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ImathMatrixAlgo.h

///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
// 
// All rights reserved.
// 
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
// *       Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// *       Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following disclaimer
// in the documentation and/or other materials provided with the
// distribution.
// *       Neither the name of Industrial Light & Magic nor the names of
// its contributors may be used to endorse or promote products derived
// from this software without specific prior written permission. 
// 
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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///////////////////////////////////////////////////////////////////////////


#ifndef INCLUDED_IMATHMATRIXALGO_H
#define INCLUDED_IMATHMATRIXALGO_H

//-------------------------------------------------------------------------
//
//      This file contains algorithms applied to or in conjunction with
//    transformation matrices (Imath::Matrix33 and Imath::Matrix44).
//    The assumption made is that these functions are called much less
//    often than the basic point functions or these functions require
//    more support classes.
//
//    This file also defines a few predefined constant matrices.
//
//-------------------------------------------------------------------------

#include <ImathMatrix.h>
#include <ImathQuat.h>
#include <ImathEuler.h>
#include <ImathExc.h>
#include <ImathVec.h>
#include <math.h>


#ifdef OPENEXR_DLL
    #ifdef IMATH_EXPORTS
        #define IMATH_EXPORT_CONST extern __declspec(dllexport)
    #else
      #define IMATH_EXPORT_CONST extern __declspec(dllimport)
    #endif
#else
    #define IMATH_EXPORT_CONST extern const
#endif


namespace Imath {

//------------------
// Identity matrices
//------------------

IMATH_EXPORT_CONST M33f identity33f;
IMATH_EXPORT_CONST M44f identity44f;
IMATH_EXPORT_CONST M33d identity33d;
IMATH_EXPORT_CONST M44d identity44d;

//----------------------------------------------------------------------
// Extract scale, shear, rotation, and translation values from a matrix:
// 
// Notes:
//
// This implementation follows the technique described in the paper by
// Spencer W. Thomas in the Graphics Gems II article: "Decomposing a 
// Matrix into Simple Transformations", p. 320.
//
// - Some of the functions below have an optional exc parameter
//   that determines the functions' behavior when the matrix'
//   scaling is very close to zero:
//
//   If exc is true, the functions throw an Imath::ZeroScale exception.
//
//   If exc is false:
//
//      extractScaling (m, s)            returns false, s is invalid
//    sansScaling (m)                  returns m
//    removeScaling (m)          returns false, m is unchanged
//      sansScalingAndShear (m)          returns m
//      removeScalingAndShear (m)        returns false, m is unchanged
//      extractAndRemoveScalingAndShear (m, s, h)  
//                                       returns false, m is unchanged, 
//                                                      (sh) are invalid
//      checkForZeroScaleInRow ()        returns false
//    extractSHRT (m, s, h, r, t)      returns false, (shrt) are invalid
//
// - Functions extractEuler(), extractEulerXYZ() and extractEulerZYX() 
//   assume that the matrix does not include shear or non-uniform scaling, 
//   but they do not examine the matrix to verify this assumption.  
//   Matrices with shear or non-uniform scaling are likely to produce 
//   meaningless results.  Therefore, you should use the 
//   removeScalingAndShear() routine, if necessary, prior to calling
//   extractEuler...() .
//
// - All functions assume that the matrix does not include perspective
//   transformation(s), but they do not examine the matrix to verify 
//   this assumption.  Matrices with perspective transformations are 
//   likely to produce meaningless results.
//
//----------------------------------------------------------------------


//
// Declarations for 4x4 matrix.
//

template <class T>  bool        extractScaling 
                                            (const Matrix44<T> &mat,
                                   Vec3<T> &scl,
                                   bool exc = true);
  
template <class T>  Matrix44<T> sansScaling (const Matrix44<T> &mat, 
                                   bool exc = true);

template <class T>  bool        removeScaling 
                                            (Matrix44<T> &mat, 
                                   bool exc = true);

template <class T>  bool        extractScalingAndShear 
                                            (const Matrix44<T> &mat,
                                   Vec3<T> &scl,
                                   Vec3<T> &shr,
                                   bool exc = true);
  
template <class T>  Matrix44<T> sansScalingAndShear 
                                            (const Matrix44<T> &mat, 
                                   bool exc = true);

template <class T>  bool        removeScalingAndShear 
                                            (Matrix44<T> &mat,
                                   bool exc = true);

template <class T>  bool        extractAndRemoveScalingAndShear
                                            (Matrix44<T> &mat,
                                   Vec3<T>     &scl,
                                   Vec3<T>     &shr,
                                   bool exc = true);

template <class T>  void      extractEulerXYZ 
                                            (const Matrix44<T> &mat,
                                   Vec3<T> &rot);

template <class T>  void      extractEulerZYX 
                                            (const Matrix44<T> &mat,
                                   Vec3<T> &rot);

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

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);

template <class T>  bool      extractSHRT 
                                    (const Matrix44<T> &mat,
                             Vec3<T> &s,
                             Vec3<T> &h,
                             Vec3<T> &r,
                             Vec3<T> &t,
                             bool exc = true);

template <class T>  bool      extractSHRT 
                                    (const Matrix44<T> &mat,
                             Vec3<T> &s,
                             Vec3<T> &h,
                             Euler<T> &r,
                             Vec3<T> &t,
                             bool exc = true);

//
// Internal utility function.
//

template <class T>  bool      checkForZeroScaleInRow
                                            (const T       &scl, 
                                   const Vec3<T> &row,
                                   bool exc = true);

//
// Returns a matrix that rotates "fromDirection" vector to "toDirection"
// vector.
//

template <class T> Matrix44<T>      rotationMatrix (const Vec3<T> &fromDirection,
                                    const Vec3<T> &toDirection);



//
// Returns a matrix that rotates the "fromDir" vector 
// so that it points towards "toDir".  You may also 
// specify that you want the up vector to be pointing 
// in a certain direction "upDir".
//

template <class T> Matrix44<T>      rotationMatrixWithUpDir 
                                            (const Vec3<T> &fromDir,
                                   const Vec3<T> &toDir,
                                   const Vec3<T> &upDir);


//
// Returns a matrix that rotates the z-axis so that it 
// points towards "targetDir".  You must also specify 
// that you want the up vector to be pointing in a 
// certain direction "upDir".
//
// Notes: The following degenerate cases are handled:
//        (a) when the directions given by "toDir" and "upDir" 
//            are parallel or opposite;
//            (the direction vectors must have a non-zero cross product)
//        (b) when any of the given direction vectors have zero length
//

template <class T> Matrix44<T>      alignZAxisWithTargetDir 
                                            (Vec3<T> targetDir, 
                                   Vec3<T> upDir);


//----------------------------------------------------------------------


// 
// Declarations for 3x3 matrix.
//

 
template <class T>  bool        extractScaling 
                                            (const Matrix33<T> &mat,
                                   Vec2<T> &scl,
                                   bool exc = true);
  
template <class T>  Matrix33<T> sansScaling (const Matrix33<T> &mat, 
                                   bool exc = true);

template <class T>  bool        removeScaling 
                                            (Matrix33<T> &mat, 
                                   bool exc = true);

template <class T>  bool        extractScalingAndShear 
                                            (const Matrix33<T> &mat,
                                   Vec2<T> &scl,
                                   T &h,
                                   bool exc = true);
  
template <class T>  Matrix33<T> sansScalingAndShear 
                                            (const Matrix33<T> &mat, 
                                   bool exc = true);

template <class T>  bool        removeScalingAndShear 
                                            (Matrix33<T> &mat,
                                   bool exc = true);

template <class T>  bool        extractAndRemoveScalingAndShear
                                            (Matrix33<T> &mat,
                                   Vec2<T>     &scl,
                                   T           &shr,
                                   bool exc = true);

template <class T>  void      extractEuler
                                            (const Matrix33<T> &mat,
                                   T       &rot);

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

template <class T>  bool      checkForZeroScaleInRow
                                            (const T       &scl, 
                                   const Vec2<T> &row,
                                   bool exc = true);




//-----------------------------------------------------------------------------
// Implementation for 4x4 Matrix
//------------------------------


template <class T>
bool
extractScaling (const Matrix44<T> &mat, Vec3<T> &scl, bool exc)
{
    Vec3<T> shr;
    Matrix44<T> M (mat);

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


template <class T>
Matrix44<T>
sansScaling (const Matrix44<T> &mat, bool exc)
{
    Vec3<T> scl;
    Vec3<T> shr;
    Vec3<T> rot;
    Vec3<T> tran;

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

    Matrix44<T> M;
    
    M.translate (tran);
    M.rotate (rot);
    M.shear (shr);

    return M;
}


template <class T>
bool
removeScaling (Matrix44<T> &mat, bool exc)
{
    Vec3<T> scl;
    Vec3<T> shr;
    Vec3<T> rot;
    Vec3<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 Matrix44<T> &mat, 
                  Vec3<T> &scl, Vec3<T> &shr, bool exc)
{
    Matrix44<T> M (mat);

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


template <class T>
Matrix44<T>
sansScalingAndShear (const Matrix44<T> &mat, bool exc)
{
    Vec3<T> scl;
    Vec3<T> shr;
    Matrix44<T> M (mat);

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


template <class T>
bool
removeScalingAndShear (Matrix44<T> &mat, bool exc)
{
    Vec3<T> scl;
    Vec3<T> shr;

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


template <class T>
bool
extractAndRemoveScalingAndShear (Matrix44<T> &mat, 
                         Vec3<T> &scl, Vec3<T> &shr, bool exc)
{
    //
    // This implementation follows the technique described in the paper by
    // Spencer W. Thomas in the Graphics Gems II article: "Decomposing a 
    // Matrix into Simple Transformations", p. 320.
    //

    Vec3<T> row[3];

    row[0] = Vec3<T> (mat[0][0], mat[0][1], mat[0][2]);
    row[1] = Vec3<T> (mat[1][0], mat[1][1], mat[1][2]);
    row[2] = Vec3<T> (mat[2][0], mat[2][1], mat[2][2]);
    
    T maxVal = 0;
    for (int i=0; i < 3; i++)
      for (int j=0; j < 3; j++)
          if (Imath::abs (row[i][j]) > maxVal)
            maxVal = Imath::abs (row[i][j]);

    //
    // We normalize the 3x3 matrix here.
    // It was noticed that this can improve numerical stability significantly,
    // especially when many of the upper 3x3 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 < 3; 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 6 combinations (XY, XZ, YZ, YX, ZX, ZY), although we only
    // extract the first 3 because we can effect the last 3 by shearing in
    // XY, XZ, YZ combined rotations and scales.
    //
    // shear matrix <   1,  YX,  ZX,  0,
    //                 XY,   1,  ZY,  0,
    //                 XZ,  YZ,   1,  0,
    //                  0,   0,   0,  1 >

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

    // Compute XZ and YZ shears, orthogonalize 3rd row.
    shr[1]  = row[0].dot (row[2]);
    row[2] -= shr[1] * row[0];
    shr[2]  = row[1].dot (row[2]);
    row[2] -= shr[2] * row[1];

    // Next, get Z scale.
    scl.z = row[2].length ();
    if (! checkForZeroScaleInRow (scl.z, row[2], exc))
      return false;

    // Normalize 3rd row and correct the XZ and YZ shear factors for Z scaling.
    row[2] /= scl.z;
    shr[1] /= scl.z;
    shr[2] /= scl.z;

    // At this point, the upper 3x3 matrix in mat is orthonormal.
    // Check for a coordinate system flip. If the determinant
    // is less than zero, then negate the matrix and the scaling factors.
    if (row[0].dot (row[1].cross (row[2])) < 0)
      for (int  i=0; i < 3; i++)
      {
          scl[i] *= -1;
          row[i] *= -1;
      }

    // Copy over the orthonormal rows into the returned matrix.
    // The upper 3x3 matrix in mat is now a rotation matrix.
    for (int i=0; i < 3; i++)
    {
      mat[i][0] = row[i][0]; 
      mat[i][1] = row[i][1]; 
      mat[i][2] = row[i][2];
    }

    // Correct the scaling factors for the normalization step that we 
    // performed above; shear and rotation are not affected by the 
    // normalization.
    scl *= maxVal;

    return true;
}


template <class T>
void
extractEulerXYZ (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][2], M[2][2]);

    //
    // Remove the rot.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> (-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

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