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CWE-ou-minkata/Sources/Plasma/PubUtilLib/plTransform/mat_decomp.cpp

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/*==LICENSE==*
CyanWorlds.com Engine - MMOG client, server and tools
Copyright (C) 2011 Cyan Worlds, Inc.
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
Additional permissions under GNU GPL version 3 section 7
If you modify this Program, or any covered work, by linking or
combining it with any of RAD Game Tools Bink SDK, Autodesk 3ds Max SDK,
NVIDIA PhysX SDK, Microsoft DirectX SDK, OpenSSL library, Independent
JPEG Group JPEG library, Microsoft Windows Media SDK, or Apple QuickTime SDK
(or a modified version of those libraries),
containing parts covered by the terms of the Bink SDK EULA, 3ds Max EULA,
PhysX SDK EULA, DirectX SDK EULA, OpenSSL and SSLeay licenses, IJG
JPEG Library README, Windows Media SDK EULA, or QuickTime SDK EULA, the
licensors of this Program grant you additional
permission to convey the resulting work. Corresponding Source for a
non-source form of such a combination shall include the source code for
the parts of OpenSSL and IJG JPEG Library used as well as that of the covered
work.
You can contact Cyan Worlds, Inc. by email legal@cyan.com
or by snail mail at:
Cyan Worlds, Inc.
14617 N Newport Hwy
Mead, WA 99021
*==LICENSE==*/
/**** Decompose.c ****/
/* Ken Shoemake, 1993 */
//
// Gems IV. Polar Decomp
//
#include <cmath>
#include "mat_decomp.h"
/******* 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(const HMatrix A, const 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 **/
float vdot(float *va, float *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(float *va, float *vb, float *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. */
gemQuat Qt_(float x, float y, float z, float w)
{
gemQuat qq;
qq.x = x; qq.y = y; qq.z = z; qq.w = w;
return (qq);
}
/* Return conjugate of quaternion. */
gemQuat Qt_Conj(gemQuat q)
{
gemQuat 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. */
gemQuat Qt_Mul(gemQuat qL, gemQuat qR)
{
gemQuat 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. */
gemQuat Qt_Scale(gemQuat q, float w)
{
gemQuat 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. */
gemQuat 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. */
gemQuat qu;
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 = static_cast<float>(s*0.5);
s = 0.5 / s;
qu.x = static_cast<float>((mat[Z][Y] - mat[Y][Z]) * s);
qu.y = static_cast<float>((mat[X][Z] - mat[Z][X]) * s);
qu.z = static_cast<float>((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 = static_cast<float>(s*0.5);\
s = 0.5 / s;\
qu.j = static_cast<float>((mat[I][J] + mat[J][I]) * s);\
qu.k = static_cast<float>((mat[K][I] + mat[I][K]) * s);\
qu.w = static_cast<float>((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, static_cast<float>(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 **/
float mat_norm(HMatrix M, int tpose)
{
int i;
float sum, max;
max = 0.0;
for (i=0; i<3; i++) {
if (tpose) sum = static_cast<float>(fabs(M[0][i])+fabs(M[1][i])+fabs(M[2][i]));
else sum = static_cast<float>(fabs(M[i][0])+fabs(M[i][1])+fabs(M[i][2]));
if (max<sum) max = sum;
}
return max;
}
float norm_inf(HMatrix M) {return mat_norm(M, 0);}
float 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)
{
float 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(float *v, float *u)
{
float s = static_cast<float>(sqrt(vdot(v, v)));
u[0] = v[0]; u[1] = v[1];
u[2] = v[2] + ((v[2]<0.0) ? -s : s);
s = static_cast<float>(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, float *u)
{
int i, j;
for (i=0; i<3; i++) {
float 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, float *u)
{
int i, j;
for (i=0; i<3; i++) {
float 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)
{
float 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)
{
float v1[3], v2[3];
float 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 = static_cast<float>(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 = static_cast<float>(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.
*/
float polar_decomp(const HMatrix M, HMatrix Q, HMatrix S)
{
#define TOL 1.0e-6
HMatrix Mk, MadjTk, Ek;
float 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 = static_cast<float>(sqrt(sqrt((MadjT_one*MadjT_inf)/(M_one*M_inf))/fabs(det)));
g1 = gamma*0.5f;
g2 = 0.5f/(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] = 0.5f*(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--) {
float sm = static_cast<float>(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] -= static_cast<float>(s*(b + tau*a));
U[j][q] += static_cast<float>(s*(a - tau*b));
}
}
}
}
kv.x = static_cast<float>(Diag[X]);
kv.y = static_cast<float>(Diag[Y]);
kv.z = static_cast<float>(Diag[Z]);
kv.w = 1.0f;
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.
*/
gemQuat snuggle(gemQuat q, HVect *k)
{
#define SQRTHALF (0.7071067811865475244f)
#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];}
gemQuat p;
float 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) {
gemQuat qtoz, qp;
unsigned neg[3], win;
double mag[3], t;
static gemQuat qxtoz = {0,SQRTHALF,0,SQRTHALF};
static gemQuat qytoz = {SQRTHALF,0,0,SQRTHALF};
static gemQuat qppmm = { 0.5, 0.5,-0.5,-0.5};
static gemQuat qpppp = { 0.5, 0.5, 0.5, 0.5};
static gemQuat qmpmm = {-0.5, 0.5,-0.5,-0.5};
static gemQuat qpppm = { 0.5, 0.5, 0.5,-0.5};
static gemQuat q0001 = { 0.0, 0.0, 0.0, 1.0};
static gemQuat 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,static_cast<float>(-qp.z/t),static_cast<float>(qp.w/t)));
p = Qt_Mul(qtoz, Qt_Conj(p));
} else {
float 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] = static_cast<float>(sgn(neg[i], 0.5));}
cycle(ka,par)
} else {/*big*/ pa[hi] = static_cast<float>(sgn(neg[hi],1.0));}
} else {
if (two>big) {/*two*/
pa[hi] = static_cast<float>(sgn(neg[hi],SQRTHALF));
pa[lo] = static_cast<float>(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] = static_cast<float>(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 *******/
/* 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(const HMatrix A, gemAffineParts *parts)
{
HMatrix Q, S, U;
gemQuat p;
float 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(gemAffineParts *parts, gemAffineParts *inverse)
{
gemQuat t, p;
inverse->f = parts->f;
inverse->q = Qt_Conj(parts->q);
inverse->u = Qt_Mul(parts->q, parts->u);
inverse->k.x = static_cast<float>((parts->k.x==0.0) ? 0.0 : 1.0/parts->k.x);
inverse->k.y = static_cast<float>((parts->k.y==0.0) ? 0.0 : 1.0/parts->k.y);
inverse->k.z = static_cast<float>((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);
}