imathmatrixalgo.h

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    N.rotate (Vec3<T> (-rot.x, 0, 0));
    N = N * M;

    //
    // Extract the other two angles, rot.y and rot.z, from N.
    //

    T cy = Math<T>::sqrt (N[0][0]*N[0][0] + N[0][1]*N[0][1]);
    rot.y = Math<T>::atan2 (-N[0][2], cy);
    rot.z = Math<T>::atan2 (-N[1][0], N[1][1]);
}


template <class T>
void
extractEulerZYX (const Matrix44<T> &mat, Vec3<T> &rot)
{
    //
    // Normalize the local x, y and z axes to remove scaling.
    //

    Vec3<T> i (mat[0][0], mat[0][1], mat[0][2]);
    Vec3<T> j (mat[1][0], mat[1][1], mat[1][2]);
    Vec3<T> k (mat[2][0], mat[2][1], mat[2][2]);

    i.normalize();
    j.normalize();
    k.normalize();

    Matrix44<T> M (i[0], i[1], i[2], 0, 
		   j[0], j[1], j[2], 0, 
		   k[0], k[1], k[2], 0, 
		   0,    0,    0,    1);

    //
    // Extract the first angle, rot.x.
    // 

    rot.x = -Math<T>::atan2 (M[1][0], M[0][0]);

    //
    // Remove the x rotation from M, so that the remaining
    // rotation, N, is only around two axes, and gimbal lock
    // cannot occur.
    //

    Matrix44<T> N;
    N.rotate (Vec3<T> (0, 0, -rot.x));
    N = N * M;

    //
    // Extract the other two angles, rot.y and rot.z, from N.
    //

    T cy = Math<T>::sqrt (N[2][2]*N[2][2] + N[2][1]*N[2][1]);
    rot.y = -Math<T>::atan2 (-N[2][0], cy);
    rot.z = -Math<T>::atan2 (-N[1][2], N[1][1]);
}


template <class T>
Quat<T>
extractQuat (const Matrix44<T> &mat)
{
  Matrix44<T> rot;

  T        tr, s;
  T         q[4];
  int    i, j, k;
  Quat<T>   quat;

  int nxt[3] = {1, 2, 0};
  tr = mat[0][0] + mat[1][1] + mat[2][2];

  // check the diagonal
  if (tr > 0.0) {
     s = Math<T>::sqrt (tr + 1.0);
     quat.r = s / 2.0;
     s = 0.5 / s;

     quat.v.x = (mat[1][2] - mat[2][1]) * s;
     quat.v.y = (mat[2][0] - mat[0][2]) * s;
     quat.v.z = (mat[0][1] - mat[1][0]) * s;
  } 
  else {      
     // diagonal is negative
     i = 0;
     if (mat[1][1] > mat[0][0]) 
        i=1;
     if (mat[2][2] > mat[i][i]) 
        i=2;
    
     j = nxt[i];
     k = nxt[j];
     s = Math<T>::sqrt ((mat[i][i] - (mat[j][j] + mat[k][k])) + 1.0);
      
     q[i] = s * 0.5;
     if (s != 0.0) 
        s = 0.5 / s;

     q[3] = (mat[j][k] - mat[k][j]) * s;
     q[j] = (mat[i][j] + mat[j][i]) * s;
     q[k] = (mat[i][k] + mat[k][i]) * s;

     quat.v.x = q[0];
     quat.v.y = q[1];
     quat.v.z = q[2];
     quat.r = q[3];
 }

  return quat;
}

template <class T>
bool 
extractSHRT (const Matrix44<T> &mat,
	     Vec3<T> &s,
	     Vec3<T> &h,
	     Vec3<T> &r,
	     Vec3<T> &t,
	     bool exc /* = true */ ,
	     typename Euler<T>::Order rOrder /* = Euler<T>::XYZ */ )
{
    Matrix44<T> rot;

    rot = mat;
    if (! extractAndRemoveScalingAndShear (rot, s, h, exc))
	return false;

    extractEulerXYZ (rot, r);

    t.x = mat[3][0];
    t.y = mat[3][1];
    t.z = mat[3][2];

    if (rOrder != Euler<T>::XYZ)
    {
	Imath::Euler<T> eXYZ (r, Imath::Euler<T>::XYZ);
	Imath::Euler<T> e (eXYZ, rOrder);
	r = e.toXYZVector ();
    }

    return true;
}

template <class T>
bool 
extractSHRT (const Matrix44<T> &mat,
	     Vec3<T> &s,
	     Vec3<T> &h,
	     Vec3<T> &r,
	     Vec3<T> &t,
	     bool exc)
{
    return extractSHRT(mat, s, h, r, t, exc, Imath::Euler<T>::XYZ);
}

template <class T>
bool 
extractSHRT (const Matrix44<T> &mat,
	     Vec3<T> &s,
	     Vec3<T> &h,
	     Euler<T> &r,
	     Vec3<T> &t,
	     bool exc /* = true */)
{
    return extractSHRT (mat, s, h, r, t, exc, r.order ());
}


template <class T> 
bool		
checkForZeroScaleInRow (const T& scl, 
			const Vec3<T> &row,
			bool exc /* = true */ )
{
    for (int i = 0; i < 3; i++)
    {
	if ((abs (scl) < 1 && abs (row[i]) >= limits<T>::max() * abs (scl)))
	{
	    if (exc)
		throw Imath::ZeroScaleExc ("Cannot remove zero scaling "
					   "from matrix.");
	    else
		return false;
	}
    }

    return true;
}


template <class T>
Matrix44<T>
rotationMatrix (const Vec3<T> &from, const Vec3<T> &to)
{
    Quat<T> q;
    q.setRotation(from, to);
    return q.toMatrix44();
}


template <class T>
Matrix44<T>	
rotationMatrixWithUpDir (const Vec3<T> &fromDir,
			 const Vec3<T> &toDir,
			 const Vec3<T> &upDir)
{
    //
    // The goal is to obtain a rotation matrix that takes 
    // "fromDir" to "toDir".  We do this in two steps and 
    // compose the resulting rotation matrices; 
    //    (a) rotate "fromDir" into the z-axis
    //    (b) rotate the z-axis into "toDir"
    //

    // The from direction must be non-zero; but we allow zero to and up dirs.
    if (fromDir.length () == 0)
	return Matrix44<T> ();

    else
    {
	Matrix44<T> zAxis2FromDir  = alignZAxisWithTargetDir 
	                                 (fromDir, Vec3<T> (0, 1, 0));

	Matrix44<T> fromDir2zAxis  = zAxis2FromDir.transposed ();
	
	Matrix44<T> zAxis2ToDir    = alignZAxisWithTargetDir (toDir, upDir);

	return fromDir2zAxis * zAxis2ToDir;
    }
}


template <class T>
Matrix44<T>
alignZAxisWithTargetDir (Vec3<T> targetDir, Vec3<T> upDir)
{
    //
    // Ensure that the target direction is non-zero.
    //

    if ( targetDir.length () == 0 )
	targetDir = Vec3<T> (0, 0, 1);

    //
    // Ensure that the up direction is non-zero.
    //

    if ( upDir.length () == 0 )
	upDir = Vec3<T> (0, 1, 0);

    //
    // Check for degeneracies.  If the upDir and targetDir are parallel 
    // or opposite, then compute a new, arbitrary up direction that is
    // not parallel or opposite to the targetDir.
    //

    if (upDir.cross (targetDir).length () == 0)
    {
	upDir = targetDir.cross (Vec3<T> (1, 0, 0));
	if (upDir.length() == 0)
	    upDir = targetDir.cross(Vec3<T> (0, 0, 1));
    }

    //
    // Compute the x-, y-, and z-axis vectors of the new coordinate system.
    //

    Vec3<T> targetPerpDir = upDir.cross (targetDir);    
    Vec3<T> targetUpDir   = targetDir.cross (targetPerpDir);
    
    //
    // Rotate the x-axis into targetPerpDir (row 0),
    // rotate the y-axis into targetUpDir   (row 1),
    // rotate the z-axis into targetDir     (row 2).
    //
    
    Vec3<T> row[3];
    row[0] = targetPerpDir.normalized ();
    row[1] = targetUpDir  .normalized ();
    row[2] = targetDir    .normalized ();
    
    Matrix44<T> mat ( row[0][0],  row[0][1],  row[0][2],  0,
		      row[1][0],  row[1][1],  row[1][2],  0,
		      row[2][0],  row[2][1],  row[2][2],  0,
		      0,          0,          0,          1 );
    
    return mat;
}



//-----------------------------------------------------------------------------
// Implementation for 3x3 Matrix
//------------------------------


template <class T>
bool
extractScaling (const Matrix33<T> &mat, Vec2<T> &scl, bool exc)
{
    T shr;
    Matrix33<T> M (mat);

    if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
	return false;

    return true;
}


template <class T>
Matrix33<T>
sansScaling (const Matrix33<T> &mat, bool exc)
{
    Vec2<T> scl;
    T shr;
    T rot;
    Vec2<T> tran;

    if (! extractSHRT (mat, scl, shr, rot, tran, exc))
	return mat;

    Matrix33<T> M;
    
    M.translate (tran);
    M.rotate (rot);
    M.shear (shr);

    return M;
}


template <class T>
bool
removeScaling (Matrix33<T> &mat, bool exc)
{
    Vec2<T> scl;
    T shr;
    T rot;
    Vec2<T> tran;

    if (! extractSHRT (mat, scl, shr, rot, tran, exc))
	return false;

    mat.makeIdentity ();
    mat.translate (tran);
    mat.rotate (rot);
    mat.shear (shr);

    return true;
}


template <class T>
bool
extractScalingAndShear (const Matrix33<T> &mat, Vec2<T> &scl, T &shr, bool exc)
{
    Matrix33<T> M (mat);

    if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
	return false;

    return true;
}


template <class T>
Matrix33<T>
sansScalingAndShear (const Matrix33<T> &mat, bool exc)
{
    Vec2<T> scl;
    T shr;
    Matrix33<T> M (mat);

    if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
	return mat;
    
    return M;
}


template <class T>
bool
removeScalingAndShear (Matrix33<T> &mat, bool exc)
{
    Vec2<T> scl;
    T shr;

    if (! extractAndRemoveScalingAndShear (mat, scl, shr, exc))
	return false;
    
    return true;
}

template <class T>
bool
extractAndRemoveScalingAndShear (Matrix33<T> &mat, 
				 Vec2<T> &scl, T &shr, bool exc)
{
    Vec2<T> row[2];

    row[0] = Vec2<T> (mat[0][0], mat[0][1]);
    row[1] = Vec2<T> (mat[1][0], mat[1][1]);
    
    T maxVal = 0;
    for (int i=0; i < 2; i++)
	for (int j=0; j < 2; j++)
	    if (Imath::abs (row[i][j]) > maxVal)
		maxVal = Imath::abs (row[i][j]);

    //
    // We normalize the 2x2 matrix here.
    // It was noticed that this can improve numerical stability significantly,
    // especially when many of the upper 2x2 matrix's coefficients are very
    // close to zero; we correct for this step at the end by multiplying the 
    // scaling factors by maxVal at the end (shear and rotation are not 
    // affected by the normalization).

    if (maxVal != 0)
    {
	for (int i=0; i < 2; i++)
	    if (! checkForZeroScaleInRow (maxVal, row[i], exc))
		return false;
	    else
		row[i] /= maxVal;
    }

    // Compute X scale factor. 
    scl.x = row[0].length ();
    if (! checkForZeroScaleInRow (scl.x, row[0], exc))
	return false;

    // Normalize first row.
    row[0] /= scl.x;

    // An XY shear factor will shear the X coord. as the Y coord. changes.
    // There are 2 combinations (XY, YX), although we only extract the XY 
    // shear factor because we can effect the an YX shear factor by 
    // shearing in XY combined with rotations and scales.
    //
    // shear matrix <   1,  YX,  0,
    //                 XY,   1,  0,
    //                  0,   0,  1 >

    // Compute XY shear factor and make 2nd row orthogonal to 1st.
    shr     = row[0].dot (row[1]);
    row[1] -= shr * row[0];

    // Now, compute Y scale.
    scl.y = row[1].length ();
    if (! checkForZeroScaleInRow (scl.y, row[1], exc))
	return false;

    // Normalize 2nd row and correct the XY shear factor for Y scaling.
    row[1] /= scl.y; 
    shr    /= scl.y;

    // At this point, the upper 2x2 matrix in mat is orthonormal.
    // Check for a coordinate system flip. If the determinant
    // is -1, then flip the rotation matrix and adjust the scale(Y) 
    // and shear(XY) factors to compensate.
    if (row[0][0] * row[1][1] - row[0][1] * row[1][0] < 0)
    {
	row[1][0] *= -1;
	row[1][1] *= -1;
	scl[1] *= -1;
	shr *= -1;
    }

    // Copy over the orthonormal rows into the returned matrix.
    // The upper 2x2 matrix in mat is now a rotation matrix.
    for (int i=0; i < 2; i++)
    {
	mat[i][0] = row[i][0]; 
	mat[i][1] = row[i][1]; 
    }

    scl *= maxVal;

    return true;
}


template <class T>
void
extractEuler (const Matrix33<T> &mat, T &rot)
{
    //
    // Normalize the local x and y axes to remove scaling.
    //

    Vec2<T> i (mat[0][0], mat[0][1]);
    Vec2<T> j (mat[1][0], mat[1][1]);

    i.normalize();
    j.normalize();

    //
    // Extract the angle, rot.
    // 

    rot = - Math<T>::atan2 (j[0], i[0]);
}


template <class T>
bool 
extractSHRT (const Matrix33<T> &mat,
	     Vec2<T> &s,
	     T       &h,
	     T       &r,
	     Vec2<T> &t,
	     bool    exc)
{
    Matrix33<T> rot;

    rot = mat;
    if (! extractAndRemoveScalingAndShear (rot, s, h, exc))
	return false;

    extractEuler (rot, r);

    t.x = mat[2][0];
    t.y = mat[2][1];

    return true;
}


template <class T> 
bool		
checkForZeroScaleInRow (const T& scl, 
			const Vec2<T> &row,
			bool exc /* = true */ )
{
    for (int i = 0; i < 2; i++)
    {
	if ((abs (scl) < 1 && abs (row[i]) >= limits<T>::max() * abs (scl)))
	{
	    if (exc)
		throw Imath::ZeroScaleExc ("Cannot remove zero scaling "
					   "from matrix.");
	    else
		return false;
	}
    }

    return true;
}


} // namespace Imath

#endif