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ImathMatrix.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
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// *       Redistributions in binary form must reproduce the above
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// distribution.
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// its contributors may be used to endorse or promote products derived
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#ifndef INCLUDED_IMATHMATRIX_H
#define INCLUDED_IMATHMATRIX_H

//----------------------------------------------------------------
//
//    2D (3x3) and 3D (4x4) transformation matrix templates.
//
//----------------------------------------------------------------

#include <ImathPlatform.h>
#include <ImathFun.h>
#include <ImathExc.h>
#include <ImathVec.h>
#include <ImathShear.h>

#include <iostream>
#include <iomanip>


namespace Imath {


00059 template <class T> class Matrix33
{
  public:

    //-------------------
    // Access to elements
    //-------------------

    T       x[3][3];

    T *           operator [] (int i);
    const T *     operator [] (int i) const;


    //-------------
    // Constructors
    //-------------

    Matrix33 ();
                        // 1 0 0
                        // 0 1 0
                        // 0 0 1

    Matrix33 (T a);
                        // a a a
                        // a a a
                        // a a a

    Matrix33 (const T a[3][3]);
                        // a[0][0] a[0][1] a[0][2]
                        // a[1][0] a[1][1] a[1][2]
                        // a[2][0] a[2][1] a[2][2]

    Matrix33 (T a, T b, T c, T d, T e, T f, T g, T h, T i);

                        // a b c
                        // d e f
                        // g h i


    //--------------------------------
    // Copy constructor and assignment
    //--------------------------------

    Matrix33 (const Matrix33 &v);

    const Matrix33 &    operator = (const Matrix33 &v);
    const Matrix33 &    operator = (T a);


    //----------------------
    // Compatibility with Sb
    //----------------------
    
    T *                 getValue ();
    const T *           getValue () const;

    template <class S>
    void          getValue (Matrix33<S> &v) const;
    template <class S>
    Matrix33 &          setValue (const Matrix33<S> &v);

    template <class S>
    Matrix33 &          setTheMatrix (const Matrix33<S> &v);


    //---------
    // Identity
    //---------

    void                makeIdentity();


    //---------
    // Equality
    //---------

    bool          operator == (const Matrix33 &v) const;
    bool          operator != (const Matrix33 &v) const;

    //-----------------------------------------------------------------------
    // Compare two matrices and test if they are "approximately equal":
    //
    // equalWithAbsError (m, e)
    //
    //          Returns true if the coefficients of this and m are the same with
    //          an absolute error of no more than e, i.e., for all i, j
    //
    //      abs (this[i][j] - m[i][j]) <= e
    //
    // equalWithRelError (m, e)
    //
    //          Returns true if the coefficients of this and m are the same with
    //          a relative error of no more than e, i.e., for all i, j
    //
    //      abs (this[i] - v[i][j]) <= e * abs (this[i][j])
    //-----------------------------------------------------------------------

    bool          equalWithAbsError (const Matrix33<T> &v, T e) const;
    bool          equalWithRelError (const Matrix33<T> &v, T e) const;


    //------------------------
    // Component-wise addition
    //------------------------

    const Matrix33 &    operator += (const Matrix33 &v);
    const Matrix33 &    operator += (T a);
    Matrix33            operator + (const Matrix33 &v) const;


    //---------------------------
    // Component-wise subtraction
    //---------------------------

    const Matrix33 &    operator -= (const Matrix33 &v);
    const Matrix33 &    operator -= (T a);
    Matrix33            operator - (const Matrix33 &v) const;


    //------------------------------------
    // Component-wise multiplication by -1
    //------------------------------------

    Matrix33            operator - () const;
    const Matrix33 &    negate ();


    //------------------------------
    // Component-wise multiplication
    //------------------------------

    const Matrix33 &    operator *= (T a);
    Matrix33            operator * (T a) const;


    //-----------------------------------
    // Matrix-times-matrix multiplication
    //-----------------------------------

    const Matrix33 &    operator *= (const Matrix33 &v);
    Matrix33            operator * (const Matrix33 &v) const;


    //---------------------------------------------
    // Vector-times-matrix multiplication; see also
    // the "operator *" functions defined below.
    //---------------------------------------------

    template <class S>
    void          multVecMatrix(const Vec2<S> &src, Vec2<S> &dst) const;

    template <class S>
    void          multDirMatrix(const Vec2<S> &src, Vec2<S> &dst) const;


    //------------------------
    // Component-wise division
    //------------------------

    const Matrix33 &    operator /= (T a);
    Matrix33            operator / (T a) const;


    //------------------
    // Transposed matrix
    //------------------

    const Matrix33 &    transpose ();
    Matrix33            transposed () const;


    //------------------------------------------------------------
    // Inverse matrix: If singExc is false, inverting a singular
    // matrix produces an identity matrix.  If singExc is true,
    // inverting a singular matrix throws a SingMatrixExc.
    //
    // inverse() and invert() invert matrices using determinants;
    // gjInverse() and gjInvert() use the Gauss-Jordan method.
    //
    // inverse() and invert() are significantly faster than
    // gjInverse() and gjInvert(), but the results may be slightly
    // less accurate.
    // 
    //------------------------------------------------------------

    const Matrix33 &    invert (bool singExc = false)
                  throw (Iex::MathExc);

    Matrix33<T>         inverse (bool singExc = false) const
                  throw (Iex::MathExc);

    const Matrix33 &    gjInvert (bool singExc = false)
                  throw (Iex::MathExc);

    Matrix33<T>         gjInverse (bool singExc = false) const
                  throw (Iex::MathExc);


    //-----------------------------------------
    // Set matrix to rotation by r (in radians)
    //-----------------------------------------

    template <class S>
    const Matrix33 &    setRotation (S r);


    //-----------------------------
    // Rotate the given matrix by r
    //-----------------------------

    template <class S>
    const Matrix33 &    rotate (S r);


    //--------------------------------------------
    // Set matrix to scale by given uniform factor
    //--------------------------------------------

    const Matrix33 &    setScale (T s);


    //------------------------------------
    // Set matrix to scale by given vector
    //------------------------------------

    template <class S>
    const Matrix33 &    setScale (const Vec2<S> &s);


    //----------------------
    // Scale the matrix by s
    //----------------------

    template <class S>
    const Matrix33 &    scale (const Vec2<S> &s);


    //------------------------------------------
    // Set matrix to translation by given vector
    //------------------------------------------

    template <class S>
    const Matrix33 &    setTranslation (const Vec2<S> &t);


    //-----------------------------
    // Return translation component
    //-----------------------------

    Vec2<T>       translation () const;


    //--------------------------
    // Translate the matrix by t
    //--------------------------

    template <class S>
    const Matrix33 &    translate (const Vec2<S> &t);


    //-----------------------------------------------------------
    // Set matrix to shear x for each y coord. by given factor xy
    //-----------------------------------------------------------

    template <class S>
    const Matrix33 &    setShear (const S &h);


    //-------------------------------------------------------------
    // Set matrix to shear x for each y coord. by given factor h[0]
    // and to shear y for each x coord. by given factor h[1]
    //-------------------------------------------------------------

    template <class S>
    const Matrix33 &    setShear (const Vec2<S> &h);


    //-----------------------------------------------------------
    // Shear the matrix in x for each y coord. by given factor xy
    //-----------------------------------------------------------

    template <class S>
    const Matrix33 &    shear (const S &xy);


    //-----------------------------------------------------------
    // Shear the matrix in x for each y coord. by given factor xy
    // and shear y for each x coord. by given factor yx
    //-----------------------------------------------------------

    template <class S>
    const Matrix33 &    shear (const Vec2<S> &h);


    //-------------------------------------------------
    // Limitations of type T (see also class limits<T>)
    //-------------------------------------------------

    static T            baseTypeMin()           {return limits<T>::min();}
    static T            baseTypeMax()           {return limits<T>::max();}
    static T            baseTypeSmallest()      {return limits<T>::smallest();}
    static T            baseTypeEpsilon() {return limits<T>::epsilon();}
};


00365 template <class T> class Matrix44
{
  public:

    //-------------------
    // Access to elements
    //-------------------

    T       x[4][4];

    T *           operator [] (int i);
    const T *     operator [] (int i) const;


    //-------------
    // Constructors
    //-------------

    Matrix44 ();
                        // 1 0 0 0
                        // 0 1 0 0
                        // 0 0 1 0
                        // 0 0 0 1

    Matrix44 (T a);
                        // a a a a
                        // a a a a
                        // a a a a
                        // a a a a

    Matrix44 (const T a[4][4]) ;
                        // a[0][0] a[0][1] a[0][2] a[0][3]
                        // a[1][0] a[1][1] a[1][2] a[1][3]
                        // a[2][0] a[2][1] a[2][2] a[2][3]
                        // a[3][0] a[3][1] a[3][2] a[3][3]

    Matrix44 (T a, T b, T c, T d, T e, T f, T g, T h,
            T i, T j, T k, T l, T m, T n, T o, T p);

                        // a b c d
                        // e f g h
                        // i j k l
                        // m n o p

    Matrix44 (Matrix33<T> r, Vec3<T> t);
                        // r r r 0
                        // r r r 0
                        // r r r 0
                        // t t t 1


    //--------------------------------
    // Copy constructor and assignment
    //--------------------------------

    Matrix44 (const Matrix44 &v);

    const Matrix44 &    operator = (const Matrix44 &v);
    const Matrix44 &    operator = (T a);


    //----------------------
    // Compatibility with Sb
    //----------------------
    
    T *                 getValue ();
    const T *           getValue () const;

    template <class S>
    void          getValue (Matrix44<S> &v) const;
    template <class S>
    Matrix44 &          setValue (const Matrix44<S> &v);

    template <class S>
    Matrix44 &          setTheMatrix (const Matrix44<S> &v);

    //---------
    // Identity
    //---------

    void                makeIdentity();


    //---------
    // Equality
    //---------

    bool          operator == (const Matrix44 &v) const;
    bool          operator != (const Matrix44 &v) const;

    //-----------------------------------------------------------------------
    // Compare two matrices and test if they are "approximately equal":
    //
    // equalWithAbsError (m, e)
    //
    //          Returns true if the coefficients of this and m are the same with
    //          an absolute error of no more than e, i.e., for all i, j
    //
    //      abs (this[i][j] - m[i][j]) <= e
    //
    // equalWithRelError (m, e)
    //
    //          Returns true if the coefficients of this and m are the same with
    //          a relative error of no more than e, i.e., for all i, j
    //
    //      abs (this[i] - v[i][j]) <= e * abs (this[i][j])
    //-----------------------------------------------------------------------

    bool          equalWithAbsError (const Matrix44<T> &v, T e) const;
    bool          equalWithRelError (const Matrix44<T> &v, T e) const;


    //------------------------
    // Component-wise addition
    //------------------------

    const Matrix44 &    operator += (const Matrix44 &v);
    const Matrix44 &    operator += (T a);
    Matrix44            operator + (const Matrix44 &v) const;


    //---------------------------
    // Component-wise subtraction
    //---------------------------

    const Matrix44 &    operator -= (const Matrix44 &v);
    const Matrix44 &    operator -= (T a);
    Matrix44            operator - (const Matrix44 &v) const;


    //------------------------------------
    // Component-wise multiplication by -1
    //------------------------------------

    Matrix44            operator - () const;
    const Matrix44 &    negate ();


    //------------------------------
    // Component-wise multiplication
    //------------------------------

    const Matrix44 &    operator *= (T a);
    Matrix44            operator * (T a) const;


    //-----------------------------------
    // Matrix-times-matrix multiplication
    //-----------------------------------

    const Matrix44 &    operator *= (const Matrix44 &v);
    Matrix44            operator * (const Matrix44 &v) const;

    static void         multiply (const Matrix44 &a,  // assumes that
                          const Matrix44 &b,    // &a != &c and
                          Matrix44 &c);         // &b != &c.


    //---------------------------------------------
    // Vector-times-matrix multiplication; see also
    // the "operator *" functions defined below.
    //---------------------------------------------

    template <class S>
    void          multVecMatrix(const Vec3<S> &src, Vec3<S> &dst) const;

    template <class S>
    void          multDirMatrix(const Vec3<S> &src, Vec3<S> &dst) const;


    //------------------------
    // Component-wise division
    //------------------------

    const Matrix44 &    operator /= (T a);
    Matrix44            operator / (T a) const;


    //------------------
    // Transposed matrix
    //------------------

    const Matrix44 &    transpose ();
    Matrix44            transposed () const;


    //------------------------------------------------------------
    // Inverse matrix: If singExc is false, inverting a singular
    // matrix produces an identity matrix.  If singExc is true,
    // inverting a singular matrix throws a SingMatrixExc.
    //
    // inverse() and invert() invert matrices using determinants;
    // gjInverse() and gjInvert() use the Gauss-Jordan method.
    //
    // inverse() and invert() are significantly faster than
    // gjInverse() and gjInvert(), but the results may be slightly
    // less accurate.
    // 
    //------------------------------------------------------------

    const Matrix44 &    invert (bool singExc = false)
                  throw (Iex::MathExc);

    Matrix44<T>         inverse (bool singExc = false) const
                  throw (Iex::MathExc);

    const Matrix44 &    gjInvert (bool singExc = false)
                  throw (Iex::MathExc);

    Matrix44<T>         gjInverse (bool singExc = false) const
                  throw (Iex::MathExc);


    //--------------------------------------------------------
    // Set matrix to rotation by XYZ euler angles (in radians)
    //--------------------------------------------------------

    template <class S>
    const Matrix44 &    setEulerAngles (const Vec3<S>& r);


    //--------------------------------------------------------
    // Set matrix to rotation around given axis by given angle
    //--------------------------------------------------------

    template <class S>
    const Matrix44 &    setAxisAngle (const Vec3<S>& ax, S ang);


    //-------------------------------------------
    // Rotate the matrix by XYZ euler angles in r
    //-------------------------------------------

    template <class S>
    const Matrix44 &    rotate (const Vec3<S> &r);


    //--------------------------------------------
    // Set matrix to scale by given uniform factor
    //--------------------------------------------

    const Matrix44 &    setScale (T s);


    //------------------------------------
    // Set matrix to scale by given vector
    //------------------------------------

    template <class S>
    const Matrix44 &    setScale (const Vec3<S> &s);


    //----------------------
    // Scale the matrix by s
    //----------------------

    template <class S>
    const Matrix44 &    scale (const Vec3<S> &s);


    //------------------------------------------
    // Set matrix to translation by given vector
    //------------------------------------------

    template <class S>
    const Matrix44 &    setTranslation (const Vec3<S> &t);


    //-----------------------------
    // Return translation component
    //-----------------------------

    const Vec3<T> translation () const;


    //--------------------------
    // Translate the matrix by t
    //--------------------------

    template <class S>
    const Matrix44 &    translate (const Vec3<S> &t);


    //-------------------------------------------------------------
    // Set matrix to shear by given vector h.  The resulting matrix
    //    will shear x for each y coord. by a factor of h[0] ;
    //    will shear x for each z coord. by a factor of h[1] ;
    //    will shear y for each z coord. by a factor of h[2] .
    //-------------------------------------------------------------

    template <class S>
    const Matrix44 &    setShear (const Vec3<S> &h);


    //------------------------------------------------------------
    // Set matrix to shear by given factors.  The resulting matrix
    //    will shear x for each y coord. by a factor of h.xy ;
    //    will shear x for each z coord. by a factor of h.xz ;
    //    will shear y for each z coord. by a factor of h.yz ; 
    //    will shear y for each x coord. by a factor of h.yx ;
    //    will shear z for each x coord. by a factor of h.zx ;
    //    will shear z for each y coord. by a factor of h.zy .
    //------------------------------------------------------------

    template <class S>
    const Matrix44 &    setShear (const Shear6<S> &h);


    //--------------------------------------------------------
    // Shear the matrix by given vector.  The composed matrix 
    // will be <shear> * <this>, where the shear matrix ...
    //    will shear x for each y coord. by a factor of h[0] ;
    //    will shear x for each z coord. by a factor of h[1] ;
    //    will shear y for each z coord. by a factor of h[2] .
    //--------------------------------------------------------

    template <class S>
    const Matrix44 &    shear (const Vec3<S> &h);


    //------------------------------------------------------------
    // Shear the matrix by the given factors.  The composed matrix 
    // will be <shear> * <this>, where the shear matrix ...
    //    will shear x for each y coord. by a factor of h.xy ;
    //    will shear x for each z coord. by a factor of h.xz ;
    //    will shear y for each z coord. by a factor of h.yz ;
    //    will shear y for each x coord. by a factor of h.yx ;
    //    will shear z for each x coord. by a factor of h.zx ;
    //    will shear z for each y coord. by a factor of h.zy .
    //------------------------------------------------------------

    template <class S>
    const Matrix44 &    shear (const Shear6<S> &h);


    //-------------------------------------------------
    // Limitations of type T (see also class limits<T>)
    //-------------------------------------------------

    static T            baseTypeMin()           {return limits<T>::min();}
    static T            baseTypeMax()           {return limits<T>::max();}
    static T            baseTypeSmallest()      {return limits<T>::smallest();}
    static T            baseTypeEpsilon() {return limits<T>::epsilon();}
};


//--------------
// Stream output
//--------------

template <class T>
std::ostream &    operator << (std::ostream & s, const Matrix33<T> &m); 

template <class T>
std::ostream &    operator << (std::ostream & s, const Matrix44<T> &m); 


//---------------------------------------------
// Vector-times-matrix multiplication operators
//---------------------------------------------

template <class S, class T>
const Vec2<S> &            operator *= (Vec2<S> &v, const Matrix33<T> &m);

template <class S, class T>
Vec2<S>                    operator * (const Vec2<S> &v, const Matrix33<T> &m);

template <class S, class T>
const Vec3<S> &            operator *= (Vec3<S> &v, const Matrix33<T> &m);

template <class S, class T>
Vec3<S>                    operator * (const Vec3<S> &v, const Matrix33<T> &m);

template <class S, class T>
const Vec3<S> &            operator *= (Vec3<S> &v, const Matrix44<T> &m);

template <class S, class T>
Vec3<S>                    operator * (const Vec3<S> &v, const Matrix44<T> &m);


//-------------------------
// Typedefs for convenience
//-------------------------

typedef Matrix33 <float>  M33f;
typedef Matrix33 <double> M33d;
typedef Matrix44 <float>  M44f;
typedef Matrix44 <double> M44d;


//---------------------------
// Implementation of Matrix33
//---------------------------

template <class T>
inline T *
Matrix33<T>::operator [] (int i)
{
    return x[i];
}

template <class T>
inline const T *
Matrix33<T>::operator [] (int i) const
{
    return x[i];
}

template <class T>
inline
Matrix33<T>::Matrix33 ()
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;
    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;
}

template <class T>
inline
Matrix33<T>::Matrix33 (T a)
{
    x[0][0] = a;
    x[0][1] = a;
    x[0][2] = a;
    x[1][0] = a;
    x[1][1] = a;
    x[1][2] = a;
    x[2][0] = a;
    x[2][1] = a;
    x[2][2] = a;
}

template <class T>
inline
Matrix33<T>::Matrix33 (const T a[3][3]) 
{
    x[0][0] = a[0][0];
    x[0][1] = a[0][1];
    x[0][2] = a[0][2];
    x[1][0] = a[1][0];
    x[1][1] = a[1][1];
    x[1][2] = a[1][2];
    x[2][0] = a[2][0];
    x[2][1] = a[2][1];
    x[2][2] = a[2][2];
}

template <class T>
inline
Matrix33<T>::Matrix33 (T a, T b, T c, T d, T e, T f, T g, T h, T i)
{
    x[0][0] = a;
    x[0][1] = b;
    x[0][2] = c;
    x[1][0] = d;
    x[1][1] = e;
    x[1][2] = f;
    x[2][0] = g;
    x[2][1] = h;
    x[2][2] = i;
}

template <class T>
inline
Matrix33<T>::Matrix33 (const Matrix33 &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
}

template <class T>
inline const Matrix33<T> &
Matrix33<T>::operator = (const Matrix33 &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    return *this;
}

template <class T>
inline const Matrix33<T> &
Matrix33<T>::operator = (T a)
{
    x[0][0] = a;
    x[0][1] = a;
    x[0][2] = a;
    x[1][0] = a;
    x[1][1] = a;
    x[1][2] = a;
    x[2][0] = a;
    x[2][1] = a;
    x[2][2] = a;
    return *this;
}

template <class T>
inline T *
Matrix33<T>::getValue ()
{
    return (T *) &x[0][0];
}

template <class T>
inline const T *
Matrix33<T>::getValue () const
{
    return (const T *) &x[0][0];
}

template <class T>
template <class S>
inline void
Matrix33<T>::getValue (Matrix33<S> &v) const
{
    v.x[0][0] = x[0][0];
    v.x[0][1] = x[0][1];
    v.x[0][2] = x[0][2];
    v.x[1][0] = x[1][0];
    v.x[1][1] = x[1][1];
    v.x[1][2] = x[1][2];
    v.x[2][0] = x[2][0];
    v.x[2][1] = x[2][1];
    v.x[2][2] = x[2][2];
}

template <class T>
template <class S>
inline Matrix33<T> &
Matrix33<T>::setValue (const Matrix33<S> &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    return *this;
}

template <class T>
template <class S>
inline Matrix33<T> &
Matrix33<T>::setTheMatrix (const Matrix33<S> &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    return *this;
}

template <class T>
inline void
Matrix33<T>::makeIdentity()
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;
    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;
}

template <class T>
bool
Matrix33<T>::operator == (const Matrix33 &v) const
{
    return x[0][0] == v.x[0][0] &&
         x[0][1] == v.x[0][1] &&
         x[0][2] == v.x[0][2] &&
         x[1][0] == v.x[1][0] &&
         x[1][1] == v.x[1][1] &&
         x[1][2] == v.x[1][2] &&
         x[2][0] == v.x[2][0] &&
         x[2][1] == v.x[2][1] &&
         x[2][2] == v.x[2][2];
}

template <class T>
bool
Matrix33<T>::operator != (const Matrix33 &v) const
{
    return x[0][0] != v.x[0][0] ||
         x[0][1] != v.x[0][1] ||
         x[0][2] != v.x[0][2] ||
         x[1][0] != v.x[1][0] ||
         x[1][1] != v.x[1][1] ||
         x[1][2] != v.x[1][2] ||
         x[2][0] != v.x[2][0] ||
         x[2][1] != v.x[2][1] ||
         x[2][2] != v.x[2][2];
}

template <class T>
bool
Matrix33<T>::equalWithAbsError (const Matrix33<T> &m, T e) const
{
    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
          if (!Imath::equalWithAbsError ((*this)[i][j], m[i][j], e))
            return false;

    return true;
}

template <class T>
bool
Matrix33<T>::equalWithRelError (const Matrix33<T> &m, T e) const
{
    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
          if (!Imath::equalWithRelError ((*this)[i][j], m[i][j], e))
            return false;

    return true;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator += (const Matrix33<T> &v)
{
    x[0][0] += v.x[0][0];
    x[0][1] += v.x[0][1];
    x[0][2] += v.x[0][2];
    x[1][0] += v.x[1][0];
    x[1][1] += v.x[1][1];
    x[1][2] += v.x[1][2];
    x[2][0] += v.x[2][0];
    x[2][1] += v.x[2][1];
    x[2][2] += v.x[2][2];

    return *this;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator += (T a)
{
    x[0][0] += a;
    x[0][1] += a;
    x[0][2] += a;
    x[1][0] += a;
    x[1][1] += a;
    x[1][2] += a;
    x[2][0] += a;
    x[2][1] += a;
    x[2][2] += a;
  
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::operator + (const Matrix33<T> &v) const
{
    return Matrix33 (x[0][0] + v.x[0][0],
                 x[0][1] + v.x[0][1],
                 x[0][2] + v.x[0][2],
                 x[1][0] + v.x[1][0],
                 x[1][1] + v.x[1][1],
                 x[1][2] + v.x[1][2],
                 x[2][0] + v.x[2][0],
                 x[2][1] + v.x[2][1],
                 x[2][2] + v.x[2][2]);
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator -= (const Matrix33<T> &v)
{
    x[0][0] -= v.x[0][0];
    x[0][1] -= v.x[0][1];
    x[0][2] -= v.x[0][2];
    x[1][0] -= v.x[1][0];
    x[1][1] -= v.x[1][1];
    x[1][2] -= v.x[1][2];
    x[2][0] -= v.x[2][0];
    x[2][1] -= v.x[2][1];
    x[2][2] -= v.x[2][2];
  
    return *this;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator -= (T a)
{
    x[0][0] -= a;
    x[0][1] -= a;
    x[0][2] -= a;
    x[1][0] -= a;
    x[1][1] -= a;
    x[1][2] -= a;
    x[2][0] -= a;
    x[2][1] -= a;
    x[2][2] -= a;
  
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::operator - (const Matrix33<T> &v) const
{
    return Matrix33 (x[0][0] - v.x[0][0],
                 x[0][1] - v.x[0][1],
                 x[0][2] - v.x[0][2],
                 x[1][0] - v.x[1][0],
                 x[1][1] - v.x[1][1],
                 x[1][2] - v.x[1][2],
                 x[2][0] - v.x[2][0],
                 x[2][1] - v.x[2][1],
                 x[2][2] - v.x[2][2]);
}

template <class T>
Matrix33<T>
Matrix33<T>::operator - () const
{
    return Matrix33 (-x[0][0],
                 -x[0][1],
                 -x[0][2],
                 -x[1][0],
                 -x[1][1],
                 -x[1][2],
                 -x[2][0],
                 -x[2][1],
                 -x[2][2]);
}

template <class T>
const Matrix33<T> &
Matrix33<T>::negate ()
{
    x[0][0] = -x[0][0];
    x[0][1] = -x[0][1];
    x[0][2] = -x[0][2];
    x[1][0] = -x[1][0];
    x[1][1] = -x[1][1];
    x[1][2] = -x[1][2];
    x[2][0] = -x[2][0];
    x[2][1] = -x[2][1];
    x[2][2] = -x[2][2];

    return *this;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator *= (T a)
{
    x[0][0] *= a;
    x[0][1] *= a;
    x[0][2] *= a;
    x[1][0] *= a;
    x[1][1] *= a;
    x[1][2] *= a;
    x[2][0] *= a;
    x[2][1] *= a;
    x[2][2] *= a;
  
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::operator * (T a) const
{
    return Matrix33 (x[0][0] * a,
                 x[0][1] * a,
                 x[0][2] * a,
                 x[1][0] * a,
                 x[1][1] * a,
                 x[1][2] * a,
                 x[2][0] * a,
                 x[2][1] * a,
                 x[2][2] * a);
}

template <class T>
inline Matrix33<T>
operator * (T a, const Matrix33<T> &v)
{
    return v * a;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator *= (const Matrix33<T> &v)
{
    Matrix33 tmp (T (0));

    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
          for (int k = 0; k < 3; k++)
            tmp.x[i][j] += x[i][k] * v.x[k][j];

    *this = tmp;
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::operator * (const Matrix33<T> &v) const
{
    Matrix33 tmp (T (0));

    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
          for (int k = 0; k < 3; k++)
            tmp.x[i][j] += x[i][k] * v.x[k][j];

    return tmp;
}

template <class T>
template <class S>
void
Matrix33<T>::multVecMatrix(const Vec2<S> &src, Vec2<S> &dst) const
{
    S a, b, w;

    a = src[0] * x[0][0] + src[1] * x[1][0] + x[2][0];
    b = src[0] * x[0][1] + src[1] * x[1][1] + x[2][1];
    w = src[0] * x[0][2] + src[1] * x[1][2] + x[2][2];

    dst.x = a / w;
    dst.y = b / w;
}

template <class T>
template <class S>
void
Matrix33<T>::multDirMatrix(const Vec2<S> &src, Vec2<S> &dst) const
{
    S a, b;

    a = src[0] * x[0][0] + src[1] * x[1][0];
    b = src[0] * x[0][1] + src[1] * x[1][1];

    dst.x = a;
    dst.y = b;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::operator /= (T a)
{
    x[0][0] /= a;
    x[0][1] /= a;
    x[0][2] /= a;
    x[1][0] /= a;
    x[1][1] /= a;
    x[1][2] /= a;
    x[2][0] /= a;
    x[2][1] /= a;
    x[2][2] /= a;
    x[2][3] /= a;
  
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::operator / (T a) const
{
    return Matrix33 (x[0][0] / a,
                 x[0][1] / a,
                 x[0][2] / a,
                 x[1][0] / a,
                 x[1][1] / a,
                 x[1][2] / a,
                 x[2][0] / a,
                 x[2][1] / a,
                 x[2][2] / a);
}

template <class T>
const Matrix33<T> &
Matrix33<T>::transpose ()
{
    Matrix33 tmp (x[0][0],
              x[1][0],
              x[2][0],
              x[0][1],
              x[1][1],
              x[2][1],
              x[0][2],
              x[1][2],
              x[2][2]);
    *this = tmp;
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::transposed () const
{
    return Matrix33 (x[0][0],
                 x[1][0],
                 x[2][0],
                 x[0][1],
                 x[1][1],
                 x[2][1],
                 x[0][2],
                 x[1][2],
                 x[2][2]);
}

template <class T>
const Matrix33<T> &
Matrix33<T>::gjInvert (bool singExc) throw (Iex::MathExc)
{
    *this = gjInverse (singExc);
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::gjInverse (bool singExc) const throw (Iex::MathExc)
{
    int i, j, k;
    Matrix33 s;
    Matrix33 t (*this);

    // Forward elimination

    for (i = 0; i < 2 ; i++)
    {
      int pivot = i;

      T pivotsize = t[i][i];

      if (pivotsize < 0)
          pivotsize = -pivotsize;

      for (j = i + 1; j < 3; j++)
      {
          T tmp = t[j][i];

          if (tmp < 0)
            tmp = -tmp;

          if (tmp > pivotsize)
          {
            pivot = j;
            pivotsize = tmp;
          }
      }

      if (pivotsize == 0)
      {
          if (singExc)
            throw ::Imath::SingMatrixExc ("Cannot invert singular matrix.");

          return Matrix33();
      }

      if (pivot != i)
      {
          for (j = 0; j < 3; j++)
          {
            T tmp;

            tmp = t[i][j];
            t[i][j] = t[pivot][j];
            t[pivot][j] = tmp;

            tmp = s[i][j];
            s[i][j] = s[pivot][j];
            s[pivot][j] = tmp;
          }
      }

      for (j = i + 1; j < 3; j++)
      {
          T f = t[j][i] / t[i][i];

          for (k = 0; k < 3; k++)
          {
            t[j][k] -= f * t[i][k];
            s[j][k] -= f * s[i][k];
          }
      }
    }

    // Backward substitution

    for (i = 2; i >= 0; --i)
    {
      T f;

      if ((f = t[i][i]) == 0)
      {
          if (singExc)
            throw ::Imath::SingMatrixExc ("Cannot invert singular matrix.");

          return Matrix33();
      }

      for (j = 0; j < 3; j++)
      {
          t[i][j] /= f;
          s[i][j] /= f;
      }

      for (j = 0; j < i; j++)
      {
          f = t[j][i];

          for (k = 0; k < 3; k++)
          {
            t[j][k] -= f * t[i][k];
            s[j][k] -= f * s[i][k];
          }
      }
    }

    return s;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::invert (bool singExc) throw (Iex::MathExc)
{
    *this = inverse (singExc);
    return *this;
}

template <class T>
Matrix33<T>
Matrix33<T>::inverse (bool singExc) const throw (Iex::MathExc)
{
    if (x[0][2] != 0 || x[1][2] != 0 || x[2][2] != 1)
    {
      Matrix33 s (x[1][1] * x[2][2] - x[2][1] * x[1][2],
                x[2][1] * x[0][2] - x[0][1] * x[2][2],
                x[0][1] * x[1][2] - x[1][1] * x[0][2],

                x[2][0] * x[1][2] - x[1][0] * x[2][2],
                x[0][0] * x[2][2] - x[2][0] * x[0][2],
                x[1][0] * x[0][2] - x[0][0] * x[1][2],

                x[1][0] * x[2][1] - x[2][0] * x[1][1],
                x[2][0] * x[0][1] - x[0][0] * x[2][1],
                x[0][0] * x[1][1] - x[1][0] * x[0][1]);

      T r = x[0][0] * s[0][0] + x[0][1] * s[1][0] + x[0][2] * s[2][0];

      if (Imath::abs (r) >= 1)
      {
          for (int i = 0; i < 3; ++i)
          {
            for (int j = 0; j < 3; ++j)
            {
                s[i][j] /= r;
            }
          }
      }
      else
      {
          T mr = Imath::abs (r) / limits<T>::smallest();

          for (int i = 0; i < 3; ++i)
          {
            for (int j = 0; j < 3; ++j)
            {
                if (mr > Imath::abs (s[i][j]))
                {
                  s[i][j] /= r;
                }
                else
                {
                  if (singExc)
                      throw SingMatrixExc ("Cannot invert "
                                     "singular matrix.");
                  return Matrix33();
                }
            }
          }
      }

      return s;
    }
    else
    {
      Matrix33 s ( x[1][1],
                -x[0][1],
                 0, 

                -x[1][0],
                 x[0][0],
                 0,

                 0,
                 0,
                 1);

      T r = x[0][0] * x[1][1] - x[1][0] * x[0][1];

      if (Imath::abs (r) >= 1)
      {
          for (int i = 0; i < 2; ++i)
          {
            for (int j = 0; j < 2; ++j)
            {
                s[i][j] /= r;
            }
          }
      }
      else
      {
          T mr = Imath::abs (r) / limits<T>::smallest();

          for (int i = 0; i < 2; ++i)
          {
            for (int j = 0; j < 2; ++j)
            {
                if (mr > Imath::abs (s[i][j]))
                {
                  s[i][j] /= r;
                }
                else
                {
                  if (singExc)
                      throw SingMatrixExc ("Cannot invert "
                                     "singular matrix.");
                  return Matrix33();
                }
            }
          }
      }

      s[2][0] = -x[2][0] * s[0][0] - x[2][1] * s[1][0];
      s[2][1] = -x[2][0] * s[0][1] - x[2][1] * s[1][1];

      return s;
    }
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::setRotation (S r)
{
    S cos_r, sin_r;

    cos_r = Math<T>::cos (r);
    sin_r = Math<T>::sin (r);

    x[0][0] =  cos_r;
    x[0][1] =  sin_r;
    x[0][2] =  0;

    x[1][0] =  -sin_r;
    x[1][1] =  cos_r;
    x[1][2] =  0;

    x[2][0] =  0;
    x[2][1] =  0;
    x[2][2] =  1;

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::rotate (S r)
{
    *this *= Matrix33<T>().setRotation (r);
    return *this;
}

template <class T>
const Matrix33<T> &
Matrix33<T>::setScale (T s)
{
    x[0][0] = s;
    x[0][1] = 0;
    x[0][2] = 0;

    x[1][0] = 0;
    x[1][1] = s;
    x[1][2] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::setScale (const Vec2<S> &s)
{
    x[0][0] = s[0];
    x[0][1] = 0;
    x[0][2] = 0;

    x[1][0] = 0;
    x[1][1] = s[1];
    x[1][2] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::scale (const Vec2<S> &s)
{
    x[0][0] *= s[0];
    x[0][1] *= s[0];
    x[0][2] *= s[0];

    x[1][0] *= s[1];
    x[1][1] *= s[1];
    x[1][2] *= s[1];

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::setTranslation (const Vec2<S> &t)
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;

    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;

    x[2][0] = t[0];
    x[2][1] = t[1];
    x[2][2] = 1;

    return *this;
}

template <class T>
inline Vec2<T> 
Matrix33<T>::translation () const
{
    return Vec2<T> (x[2][0], x[2][1]);
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::translate (const Vec2<S> &t)
{
    x[2][0] += t[0] * x[0][0] + t[1] * x[1][0];
    x[2][1] += t[0] * x[0][1] + t[1] * x[1][1];
    x[2][2] += t[0] * x[0][2] + t[1] * x[1][2];

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::setShear (const S &xy)
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;

    x[1][0] = xy;
    x[1][1] = 1;
    x[1][2] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::setShear (const Vec2<S> &h)
{
    x[0][0] = 1;
    x[0][1] = h[1];
    x[0][2] = 0;

    x[1][0] = h[0];
    x[1][1] = 1;
    x[1][2] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::shear (const S &xy)
{
    //
    // In this case, we don't need a temp. copy of the matrix 
    // because we never use a value on the RHS after we've 
    // changed it on the LHS.
    // 

    x[1][0] += xy * x[0][0];
    x[1][1] += xy * x[0][1];
    x[1][2] += xy * x[0][2];

    return *this;
}

template <class T>
template <class S>
const Matrix33<T> &
Matrix33<T>::shear (const Vec2<S> &h)
{
    Matrix33<T> P (*this);
    
    x[0][0] = P[0][0] + h[1] * P[1][0];
    x[0][1] = P[0][1] + h[1] * P[1][1];
    x[0][2] = P[0][2] + h[1] * P[1][2];
    
    x[1][0] = P[1][0] + h[0] * P[0][0];
    x[1][1] = P[1][1] + h[0] * P[0][1];
    x[1][2] = P[1][2] + h[0] * P[0][2];

    return *this;
}


//---------------------------
// Implementation of Matrix44
//---------------------------

template <class T>
inline T *
Matrix44<T>::operator [] (int i)
{
    return x[i];
}

template <class T>
inline const T *
Matrix44<T>::operator [] (int i) const
{
    return x[i];
}

template <class T>
inline
Matrix44<T>::Matrix44 ()
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;
    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;
    x[1][3] = 0;
    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;
    x[2][3] = 0;
    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;
}

template <class T>
inline
Matrix44<T>::Matrix44 (T a)
{
    x[0][0] = a;
    x[0][1] = a;
    x[0][2] = a;
    x[0][3] = a;
    x[1][0] = a;
    x[1][1] = a;
    x[1][2] = a;
    x[1][3] = a;
    x[2][0] = a;
    x[2][1] = a;
    x[2][2] = a;
    x[2][3] = a;
    x[3][0] = a;
    x[3][1] = a;
    x[3][2] = a;
    x[3][3] = a;
}

template <class T>
inline
Matrix44<T>::Matrix44 (const T a[4][4]) 
{
    x[0][0] = a[0][0];
    x[0][1] = a[0][1];
    x[0][2] = a[0][2];
    x[0][3] = a[0][3];
    x[1][0] = a[1][0];
    x[1][1] = a[1][1];
    x[1][2] = a[1][2];
    x[1][3] = a[1][3];
    x[2][0] = a[2][0];
    x[2][1] = a[2][1];
    x[2][2] = a[2][2];
    x[2][3] = a[2][3];
    x[3][0] = a[3][0];
    x[3][1] = a[3][1];
    x[3][2] = a[3][2];
    x[3][3] = a[3][3];
}

template <class T>
inline
Matrix44<T>::Matrix44 (T a, T b, T c, T d, T e, T f, T g, T h,
                   T i, T j, T k, T l, T m, T n, T o, T p)
{
    x[0][0] = a;
    x[0][1] = b;
    x[0][2] = c;
    x[0][3] = d;
    x[1][0] = e;
    x[1][1] = f;
    x[1][2] = g;
    x[1][3] = h;
    x[2][0] = i;
    x[2][1] = j;
    x[2][2] = k;
    x[2][3] = l;
    x[3][0] = m;
    x[3][1] = n;
    x[3][2] = o;
    x[3][3] = p;
}


template <class T>
inline
Matrix44<T>::Matrix44 (Matrix33<T> r, Vec3<T> t)
{
    x[0][0] = r[0][0];
    x[0][1] = r[0][1];
    x[0][2] = r[0][2];
    x[0][3] = 0;
    x[1][0] = r[1][0];
    x[1][1] = r[1][1];
    x[1][2] = r[1][2];
    x[1][3] = 0;
    x[2][0] = r[2][0];
    x[2][1] = r[2][1];
    x[2][2] = r[2][2];
    x[2][3] = 0;
    x[3][0] = t[0];
    x[3][1] = t[1];
    x[3][2] = t[2];
    x[3][3] = 1;
}

template <class T>
inline
Matrix44<T>::Matrix44 (const Matrix44 &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[0][3] = v.x[0][3];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[1][3] = v.x[1][3];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    x[2][3] = v.x[2][3];
    x[3][0] = v.x[3][0];
    x[3][1] = v.x[3][1];
    x[3][2] = v.x[3][2];
    x[3][3] = v.x[3][3];
}

template <class T>
inline const Matrix44<T> &
Matrix44<T>::operator = (const Matrix44 &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[0][3] = v.x[0][3];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[1][3] = v.x[1][3];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    x[2][3] = v.x[2][3];
    x[3][0] = v.x[3][0];
    x[3][1] = v.x[3][1];
    x[3][2] = v.x[3][2];
    x[3][3] = v.x[3][3];
    return *this;
}

template <class T>
inline const Matrix44<T> &
Matrix44<T>::operator = (T a)
{
    x[0][0] = a;
    x[0][1] = a;
    x[0][2] = a;
    x[0][3] = a;
    x[1][0] = a;
    x[1][1] = a;
    x[1][2] = a;
    x[1][3] = a;
    x[2][0] = a;
    x[2][1] = a;
    x[2][2] = a;
    x[2][3] = a;
    x[3][0] = a;
    x[3][1] = a;
    x[3][2] = a;
    x[3][3] = a;
    return *this;
}

template <class T>
inline T *
Matrix44<T>::getValue ()
{
    return (T *) &x[0][0];
}

template <class T>
inline const T *
Matrix44<T>::getValue () const
{
    return (const T *) &x[0][0];
}

template <class T>
template <class S>
inline void
Matrix44<T>::getValue (Matrix44<S> &v) const
{
    v.x[0][0] = x[0][0];
    v.x[0][1] = x[0][1];
    v.x[0][2] = x[0][2];
    v.x[0][3] = x[0][3];
    v.x[1][0] = x[1][0];
    v.x[1][1] = x[1][1];
    v.x[1][2] = x[1][2];
    v.x[1][3] = x[1][3];
    v.x[2][0] = x[2][0];
    v.x[2][1] = x[2][1];
    v.x[2][2] = x[2][2];
    v.x[2][3] = x[2][3];
    v.x[3][0] = x[3][0];
    v.x[3][1] = x[3][1];
    v.x[3][2] = x[3][2];
    v.x[3][3] = x[3][3];
}

template <class T>
template <class S>
inline Matrix44<T> &
Matrix44<T>::setValue (const Matrix44<S> &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[0][3] = v.x[0][3];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[1][3] = v.x[1][3];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    x[2][3] = v.x[2][3];
    x[3][0] = v.x[3][0];
    x[3][1] = v.x[3][1];
    x[3][2] = v.x[3][2];
    x[3][3] = v.x[3][3];
    return *this;
}

template <class T>
template <class S>
inline Matrix44<T> &
Matrix44<T>::setTheMatrix (const Matrix44<S> &v)
{
    x[0][0] = v.x[0][0];
    x[0][1] = v.x[0][1];
    x[0][2] = v.x[0][2];
    x[0][3] = v.x[0][3];
    x[1][0] = v.x[1][0];
    x[1][1] = v.x[1][1];
    x[1][2] = v.x[1][2];
    x[1][3] = v.x[1][3];
    x[2][0] = v.x[2][0];
    x[2][1] = v.x[2][1];
    x[2][2] = v.x[2][2];
    x[2][3] = v.x[2][3];
    x[3][0] = v.x[3][0];
    x[3][1] = v.x[3][1];
    x[3][2] = v.x[3][2];
    x[3][3] = v.x[3][3];
    return *this;
}

template <class T>
inline void
Matrix44<T>::makeIdentity()
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;
    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;
    x[1][3] = 0;
    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;
    x[2][3] = 0;
    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;
}

template <class T>
bool
Matrix44<T>::operator == (const Matrix44 &v) const
{
    return x[0][0] == v.x[0][0] &&
         x[0][1] == v.x[0][1] &&
         x[0][2] == v.x[0][2] &&
         x[0][3] == v.x[0][3] &&
         x[1][0] == v.x[1][0] &&
         x[1][1] == v.x[1][1] &&
         x[1][2] == v.x[1][2] &&
         x[1][3] == v.x[1][3] &&
         x[2][0] == v.x[2][0] &&
         x[2][1] == v.x[2][1] &&
         x[2][2] == v.x[2][2] &&
         x[2][3] == v.x[2][3] &&
         x[3][0] == v.x[3][0] &&
         x[3][1] == v.x[3][1] &&
         x[3][2] == v.x[3][2] &&
         x[3][3] == v.x[3][3];
}

template <class T>
bool
Matrix44<T>::operator != (const Matrix44 &v) const
{
    return x[0][0] != v.x[0][0] ||
         x[0][1] != v.x[0][1] ||
         x[0][2] != v.x[0][2] ||
         x[0][3] != v.x[0][3] ||
         x[1][0] != v.x[1][0] ||
         x[1][1] != v.x[1][1] ||
         x[1][2] != v.x[1][2] ||
         x[1][3] != v.x[1][3] ||
         x[2][0] != v.x[2][0] ||
         x[2][1] != v.x[2][1] ||
         x[2][2] != v.x[2][2] ||
         x[2][3] != v.x[2][3] ||
         x[3][0] != v.x[3][0] ||
         x[3][1] != v.x[3][1] ||
         x[3][2] != v.x[3][2] ||
         x[3][3] != v.x[3][3];
}

template <class T>
bool
Matrix44<T>::equalWithAbsError (const Matrix44<T> &m, T e) const
{
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
          if (!Imath::equalWithAbsError ((*this)[i][j], m[i][j], e))
            return false;

    return true;
}

template <class T>
bool
Matrix44<T>::equalWithRelError (const Matrix44<T> &m, T e) const
{
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
          if (!Imath::equalWithRelError ((*this)[i][j], m[i][j], e))
            return false;

    return true;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator += (const Matrix44<T> &v)
{
    x[0][0] += v.x[0][0];
    x[0][1] += v.x[0][1];
    x[0][2] += v.x[0][2];
    x[0][3] += v.x[0][3];
    x[1][0] += v.x[1][0];
    x[1][1] += v.x[1][1];
    x[1][2] += v.x[1][2];
    x[1][3] += v.x[1][3];
    x[2][0] += v.x[2][0];
    x[2][1] += v.x[2][1];
    x[2][2] += v.x[2][2];
    x[2][3] += v.x[2][3];
    x[3][0] += v.x[3][0];
    x[3][1] += v.x[3][1];
    x[3][2] += v.x[3][2];
    x[3][3] += v.x[3][3];

    return *this;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator += (T a)
{
    x[0][0] += a;
    x[0][1] += a;
    x[0][2] += a;
    x[0][3] += a;
    x[1][0] += a;
    x[1][1] += a;
    x[1][2] += a;
    x[1][3] += a;
    x[2][0] += a;
    x[2][1] += a;
    x[2][2] += a;
    x[2][3] += a;
    x[3][0] += a;
    x[3][1] += a;
    x[3][2] += a;
    x[3][3] += a;

    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::operator + (const Matrix44<T> &v) const
{
    return Matrix44 (x[0][0] + v.x[0][0],
                 x[0][1] + v.x[0][1],
                 x[0][2] + v.x[0][2],
                 x[0][3] + v.x[0][3],
                 x[1][0] + v.x[1][0],
                 x[1][1] + v.x[1][1],
                 x[1][2] + v.x[1][2],
                 x[1][3] + v.x[1][3],
                 x[2][0] + v.x[2][0],
                 x[2][1] + v.x[2][1],
                 x[2][2] + v.x[2][2],
                 x[2][3] + v.x[2][3],
                 x[3][0] + v.x[3][0],
                 x[3][1] + v.x[3][1],
                 x[3][2] + v.x[3][2],
                 x[3][3] + v.x[3][3]);
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator -= (const Matrix44<T> &v)
{
    x[0][0] -= v.x[0][0];
    x[0][1] -= v.x[0][1];
    x[0][2] -= v.x[0][2];
    x[0][3] -= v.x[0][3];
    x[1][0] -= v.x[1][0];
    x[1][1] -= v.x[1][1];
    x[1][2] -= v.x[1][2];
    x[1][3] -= v.x[1][3];
    x[2][0] -= v.x[2][0];
    x[2][1] -= v.x[2][1];
    x[2][2] -= v.x[2][2];
    x[2][3] -= v.x[2][3];
    x[3][0] -= v.x[3][0];
    x[3][1] -= v.x[3][1];
    x[3][2] -= v.x[3][2];
    x[3][3] -= v.x[3][3];

    return *this;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator -= (T a)
{
    x[0][0] -= a;
    x[0][1] -= a;
    x[0][2] -= a;
    x[0][3] -= a;
    x[1][0] -= a;
    x[1][1] -= a;
    x[1][2] -= a;
    x[1][3] -= a;
    x[2][0] -= a;
    x[2][1] -= a;
    x[2][2] -= a;
    x[2][3] -= a;
    x[3][0] -= a;
    x[3][1] -= a;
    x[3][2] -= a;
    x[3][3] -= a;

    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::operator - (const Matrix44<T> &v) const
{
    return Matrix44 (x[0][0] - v.x[0][0],
                 x[0][1] - v.x[0][1],
                 x[0][2] - v.x[0][2],
                 x[0][3] - v.x[0][3],
                 x[1][0] - v.x[1][0],
                 x[1][1] - v.x[1][1],
                 x[1][2] - v.x[1][2],
                 x[1][3] - v.x[1][3],
                 x[2][0] - v.x[2][0],
                 x[2][1] - v.x[2][1],
                 x[2][2] - v.x[2][2],
                 x[2][3] - v.x[2][3],
                 x[3][0] - v.x[3][0],
                 x[3][1] - v.x[3][1],
                 x[3][2] - v.x[3][2],
                 x[3][3] - v.x[3][3]);
}

template <class T>
Matrix44<T>
Matrix44<T>::operator - () const
{
    return Matrix44 (-x[0][0],
                 -x[0][1],
                 -x[0][2],
                 -x[0][3],
                 -x[1][0],
                 -x[1][1],
                 -x[1][2],
                 -x[1][3],
                 -x[2][0],
                 -x[2][1],
                 -x[2][2],
                 -x[2][3],
                 -x[3][0],
                 -x[3][1],
                 -x[3][2],
                 -x[3][3]);
}

template <class T>
const Matrix44<T> &
Matrix44<T>::negate ()
{
    x[0][0] = -x[0][0];
    x[0][1] = -x[0][1];
    x[0][2] = -x[0][2];
    x[0][3] = -x[0][3];
    x[1][0] = -x[1][0];
    x[1][1] = -x[1][1];
    x[1][2] = -x[1][2];
    x[1][3] = -x[1][3];
    x[2][0] = -x[2][0];
    x[2][1] = -x[2][1];
    x[2][2] = -x[2][2];
    x[2][3] = -x[2][3];
    x[3][0] = -x[3][0];
    x[3][1] = -x[3][1];
    x[3][2] = -x[3][2];
    x[3][3] = -x[3][3];

    return *this;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator *= (T a)
{
    x[0][0] *= a;
    x[0][1] *= a;
    x[0][2] *= a;
    x[0][3] *= a;
    x[1][0] *= a;
    x[1][1] *= a;
    x[1][2] *= a;
    x[1][3] *= a;
    x[2][0] *= a;
    x[2][1] *= a;
    x[2][2] *= a;
    x[2][3] *= a;
    x[3][0] *= a;
    x[3][1] *= a;
    x[3][2] *= a;
    x[3][3] *= a;

    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::operator * (T a) const
{
    return Matrix44 (x[0][0] * a,
                 x[0][1] * a,
                 x[0][2] * a,
                 x[0][3] * a,
                 x[1][0] * a,
                 x[1][1] * a,
                 x[1][2] * a,
                 x[1][3] * a,
                 x[2][0] * a,
                 x[2][1] * a,
                 x[2][2] * a,
                 x[2][3] * a,
                 x[3][0] * a,
                 x[3][1] * a,
                 x[3][2] * a,
                 x[3][3] * a);
}

template <class T>
inline Matrix44<T>
operator * (T a, const Matrix44<T> &v)
{
    return v * a;
}

template <class T>
inline const Matrix44<T> &
Matrix44<T>::operator *= (const Matrix44<T> &v)
{
    Matrix44 tmp (T (0));

    multiply (*this, v, tmp);
    *this = tmp;
    return *this;
}

template <class T>
inline Matrix44<T>
Matrix44<T>::operator * (const Matrix44<T> &v) const
{
    Matrix44 tmp (T (0));

    multiply (*this, v, tmp);
    return tmp;
}

template <class T>
void
Matrix44<T>::multiply (const Matrix44<T> &a,
                   const Matrix44<T> &b,
                   Matrix44<T> &c)
{
    register const T * restrict ap = &a.x[0][0];
    register const T * restrict bp = &b.x[0][0];
    register       T * restrict cp = &c.x[0][0];

    register T a0, a1, a2, a3;

    a0 = ap[0];
    a1 = ap[1];
    a2 = ap[2];
    a3 = ap[3];

    cp[0]  = a0 * bp[0]  + a1 * bp[4]  + a2 * bp[8]  + a3 * bp[12];
    cp[1]  = a0 * bp[1]  + a1 * bp[5]  + a2 * bp[9]  + a3 * bp[13];
    cp[2]  = a0 * bp[2]  + a1 * bp[6]  + a2 * bp[10] + a3 * bp[14];
    cp[3]  = a0 * bp[3]  + a1 * bp[7]  + a2 * bp[11] + a3 * bp[15];

    a0 = ap[4];
    a1 = ap[5];
    a2 = ap[6];
    a3 = ap[7];

    cp[4]  = a0 * bp[0]  + a1 * bp[4]  + a2 * bp[8]  + a3 * bp[12];
    cp[5]  = a0 * bp[1]  + a1 * bp[5]  + a2 * bp[9]  + a3 * bp[13];
    cp[6]  = a0 * bp[2]  + a1 * bp[6]  + a2 * bp[10] + a3 * bp[14];
    cp[7]  = a0 * bp[3]  + a1 * bp[7]  + a2 * bp[11] + a3 * bp[15];

    a0 = ap[8];
    a1 = ap[9];
    a2 = ap[10];
    a3 = ap[11];

    cp[8]  = a0 * bp[0]  + a1 * bp[4]  + a2 * bp[8]  + a3 * bp[12];
    cp[9]  = a0 * bp[1]  + a1 * bp[5]  + a2 * bp[9]  + a3 * bp[13];
    cp[10] = a0 * bp[2]  + a1 * bp[6]  + a2 * bp[10] + a3 * bp[14];
    cp[11] = a0 * bp[3]  + a1 * bp[7]  + a2 * bp[11] + a3 * bp[15];

    a0 = ap[12];
    a1 = ap[13];
    a2 = ap[14];
    a3 = ap[15];

    cp[12] = a0 * bp[0]  + a1 * bp[4]  + a2 * bp[8]  + a3 * bp[12];
    cp[13] = a0 * bp[1]  + a1 * bp[5]  + a2 * bp[9]  + a3 * bp[13];
    cp[14] = a0 * bp[2]  + a1 * bp[6]  + a2 * bp[10] + a3 * bp[14];
    cp[15] = a0 * bp[3]  + a1 * bp[7]  + a2 * bp[11] + a3 * bp[15];
}

template <class T> template <class S>
void
Matrix44<T>::multVecMatrix(const Vec3<S> &src, Vec3<S> &dst) const
{
    S a, b, c, w;

    a = src[0] * x[0][0] + src[1] * x[1][0] + src[2] * x[2][0] + x[3][0];
    b = src[0] * x[0][1] + src[1] * x[1][1] + src[2] * x[2][1] + x[3][1];
    c = src[0] * x[0][2] + src[1] * x[1][2] + src[2] * x[2][2] + x[3][2];
    w = src[0] * x[0][3] + src[1] * x[1][3] + src[2] * x[2][3] + x[3][3];

    dst.x = a / w;
    dst.y = b / w;
    dst.z = c / w;
}

template <class T> template <class S>
void
Matrix44<T>::multDirMatrix(const Vec3<S> &src, Vec3<S> &dst) const
{
    S a, b, c;

    a = src[0] * x[0][0] + src[1] * x[1][0] + src[2] * x[2][0];
    b = src[0] * x[0][1] + src[1] * x[1][1] + src[2] * x[2][1];
    c = src[0] * x[0][2] + src[1] * x[1][2] + src[2] * x[2][2];

    dst.x = a;
    dst.y = b;
    dst.z = c;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::operator /= (T a)
{
    x[0][0] /= a;
    x[0][1] /= a;
    x[0][2] /= a;
    x[0][3] /= a;
    x[1][0] /= a;
    x[1][1] /= a;
    x[1][2] /= a;
    x[1][3] /= a;
    x[2][0] /= a;
    x[2][1] /= a;
    x[2][2] /= a;
    x[2][3] /= a;
    x[3][0] /= a;
    x[3][1] /= a;
    x[3][2] /= a;
    x[3][3] /= a;

    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::operator / (T a) const
{
    return Matrix44 (x[0][0] / a,
                 x[0][1] / a,
                 x[0][2] / a,
                 x[0][3] / a,
                 x[1][0] / a,
                 x[1][1] / a,
                 x[1][2] / a,
                 x[1][3] / a,
                 x[2][0] / a,
                 x[2][1] / a,
                 x[2][2] / a,
                 x[2][3] / a,
                 x[3][0] / a,
                 x[3][1] / a,
                 x[3][2] / a,
                 x[3][3] / a);
}

template <class T>
const Matrix44<T> &
Matrix44<T>::transpose ()
{
    Matrix44 tmp (x[0][0],
              x[1][0],
              x[2][0],
              x[3][0],
              x[0][1],
              x[1][1],
              x[2][1],
              x[3][1],
              x[0][2],
              x[1][2],
              x[2][2],
              x[3][2],
              x[0][3],
              x[1][3],
              x[2][3],
              x[3][3]);
    *this = tmp;
    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::transposed () const
{
    return Matrix44 (x[0][0],
                 x[1][0],
                 x[2][0],
                 x[3][0],
                 x[0][1],
                 x[1][1],
                 x[2][1],
                 x[3][1],
                 x[0][2],
                 x[1][2],
                 x[2][2],
                 x[3][2],
                 x[0][3],
                 x[1][3],
                 x[2][3],
                 x[3][3]);
}

template <class T>
const Matrix44<T> &
Matrix44<T>::gjInvert (bool singExc) throw (Iex::MathExc)
{
    *this = gjInverse (singExc);
    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::gjInverse (bool singExc) const throw (Iex::MathExc)
{
    int i, j, k;
    Matrix44 s;
    Matrix44 t (*this);

    // Forward elimination

    for (i = 0; i < 3 ; i++)
    {
      int pivot = i;

      T pivotsize = t[i][i];

      if (pivotsize < 0)
          pivotsize = -pivotsize;

      for (j = i + 1; j < 4; j++)
      {
          T tmp = t[j][i];

          if (tmp < 0)
            tmp = -tmp;

          if (tmp > pivotsize)
          {
            pivot = j;
            pivotsize = tmp;
          }
      }

      if (pivotsize == 0)
      {
          if (singExc)
            throw ::Imath::SingMatrixExc ("Cannot invert singular matrix.");

          return Matrix44();
      }

      if (pivot != i)
      {
          for (j = 0; j < 4; j++)
          {
            T tmp;

            tmp = t[i][j];
            t[i][j] = t[pivot][j];
            t[pivot][j] = tmp;

            tmp = s[i][j];
            s[i][j] = s[pivot][j];
            s[pivot][j] = tmp;
          }
      }

      for (j = i + 1; j < 4; j++)
      {
          T f = t[j][i] / t[i][i];

          for (k = 0; k < 4; k++)
          {
            t[j][k] -= f * t[i][k];
            s[j][k] -= f * s[i][k];
          }
      }
    }

    // Backward substitution

    for (i = 3; i >= 0; --i)
    {
      T f;

      if ((f = t[i][i]) == 0)
      {
          if (singExc)
            throw ::Imath::SingMatrixExc ("Cannot invert singular matrix.");

          return Matrix44();
      }

      for (j = 0; j < 4; j++)
      {
          t[i][j] /= f;
          s[i][j] /= f;
      }

      for (j = 0; j < i; j++)
      {
          f = t[j][i];

          for (k = 0; k < 4; k++)
          {
            t[j][k] -= f * t[i][k];
            s[j][k] -= f * s[i][k];
          }
      }
    }

    return s;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::invert (bool singExc) throw (Iex::MathExc)
{
    *this = inverse (singExc);
    return *this;
}

template <class T>
Matrix44<T>
Matrix44<T>::inverse (bool singExc) const throw (Iex::MathExc)
{
    if (x[0][3] != 0 || x[1][3] != 0 || x[2][3] != 0 || x[3][3] != 1)
      return gjInverse(singExc);

    Matrix44 s (x[1][1] * x[2][2] - x[2][1] * x[1][2],
            x[2][1] * x[0][2] - x[0][1] * x[2][2],
            x[0][1] * x[1][2] - x[1][1] * x[0][2],
            0,

            x[2][0] * x[1][2] - x[1][0] * x[2][2],
            x[0][0] * x[2][2] - x[2][0] * x[0][2],
            x[1][0] * x[0][2] - x[0][0] * x[1][2],
            0,

            x[1][0] * x[2][1] - x[2][0] * x[1][1],
            x[2][0] * x[0][1] - x[0][0] * x[2][1],
            x[0][0] * x[1][1] - x[1][0] * x[0][1],
            0,

            0,
            0,
            0,
            1);

    T r = x[0][0] * s[0][0] + x[0][1] * s[1][0] + x[0][2] * s[2][0];

    if (Imath::abs (r) >= 1)
    {
      for (int i = 0; i < 3; ++i)
      {
          for (int j = 0; j < 3; ++j)
          {
            s[i][j] /= r;
          }
      }
    }
    else
    {
      T mr = Imath::abs (r) / limits<T>::smallest();

      for (int i = 0; i < 3; ++i)
      {
          for (int j = 0; j < 3; ++j)
          {
            if (mr > Imath::abs (s[i][j]))
            {
                s[i][j] /= r;
            }
            else
            {
                if (singExc)
                  throw SingMatrixExc ("Cannot invert singular matrix.");

                return Matrix44();
            }
          }
      }
    }

    s[3][0] = -x[3][0] * s[0][0] - x[3][1] * s[1][0] - x[3][2] * s[2][0];
    s[3][1] = -x[3][0] * s[0][1] - x[3][1] * s[1][1] - x[3][2] * s[2][1];
    s[3][2] = -x[3][0] * s[0][2] - x[3][1] * s[1][2] - x[3][2] * s[2][2];

    return s;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setEulerAngles (const Vec3<S>& r)
{
    S cos_rz, sin_rz, cos_ry, sin_ry, cos_rx, sin_rx;
    
    cos_rz = Math<T>::cos (r[2]);
    cos_ry = Math<T>::cos (r[1]);
    cos_rx = Math<T>::cos (r[0]);
    
    sin_rz = Math<T>::sin (r[2]);
    sin_ry = Math<T>::sin (r[1]);
    sin_rx = Math<T>::sin (r[0]);
    
    x[0][0] =  cos_rz * cos_ry;
    x[0][1] =  sin_rz * cos_ry;
    x[0][2] = -sin_ry;
    x[0][3] =  0;
    
    x[1][0] = -sin_rz * cos_rx + cos_rz * sin_ry * sin_rx;
    x[1][1] =  cos_rz * cos_rx + sin_rz * sin_ry * sin_rx;
    x[1][2] =  cos_ry * sin_rx;
    x[1][3] =  0;
    
    x[2][0] =  sin_rz * sin_rx + cos_rz * sin_ry * cos_rx;
    x[2][1] = -cos_rz * sin_rx + sin_rz * sin_ry * cos_rx;
    x[2][2] =  cos_ry * cos_rx;
    x[2][3] =  0;

    x[3][0] =  0;
    x[3][1] =  0;
    x[3][2] =  0;
    x[3][3] =  1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setAxisAngle (const Vec3<S>& axis, S angle)
{
    Vec3<S> unit (axis.normalized());
    S sine   = Math<T>::sin (angle);
    S cosine = Math<T>::cos (angle);

    x[0][0] = unit[0] * unit[0] * (1 - cosine) + cosine;
    x[0][1] = unit[0] * unit[1] * (1 - cosine) + unit[2] * sine;
    x[0][2] = unit[0] * unit[2] * (1 - cosine) - unit[1] * sine;
    x[0][3] = 0;

    x[1][0] = unit[0] * unit[1] * (1 - cosine) - unit[2] * sine;
    x[1][1] = unit[1] * unit[1] * (1 - cosine) + cosine;
    x[1][2] = unit[1] * unit[2] * (1 - cosine) + unit[0] * sine;
    x[1][3] = 0;

    x[2][0] = unit[0] * unit[2] * (1 - cosine) + unit[1] * sine;
    x[2][1] = unit[1] * unit[2] * (1 - cosine) - unit[0] * sine;
    x[2][2] = unit[2] * unit[2] * (1 - cosine) + cosine;
    x[2][3] = 0;

    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::rotate (const Vec3<S> &r)
{
    S cos_rz, sin_rz, cos_ry, sin_ry, cos_rx, sin_rx;
    S m00, m01, m02;
    S m10, m11, m12;
    S m20, m21, m22;

    cos_rz = Math<S>::cos (r[2]);
    cos_ry = Math<S>::cos (r[1]);
    cos_rx = Math<S>::cos (r[0]);
    
    sin_rz = Math<S>::sin (r[2]);
    sin_ry = Math<S>::sin (r[1]);
    sin_rx = Math<S>::sin (r[0]);

    m00 =  cos_rz *  cos_ry;
    m01 =  sin_rz *  cos_ry;
    m02 = -sin_ry;
    m10 = -sin_rz *  cos_rx + cos_rz * sin_ry * sin_rx;
    m11 =  cos_rz *  cos_rx + sin_rz * sin_ry * sin_rx;
    m12 =  cos_ry *  sin_rx;
    m20 = -sin_rz * -sin_rx + cos_rz * sin_ry * cos_rx;
    m21 =  cos_rz * -sin_rx + sin_rz * sin_ry * cos_rx;
    m22 =  cos_ry *  cos_rx;

    Matrix44<T> P (*this);

    x[0][0] = P[0][0] * m00 + P[1][0] * m01 + P[2][0] * m02;
    x[0][1] = P[0][1] * m00 + P[1][1] * m01 + P[2][1] * m02;
    x[0][2] = P[0][2] * m00 + P[1][2] * m01 + P[2][2] * m02;
    x[0][3] = P[0][3] * m00 + P[1][3] * m01 + P[2][3] * m02;

    x[1][0] = P[0][0] * m10 + P[1][0] * m11 + P[2][0] * m12;
    x[1][1] = P[0][1] * m10 + P[1][1] * m11 + P[2][1] * m12;
    x[1][2] = P[0][2] * m10 + P[1][2] * m11 + P[2][2] * m12;
    x[1][3] = P[0][3] * m10 + P[1][3] * m11 + P[2][3] * m12;

    x[2][0] = P[0][0] * m20 + P[1][0] * m21 + P[2][0] * m22;
    x[2][1] = P[0][1] * m20 + P[1][1] * m21 + P[2][1] * m22;
    x[2][2] = P[0][2] * m20 + P[1][2] * m21 + P[2][2] * m22;
    x[2][3] = P[0][3] * m20 + P[1][3] * m21 + P[2][3] * m22;

    return *this;
}

template <class T>
const Matrix44<T> &
Matrix44<T>::setScale (T s)
{
    x[0][0] = s;
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;

    x[1][0] = 0;
    x[1][1] = s;
    x[1][2] = 0;
    x[1][3] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = s;
    x[2][3] = 0;

    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setScale (const Vec3<S> &s)
{
    x[0][0] = s[0];
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;

    x[1][0] = 0;
    x[1][1] = s[1];
    x[1][2] = 0;
    x[1][3] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = s[2];
    x[2][3] = 0;

    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::scale (const Vec3<S> &s)
{
    x[0][0] *= s[0];
    x[0][1] *= s[0];
    x[0][2] *= s[0];
    x[0][3] *= s[0];

    x[1][0] *= s[1];
    x[1][1] *= s[1];
    x[1][2] *= s[1];
    x[1][3] *= s[1];

    x[2][0] *= s[2];
    x[2][1] *= s[2];
    x[2][2] *= s[2];
    x[2][3] *= s[2];

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setTranslation (const Vec3<S> &t)
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;

    x[1][0] = 0;
    x[1][1] = 1;
    x[1][2] = 0;
    x[1][3] = 0;

    x[2][0] = 0;
    x[2][1] = 0;
    x[2][2] = 1;
    x[2][3] = 0;

    x[3][0] = t[0];
    x[3][1] = t[1];
    x[3][2] = t[2];
    x[3][3] = 1;

    return *this;
}

template <class T>
inline const Vec3<T>
Matrix44<T>::translation () const
{
    return Vec3<T> (x[3][0], x[3][1], x[3][2]);
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::translate (const Vec3<S> &t)
{
    x[3][0] += t[0] * x[0][0] + t[1] * x[1][0] + t[2] * x[2][0];
    x[3][1] += t[0] * x[0][1] + t[1] * x[1][1] + t[2] * x[2][1];
    x[3][2] += t[0] * x[0][2] + t[1] * x[1][2] + t[2] * x[2][2];
    x[3][3] += t[0] * x[0][3] + t[1] * x[1][3] + t[2] * x[2][3];

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setShear (const Vec3<S> &h)
{
    x[0][0] = 1;
    x[0][1] = 0;
    x[0][2] = 0;
    x[0][3] = 0;

    x[1][0] = h[0];
    x[1][1] = 1;
    x[1][2] = 0;
    x[1][3] = 0;

    x[2][0] = h[1];
    x[2][1] = h[2];
    x[2][2] = 1;
    x[2][3] = 0;

    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::setShear (const Shear6<S> &h)
{
    x[0][0] = 1;
    x[0][1] = h.yx;
    x[0][2] = h.zx;
    x[0][3] = 0;

    x[1][0] = h.xy;
    x[1][1] = 1;
    x[1][2] = h.zy;
    x[1][3] = 0;

    x[2][0] = h.xz;
    x[2][1] = h.yz;
    x[2][2] = 1;
    x[2][3] = 0;

    x[3][0] = 0;
    x[3][1] = 0;
    x[3][2] = 0;
    x[3][3] = 1;

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::shear (const Vec3<S> &h)
{
    //
    // In this case, we don't need a temp. copy of the matrix 
    // because we never use a value on the RHS after we've 
    // changed it on the LHS.
    // 

    for (int i=0; i < 4; i++)
    {
      x[2][i] += h[1] * x[0][i] + h[2] * x[1][i];
      x[1][i] += h[0] * x[0][i];
    }

    return *this;
}

template <class T>
template <class S>
const Matrix44<T> &
Matrix44<T>::shear (const Shear6<S> &h)
{
    Matrix44<T> P (*this);

    for (int i=0; i < 4; i++)
    {
      x[0][i] = P[0][i] + h.yx * P[1][i] + h.zx * P[2][i];
      x[1][i] = h.xy * P[0][i] + P[1][i] + h.zy * P[2][i];
      x[2][i] = h.xz * P[0][i] + h.yz * P[1][i] + P[2][i];
    }

    return *this;
}


//--------------------------------
// Implementation of stream output
//--------------------------------

template <class T>
std::ostream &
operator << (std::ostream &s, const Matrix33<T> &m)
{
#ifndef HAVE_IOS_BASE
    std::ios::fmtflags oldFlags = s.flags();
    int width;

    if (s.flags() & std::ios::fixed)
    {
      s.setf (std::ios::showpoint);
      width = s.precision() + 5;
    }
    else
    {
      s.setf (std::ios::scientific);
      s.setf (std::ios::showpoint);
      width = s.precision() + 8;
    }
#else
    std::ios_base::fmtflags oldFlags = s.flags();
    int width;

    if (s.flags() & std::ios_base::fixed)
    {
      s.setf (std::ios_base::showpoint);
      width = s.precision() + 5;
    }
    else
    {
      s.setf (std::ios_base::scientific);
      s.setf (std::ios_base::showpoint);
      width = s.precision() + 8;
    }
#endif

    s << "(" << std::setw (width) << m[0][0] <<
       " " << std::setw (width) << m[0][1] <<
       " " << std::setw (width) << m[0][2] << "\n" <<

       " " << std::setw (width) << m[1][0] <<
       " " << std::setw (width) << m[1][1] <<
       " " << std::setw (width) << m[1][2] << "\n" <<

       " " << std::setw (width) << m[2][0] <<
       " " << std::setw (width) << m[2][1] <<
       " " << std::setw (width) << m[2][2] << ")\n";

    s.flags (oldFlags);
    return s;
}

template <class T>
std::ostream &
operator << (std::ostream &s, const Matrix44<T> &m)
{
#ifndef HAVE_IOS_BASE
    std::ios::fmtflags oldFlags = s.flags();
    int width;

    if (s.flags() & std::ios::fixed)
    {
      s.setf (std::ios::showpoint);
      width = s.precision() + 5;
    }
    else
    {
      s.setf (std::ios::scientific);
      s.setf (std::ios::showpoint);
      width = s.precision() + 8;
    }
#else
    std::ios_base::fmtflags oldFlags = s.flags();
    int width;

    if (s.flags() & std::ios_base::fixed)
    {
      s.setf (std::ios_base::showpoint);
      width = s.precision() + 5;
    }
    else
    {
      s.setf (std::ios_base::scientific);
      s.setf (std::ios_base::showpoint);
      width = s.precision() + 8;
    }
#endif

    s << "(" << std::setw (width) << m[0][0] <<
       " " << std::setw (width) << m[0][1] <<
       " " << std::setw (width) << m[0][2] <<
       " " << std::setw (width) << m[0][3] << "\n" <<

       " " << std::setw (width) << m[1][0] <<
       " " << std::setw (width) << m[1][1] <<
       " " << std::setw (width) << m[1][2] <<
       " " << std::setw (width) << m[1][3] << "\n" <<

       " " << std::setw (width) << m[2][0] <<
       " " << std::setw (width) << m[2][1] <<
       " " << std::setw (width) << m[2][2] <<
       " " << std::setw (width) << m[2][3] << "\n" <<

       " " << std::setw (width) << m[3][0] <<
       " " << std::setw (width) << m[3][1] <<
       " " << std::setw (width) << m[3][2] <<
       " " << std::setw (width) << m[3][3] << ")\n";

    s.flags (oldFlags);
    return s;
}


//---------------------------------------------------------------
// Implementation of vector-times-matrix multiplication operators
//---------------------------------------------------------------

template <class S, class T>
inline const Vec2<S> &
operator *= (Vec2<S> &v, const Matrix33<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + m[2][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + m[2][1];
    S w = v.x * m[0][2] + v.y * m[1][2] + m[2][2];

    v.x = x / w;
    v.y = y / w;

    return v;
}

template <class S, class T>
inline Vec2<S>
operator * (const Vec2<S> &v, const Matrix33<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + m[2][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + m[2][1];
    S w = v.x * m[0][2] + v.y * m[1][2] + m[2][2];

    return Vec2<S> (x / w, y / w);
}


template <class S, class T>
inline const Vec3<S> &
operator *= (Vec3<S> &v, const Matrix33<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + v.z * m[2][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + v.z * m[2][1];
    S z = v.x * m[0][2] + v.y * m[1][2] + v.z * m[2][2];

    v.x = x;
    v.y = y;
    v.z = z;

    return v;
}


template <class S, class T>
inline Vec3<S>
operator * (const Vec3<S> &v, const Matrix33<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + v.z * m[2][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + v.z * m[2][1];
    S z = v.x * m[0][2] + v.y * m[1][2] + v.z * m[2][2];

    return Vec3<S> (x, y, z);
}


template <class S, class T>
inline const Vec3<S> &
operator *= (Vec3<S> &v, const Matrix44<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + v.z * m[2][0] + m[3][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + v.z * m[2][1] + m[3][1];
    S z = v.x * m[0][2] + v.y * m[1][2] + v.z * m[2][2] + m[3][2];
    S w = v.x * m[0][3] + v.y * m[1][3] + v.z * m[2][3] + m[3][3];

    v.x = x / w;
    v.y = y / w;
    v.z = z / w;

    return v;
}

template <class S, class T>
inline Vec3<S>
operator * (const Vec3<S> &v, const Matrix44<T> &m)
{
    S x = v.x * m[0][0] + v.y * m[1][0] + v.z * m[2][0] + m[3][0];
    S y = v.x * m[0][1] + v.y * m[1][1] + v.z * m[2][1] + m[3][1];
    S z = v.x * m[0][2] + v.y * m[1][2] + v.z * m[2][2] + m[3][2];
    S w = v.x * m[0][3] + v.y * m[1][3] + v.z * m[2][3] + m[3][3];

    return Vec3<S> (x / w, y / w, z / w);
}

} // namespace Imath



#endif

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