = q.z*w;
return (qq);
}
/* Construct a unit quaternion from rotation matrix. Assumes matrix is
* used to multiply column vector on the left: vnew = mat vold. Works
* correctly for right-handed coordinate system and right-handed rotations.
* Translation and perspective components ignored. */
Quat Qt_FromMatrix(HMatrix mat)
{
/* This algorithm avoids near-zero divides by looking for a large component
* - first w, then x, y, or z. When the trace is greater than zero,
* |w| is greater than 1/2, which is as small as a largest component can be.
* Otherwise, the largest diagonal entry corresponds to the largest of |x|,
* |y|, or |z|, one of which must be larger than |w|, and at least 1/2. */
Quat qu;
register double tr, s;
tr = mat[X][X] + mat[Y][Y]+ mat[Z][Z];
if (tr >= 0.0) {
s = sqrt(tr + mat[W][W]);
qu.w = s*0.5;
s = 0.5 / s;
qu.x = (mat[Z][Y] - mat[Y][Z]) * s;
qu.y = (mat[X][Z] - mat[Z][X]) * s;
qu.z = (mat[Y][X] - mat[X][Y]) * s;
} else {
int h = X;
if (mat[Y][Y] > mat[X][X]) h = Y;
if (mat[Z][Z] > mat[h][h]) h = Z;
switch (h) {
#define caseMacro(i,j,k,I,J,K) \
case I:\
s = sqrt( (mat[I][I] - (mat[J][J]+mat[K][K])) + mat[W][W] );\
qu.i = s*0.5;\
s = 0.5 / s;\
qu.j = (mat[I][J] + mat[J][I]) * s;\
qu.k = (mat[K][I] + mat[I][K]) * s;\
qu.w = (mat[K][J] - mat[J][K]) * s;\
break
caseMacro(x,y,z,X,Y,Z);
caseMacro(y,z,x,Y,Z,X);
caseMacro(z,x,y,Z,X,Y);
}
}
if (mat[W][W] != 1.0) qu = Qt_Scale(qu, 1/sqrt(mat[W][W]));
return (qu);
}
/******* Decomp Auxiliaries *******/
static HMatrix mat_id = {
{1,0,0,0},
{0,1,0,0},
{0,0,1,0},
{0,0,0,1}
};
/** Compute either the 1 or infinity norm of M, depending on tpose **/
double mat_norm(HMatrix M, int tpose)
{
int i;
double sum, max;
max = 0.0;
for (i=0; i<3; i++)
{
if (tpose)
sum = fabs(M[0][i])+fabs(M[1][i])+fabs(M[2][i]);
else
sum = fabs(M[i][0])+fabs(M[i][1])+fabs(M[i][2]);
if (max<sum)
max = sum;
}
return max;
}
double norm_inf(HMatrix M) {
return mat_norm(M, 0);
}
double norm_one(HMatrix M) {
return mat_norm(M, 1);
}
/** Return index of column of M containing maximum abs entry, or -1 if M=0 **/
int find_max_col(HMatrix M)
{
double abs, max;
int i, j, col;
max = 0.0; col = -1;
for (i=0; i<3; i++) for (j=0; j<3; j++) {
abs = M[i][j]; if (abs<0.0) abs = -abs;
if (abs>max) {max = abs; col = j;}
}
return col;
}
/** Setup u for Household reflection to zero all v components but first **/
void make_reflector(double *v, double *u)
{
double s = sqrt(vdot(v, v));
u[0] = v[0]; u[1] = v[1];
u[2] = v[2] + ((v[2]<0.0) ? -s : s);
s = sqrt(2.0/vdot(u, u));
u[0] = u[0]*s; u[1] = u[1]*s; u[2] = u[2]*s;
}
/** Apply Householder reflection represented by u to column vectors of M **/
void reflect_cols(HMatrix M, double *u)
{
int i, j;
for (i=0; i<3; i++) {
double s = u[0]*M[0][i] + u[1]*M[1][i] + u[2]*M[2][i];
for (j=0; j<3; j++) M[j][i] -= u[j]*s;
}
}
/** Apply Householder reflection represented by u to row vectors of M **/
void reflect_rows(HMatrix M, double *u)
{
int i, j;
for (i=0; i<3; i++) {
double s = vdot(u, M[i]);
for (j=0; j<3; j++) M[i][j] -= u[j]*s;
}
}
/** Find orthogonal factor Q of rank 1 (or less) M **/
void do_rank1(HMatrix M, HMatrix Q)
{
double v1[3], v2[3], s;
int col;
mat_copy(Q,=,mat_id,4);
/* If rank(M) is 1, we should find a non-zero column in M */
col = find_max_col(M);
if (col<0) return; /* Rank is 0 */
v1[0] = M[0][col]; v1[1] = M[1][col]; v1[2] = M[2][col];
make_reflector(v1, v1); reflect_cols(M, v1);
v2[0] = M[2][0]; v2[1] = M[2][1]; v2[2] = M[2][2];
make_reflector(v2, v2); reflect_rows(M, v2);
s = M[2][2];
if (s<0.0) Q[2][2] = -1.0;
reflect_cols(Q, v1); reflect_rows(Q, v2);
}
/** Find orthogonal factor Q of rank 2 (or less) M using adjoint transpose **/
void do_rank2(HMatrix M, HMatrix MadjT, HMatrix Q)
{
double v1[3], v2[3];
double w, x, y, z, c, s, d;
int col;
/* If rank(M) is 2, we should find a non-zero column in MadjT */
col = find_max_col(MadjT);
if (col<0) {do_rank1(M, Q); return;} /* Rank<2 */
v1[0] = MadjT[0][col]; v1[1] = MadjT[1][col]; v1[2] = MadjT[2][col];
make_reflector(v1, v1); reflect_cols(M, v1);
vcross(M[0], M[1], v2);
make_reflector(v2, v2); reflect_rows(M, v2);
w = M[0][0]; x = M[0][1]; y = M[1][0]; z = M[1][1];
if (w*z>x*y) {
c = z+w; s = y-x; d = sqrt(c*c+s*s); c = c/d; s = s/d;
Q[0][0] = Q[1][1] = c; Q[0][1] = -(Q[1][0] = s);
} else {
c = z-w; s = y+x; d = sqrt(c*c+s*s); c = c/d; s = s/d;
Q[0][0] = -(Q[1][1] = c); Q[0][1] = Q[1][0] = s;
}
Q[0][2] = Q[2][0] = Q[1][2] = Q[2][1] = 0.0; Q[2][2] = 1.0;
reflect_cols(Q, v1); reflect_rows(Q, v2);
}
/******* Polar Decomposition *******/
/* Polar Decomposition of 3x3 matrix in 4x4,
* M = QS. See Nicholas Higham and Robert S. Schreiber,
* Fast Polar Decomposition of An Arbitrary Matrix,
* Technical Report 88-942, October 1988,
* Department of Computer Science, Cornell University.
*/
double polar_decomp(HMatrix M, HMatrix Q, HMatrix S)
{
#define TOL 1.0e-6
HMatrix Mk, MadjTk, Ek;
double det, M_one, M_inf, MadjT_one, MadjT_inf, E_one, gamma, g1, g2;
int i, j;
mat_tpose(Mk,=,M,3);
M_one = norm_one(Mk); M_inf = norm_inf(Mk);
do {
adjoint_transpose(Mk, MadjTk);
det = vdot(Mk[0], MadjTk[0]);
if (det==0.0) {do_rank2(Mk, MadjTk, Mk); break;}
MadjT_one = norm_one(MadjTk); MadjT_inf = norm_inf(MadjTk);
gamma = sqrt(sqrt((MadjT_one*MadjT_inf)/(M_one*M_inf))/fabs(det));
g1 = gamma*0.5;
g2 = 0.5/(gamma*det);
mat_copy(Ek,=,Mk,3);
mat_binop(Mk,=,g1*Mk,+,g2*MadjTk,3);
mat_copy(Ek,-=,Mk,3);
E_one = norm_one(Ek);
M_one = norm_one(Mk); M_inf = norm_inf(Mk);
} while (E_one>(M_one*TOL));
mat_tpose(Q,=,Mk,3); mat_pad(Q);
mat_mult(Mk, M, S); mat_pad(S);
for (i=0; i<3; i++) for (j=i; j<3; j++)
S[i][j] = S[j][i] = 0.5*(S[i][j]+S[j][i]);
return (det);
}
/******* Spectral Decomposition *******/
/* Compute the spectral decomposition of symmetric positive semi-definite S.
* Returns rotation in U and scale factors in result, so that if K is a diagonal
* matrix of the scale factors, then S = U K (U transpose). Uses Jacobi method.
* See Gene H. Golub and Charles F. Van Loan. Matrix Computations. Hopkins 1983.
*/
HVect spect_decomp(HMatrix S, HMatrix U)
{
HVect kv;
double Diag[3],OffD[3]; /* OffD is off-diag (by omitted index) */
double g,h,fabsh,fabsOffDi,t,theta,c,s,tau,ta,OffDq,a,b;
static char nxt[] = {Y,Z,X};
int sweep, i, j;
mat_copy(U,=,mat_id,4);
Diag[X] = S[X][X]; Diag[Y] = S[Y][Y]; Diag[Z] = S[Z][Z];
OffD[X] = S[Y][Z]; OffD[Y] = S[Z][X]; OffD[Z] = S[X][Y];
for (sweep=20; sweep>0; sweep--) {
double sm = fabs(OffD[X])+fabs(OffD[Y])+fabs(OffD[Z]);
if (sm==0.0) break;
for (i=Z; i>=X; i--) {
int p = nxt[i]; int q = nxt[p];
fabsOffDi = fabs(OffD[i]);
g = 100.0*fabsOffDi;
if (fabsOffDi>0.0) {
h = Diag[q] - Diag[p];
fabsh = fabs(h);
if (fabsh+g==fabsh) {
t = OffD[i]/h;
} else {
theta = 0.5*h/OffD[i];
t = 1.0/(fabs(theta)+sqrt(theta*theta+1.0));
if (theta<0.0) t = -t;
}
c = 1.0/sqrt(t*t+1.0); s = t*c;
tau = s/(c+1.0);
ta = t*OffD[i]; OffD[i] = 0.0;
Diag[p] -= ta; Diag[q] += ta;
OffDq = OffD[q];
OffD[q] -= s*(OffD[p] + tau*OffD[q]);
OffD[p] += s*(OffDq - tau*OffD[p]);
for (j=Z; j>=X; j--) {
a = U[j][p]; b = U[j][q];
U[j][p] -= s*(b + tau*a);
U[j][q] += s*(a - tau*b);
}
}
}
}
kv.x = Diag[X]; kv.y = Diag[Y]; kv.z = Diag[Z]; kv.w = 1.0;
return (kv);
}
/******* Spectral Axis Adjustment *******/
/* Given a unit quaternion, q, and a scale vector, k, find a unit quaternion, p,
* which permutes the axes and turns freely in the plane of duplicate scale
* factors, such that q p has the largest possible w component, i.e. the
* smallest possible angle. Permutes k's components to go with q p instead of q.
* See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
* Proceedings of Graphics Interface 1992. Details on p. 262-263.
*/
Quat snuggle(Quat q, HVect *k)
{
#define SQRTHALF (0.7071067811865475244)
#define sgn(n,v) ((n)?-(v):(v))
#define swap(a,i,j) {a[3]=a[i]; a[i]=a[j]; a[j]=a[3];}
#define cycle(a,p) if (p) {a[3]=a[0]; a[0]=a[1]; a[1]=a[2]; a[2]=a[3];}\
else {a[3]=a[2]; a[2]=a[1]; a[1]=a[0]; a[0]=a[3];}
Quat p;
double ka[4];
int i, turn = -1;
ka[X] = k->x; ka[Y] = k->y; ka[Z] = k->z;
if (ka[X]==ka[Y]) {if (ka[X]==ka[Z]) turn = W; else turn = Z;}
else {if (ka[X]==ka[Z]) turn = Y; else if (ka[Y]==ka[Z]) turn = X;}
if (turn>=0) {
Quat qtoz, qp;
unsigned neg[3], win;
double mag[3], t;
static Quat qxtoz = {0,SQRTHALF,0,SQRTHALF};
static Quat qytoz = {SQRTHALF,0,0,SQRTHALF};
static Quat qppmm = { 0.5, 0.5,-0.5,-0.5};
static Quat qpppp = { 0.5, 0.5, 0.5, 0.5};
static Quat qmpmm = {-0.5, 0.5,-0.5,-0.5};
static Quat qpppm = { 0.5, 0.5, 0.5,-0.5};
static Quat q0001 = { 0.0, 0.0, 0.0, 1.0};
static Quat q1000 = { 1.0, 0.0, 0.0, 0.0};
switch (turn) {
default: return (Qt_Conj(q));
case X: q = Qt_Mul(q, qtoz = qxtoz); swap(ka,X,Z) break;
case Y: q = Qt_Mul(q, qtoz = qytoz); swap(ka,Y,Z) break;
case Z: qtoz = q0001; break;
}
q = Qt_Conj(q);
mag[0] = (double)q.z*q.z+(double)q.w*q.w-0.5;
mag[1] = (double)q.x*q.z-(double)q.y*q.w;
mag[2] = (double)q.y*q.z+(double)q.x*q.w;
for (i=0; i<3; i++) if (neg[i] = (mag[i]<0.0)) mag[i] = -mag[i];
if (mag[0]>mag[1]) {if (mag[0]>mag[2]) win = 0; else win = 2;}
else {if (mag[1]>mag[2]) win = 1; else win = 2;}
switch (win) {
case 0: if (neg[0]) p = q1000; else p = q0001; break;
case 1: if (neg[1]) p = qppmm; else p = qpppp; cycle(ka,0) break;
case 2: if (neg[2]) p = qmpmm; else p = qpppm; cycle(ka,1) break;
}
qp = Qt_Mul(q, p);
t = sqrt(mag[win]+0.5);
p = Qt_Mul(p, Qt_(0.0,0.0,-qp.z/t,qp.w/t));
p = Qt_Mul(qtoz, Qt_Conj(p));
} else {
double qa[4], pa[4];
unsigned lo, hi, neg[4], par = 0;
double all, big, two;
qa[0] = q.x; qa[1] = q.y; qa[2] = q.z; qa[3] = q.w;
for (i=0; i<4; i++) {
pa[i] = 0.0;
if (neg[i] = (qa[i]<0.0)) qa[i] = -qa[i];
par ^= neg[i];
}
/* Find two largest components, indices in hi and lo */
if (qa[0]>qa[1]) lo = 0; else lo = 1;
if (qa[2]>qa[3]) hi = 2; else hi = 3;
if (qa[lo]>qa[hi]) {
if (qa[lo^1]>qa[hi]) {hi = lo; lo ^= 1;}
else {hi ^= lo; lo ^= hi; hi ^= lo;}
} else {if (qa[hi^1]>qa[lo]) lo = hi^1;}
all = (qa[0]+qa[1]+qa[2]+qa[3])*0.5;
two = (qa[hi]+qa[lo])*SQRTHALF;
big = qa[hi];
if (all>two) {
if (all>big) {/*all*/
{int i; for (i=0; i<4; i++) pa[i] = sgn(neg[i], 0.5);}
cycle(ka,par)
} else {/*big*/ pa[hi] = sgn(neg[hi],1.0);}
} else {
if (two>big) {/*two*/
pa[hi] = sgn(neg[hi],SQRTHALF); pa[lo] = sgn(neg[lo], SQRTHALF);
if (lo>hi) {hi ^= lo; lo ^= hi; hi ^= lo;}
if (hi==W) {hi = "\001\002\000"[lo]; lo = 3-hi-lo;}
swap(ka,hi,lo)
} else {/*big*/ pa[hi] = sgn(neg[hi],1.0);}
}
p.x = -pa[0]; p.y = -pa[1]; p.z = -pa[2]; p.w = pa[3];
}
k->x = ka[X]; k->y = ka[Y]; k->z = ka[Z];
return (p);
}
/******* Decompose Affine Matrix *******/
//-----------------------------------------------------------------------------
// Torque Game Engine
// Copyright (C) GarageGames.com, Inc.
//-----------------------------------------------------------------------------
/* Decompose 4x4 affine matrix A as TFRUK(U transpose), where t contains the
* translation components, q contains the rotation R, u contains U, k contains
* scale factors, and f contains the sign of the determinant.
* Assumes A transforms column vectors in right-handed coordinates.
* See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
* Proceedings of Graphics Interface 1992.
*/
void decomp_affine(HMatrix A, AffineParts *parts)
{
HMatrix Q, S, U;
Quat p;
double det;
parts->t = Qt_(A[X][W], A[Y][W], A[Z][W], 0);
det = polar_decomp(A, Q, S);
if (det<0.0) {
mat_copy(Q,=,-Q,3);
parts->f = -1;
} else parts->f = 1;
parts->q = Qt_FromMatrix(Q);
parts->k = spect_decomp(S, U);
parts->u = Qt_FromMatrix(U);
p = snuggle(parts->u, &parts->k);
parts->u = Qt_Mul(parts->u, p);
}
/******* Invert Affine Decomposition *******/
/* Compute inverse of affine decomposition.
*/
void invert_affine(AffineParts *parts, AffineParts *inverse)
{
Quat t, p;
inverse->f = parts->f;
inverse->q = Qt_Conj(parts->q);
inverse->u = Qt_Mul(parts->q, parts->u);
inverse->k.x = (parts->k.x==0.0) ? 0.0 : 1.0/parts->k.x;
inverse->k.y = (parts->k.y==0.0) ? 0.0 : 1.0/parts->k.y;
inverse->k.z = (parts->k.z==0.0) ? 0.0 : 1.0/parts->k.z;
inverse->k.w = parts->k.w;
t = Qt_(-parts->t.x, -parts->t.y, -parts->t.z, 0);
t = Qt_Mul(Qt_Conj(inverse->u), Qt_Mul(t, inverse->u));
t = Qt_(inverse->k.x*t.x, inverse->k.y*t.y, inverse->k.z*t.z, 0);
p = Qt_Mul(inverse->q, inverse->u);
t = Qt_Mul(p, Qt_Mul(t, Qt_Conj(p)));
inverse->t = (inverse->f>0.0) ? t : Qt_(-t.x, -t.y, -t.z, 0);
}
}; // namespace GraphicGems