matrix44.h

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// -*- Mode: C++; tab-width: 2; -*-
// vi: set ts=2:
//
// $Id: matrix44.h,v 1.55.14.1 2007/03/25 21:23:45 oliver Exp $
//

#ifndef BALL_MATHS_MATRIX44_H
#define BALL_MATHS_MATRIX44_H

#ifndef BALL_COMMON_EXCEPTION_H
# include <BALL/COMMON/exception.h>
#endif

#include <math.h>
#include <iostream>
#include <stdlib.h>

#ifndef BALL_MATHS_ANGLE_H
# include <BALL/MATHS/angle.h>
#endif

#ifndef BALL_MATHS_VECTOR3_H
# include <BALL/MATHS/vector3.h>
#endif

#ifndef BALL_MATHS_VECTOR4_H
# include <BALL/MATHS/vector4.h>
#endif

namespace BALL 
{

  template <typename T>
  class TMatrix4x4;

  template <typename T>
  std::istream& operator >> (std::istream& s, TMatrix4x4<T>& m)
    ;

  template <typename T>
  std::ostream& operator << (std::ostream& s, const TMatrix4x4<T>& m)
    ;
  
  template <typename T>
  class TMatrix4x4
  {
    public:
  
    BALL_CREATE(TMatrix4x4)

    

    TMatrix4x4()
      ;

    TMatrix4x4(const T* ptr)
      throw(Exception::NullPointer);

    TMatrix4x4(const T ptr[4][4])
      throw(Exception::NullPointer);

    TMatrix4x4(const TMatrix4x4& m)
      ;

    TMatrix4x4
      (const TVector4<T>& col1, const TVector4<T>& col2,
       const TVector4<T>& col3, const TVector4<T>& col4)
      ;

    TMatrix4x4
      (const T& m11, const T& m12, const T& m13, const T& m14, 
       const T& m21, const T& m22, const T& m23, const T& m24, 
       const T& m31, const T& m32, const T& m33, const T& m34, 
       const T& m41, const T& m42, const T& m43, const T& m44)
      ;

    virtual ~TMatrix4x4()
      
    {
    }

    virtual void clear() 
      ;



    void set(const T* ptr)
      throw(Exception::NullPointer);

    void set(const T ptr[4][4])
      throw(Exception::NullPointer);

    void set(const TMatrix4x4& m)
      ;

    void set
      (const TVector4<T>& col1, const TVector4<T>& col2,
       const TVector4<T>& col3, const TVector4<T>& col4)
      ;

    void set
      (const T& m11, const T& m12, const T& m13, const T& m14, 
       const T& m21, const T& m22, const T& m23, const T& m24, 
       const T& m31, const T& m32, const T& m33, const T& m34, 
       const T& m41, const T& m42, const T& m43, const T& m44)
      ;

    TMatrix4x4& operator = (const T* ptr)
      throw(Exception::NullPointer);

    TMatrix4x4& operator = (const T ptr[4][4])
      throw(Exception::NullPointer);

    TMatrix4x4& operator = (const TMatrix4x4& m)
      ;

    void get(T* ptr) const
      throw(Exception::NullPointer);

    void get(T ptr[4][4]) const
      throw(Exception::NullPointer);

    void get(TMatrix4x4& m) const
      ;

    void get
      (TVector4<T>& col1, TVector4<T>& col2,
       TVector4<T>& col3, TVector4<T>& col4) const
      ;

    void get
      (T& m11, T& m12, T& m13, T& m14, 
       T& m21, T& m22, T& m23, T& m24, 
       T& m31, T& m32, T& m33, T& m34, 
       T& m41, T& m42, T& m43, T& m44) const
      ;

    void swap(TMatrix4x4& m)
      ;



    T getTrace() const ;

    static const TMatrix4x4& getZero() ;

    static const TMatrix4x4& getIdentity() ;

    void setIdentity() ;

    void set(const T& t = (T)1) ;

    void transpose() ;

    TVector4<T> getRow(Position row) const
      throw(Exception::IndexOverflow);

    TVector4<T> getColumn(Position col) const
      throw(Exception::IndexOverflow);

    void setRow(Position row, const TVector4<T>& row_value)
      throw(Exception::IndexOverflow);

    void setColumn(Position col, const TVector4<T>& col_value)
      throw(Exception::IndexOverflow);

    bool isEqual(const TMatrix4x4& m) const ;

    TVector4<T> getDiagonal() const ;
    
    T& operator () (Position row, Position col)
      throw(Exception::IndexOverflow);

    const T& operator () (Position row, Position col) const
      throw(Exception::IndexOverflow);

    const T& operator [] (Position position) const
      throw(Exception::IndexOverflow);

    T& operator [] (Position position)
      throw(Exception::IndexOverflow);

    TMatrix4x4 operator + () const ;

    TMatrix4x4 operator - () const ;

    TMatrix4x4 operator + (const TMatrix4x4& m) const ;

    TMatrix4x4& operator += (const TMatrix4x4& m) ;

    TMatrix4x4 operator - (const TMatrix4x4& m) const ;

    TMatrix4x4& operator -= (const TMatrix4x4& m) ;

    TMatrix4x4 operator * (const T& scalar) const ;

    TMatrix4x4& operator *= (const T& scalar) ;

    TMatrix4x4 operator / (const T& scalar) const
      throw(Exception::DivisionByZero);

    TMatrix4x4& operator /= (const T& scalar)
      throw(Exception::DivisionByZero);

    TMatrix4x4 operator * (const TMatrix4x4& m) const ;

    TMatrix4x4& operator *= (const TMatrix4x4& m) ;

    TVector4<T> operator * (const TVector4<T>& vector) const ;

    bool invert(TMatrix4x4& inverse) const ;

    bool invert() ;

    T getDeterminant() const ;

    void translate(const T &x, const T &y, const T &z) ;

    void translate(const TVector3<T>& v) ;

    void setTranslation(const T& x, const T& y, const T& z) ;

    void setTranslation(const TVector3<T>& v) ;

    void scale(const T& x_scale, const T& y_scale, const T& z_scale) ;

    void scale(const T& scale) ;

    void scale(const TVector3<T>& v) ;

    void setScale(const T& x_scale, const T& y_scale, const T& z_scale) ;

    void setScale(const T& scale) ;

    void setScale(const TVector3<T>& v) ;

    void rotateX(const TAngle<T>& phi) ;

    void setRotationX(const TAngle<T>& phi) ;

    void rotateY(const TAngle<T>& phi) ;

    void setRotationY(const TAngle<T>& phi) ;

    void rotateZ(const TAngle<T>& phi) ;

    void setRotationZ(const TAngle<T>& phi) ;

    void rotate(const TAngle<T>& phi, const T& axis_x, const T& axis_y, const T& axis_z) ;

    void rotate(const TAngle<T>& phi, const TVector3<T>& axis) ;

    void rotate(const TAngle<T>& phi, const TVector4<T>& axis) ;

    void setRotation(const TAngle<T>& phi, const T& axis_x, const T& axis_y, const T& axis_z) ;

    void setRotation(const TAngle<T>& phi, const TVector3<T>& axis) ;

    void setRotation(const TAngle<T>& phi, const TVector4<T>& axis) ;


    bool operator == (const TMatrix4x4& m) const ; 

    bool operator != (const TMatrix4x4& m) const ;

    bool isIdentity() const ;

    bool isRegular() const ;

    bool isSingular() const ;

    bool isSymmetric() const ;

    bool isLowerTriangular() const ;

    bool isUpperTriangular() const ;

    bool isDiagonal() const ;


    bool isValid() const ;

    void dump(std::ostream& s = std::cout, Size depth = 0) const ;


    T m11;

    T m12;

    T m13;

    T m14;

    T m21;

    T m22;

    T m23;

    T m24;

    T m31;

    T m32;

    T m33;

    T m34;

    T m41;

    T m42;

    T m43;

    T m44;

    private:

    void initializeComponentPointers_()
      
    {
      T **ptr = (T **)comp_ptr_;

      *ptr++ = &m11; *ptr++ = &m12; *ptr++ = &m13; *ptr++ = &m14;
      *ptr++ = &m21; *ptr++ = &m22; *ptr++ = &m23; *ptr++ = &m24;
      *ptr++ = &m31; *ptr++ = &m32; *ptr++ = &m33; *ptr++ = &m34;
      *ptr++ = &m41; *ptr++ = &m42; *ptr++ = &m43; *ptr   = &m44;
    }

    // pointers to the components of the matrix 
    T* comp_ptr_[16];
  };

  template <typename T>
  TMatrix4x4<T>::TMatrix4x4()
    
    : m11(0), m12(0), m13(0), m14(0), 
      m21(0), m22(0), m23(0), m24(0), 
      m31(0), m32(0), m33(0), m34(0), 
      m41(0), m42(0), m43(0), m44(0)
  {
    initializeComponentPointers_();
  }

  template <typename T>
  TMatrix4x4<T>::TMatrix4x4( const T* ptr)
    throw(Exception::NullPointer)
  {
    if (ptr == 0)
    {
      throw Exception::NullPointer(__FILE__, __LINE__);
    }
    
    m11 = *ptr++; m12 = *ptr++; m13 = *ptr++; m14 = *ptr++; 
    m21 = *ptr++; m22 = *ptr++; m23 = *ptr++; m24 = *ptr++; 
    m31 = *ptr++; m32 = *ptr++; m33 = *ptr++; m34 = *ptr++; 
    m41 = *ptr++; m42 = *ptr++; m43 = *ptr++; m44 = *ptr; 

    initializeComponentPointers_();
  }

  template <typename T>
  TMatrix4x4<T>::TMatrix4x4(const T array_ptr[4][4])
    throw(Exception::NullPointer)
  {
    if (array_ptr == 0)
    {
      throw Exception::NullPointer(__FILE__, __LINE__);
    }
    
    const T *ptr = *array_ptr;
      
    m11 = *ptr++; m12 = *ptr++; m13 = *ptr++; m14 = *ptr++; 
    m21 = *ptr++; m22 = *ptr++; m23 = *ptr++; m24 = *ptr++; 
    m31 = *ptr++; m32 = *ptr++; m33 = *ptr++; m34 = *ptr++; 
    m41 = *ptr++; m42 = *ptr++; m43 = *ptr++; m44 = *ptr; 

    initializeComponentPointers_();
  }

  template <typename T>
  TMatrix4x4<T>::TMatrix4x4(const TMatrix4x4<T>& m)
    
    : m11(m.m11), m12(m.m12), m13(m.m13), m14(m.m14), 
      m21(m.m21), m22(m.m22), m23(m.m23), m24(m.m24), 
      m31(m.m31), m32(m.m32), m33(m.m33), m34(m.m34), 
      m41(m.m41), m42(m.m42), m43(m.m43), m44(m.m44)
  {
    initializeComponentPointers_();
  }


  template <typename T>
  TMatrix4x4<T>::TMatrix4x4
    (const TVector4<T>& col1, const TVector4<T>& col2,
     const TVector4<T>& col3,const TVector4<T>& col4)
    
    : m11(col1.x), m12(col1.y), m13(col1.z), m14(col1.h), 
      m21(col2.x), m22(col2.y), m23(col2.z), m24(col2.h), 
      m31(col3.x), m32(col3.y), m33(col3.z), m34(col3.h), 
      m41(col4.x), m42(col4.y), m43(col4.z), m44(col4.h)
  {
    initializeComponentPointers_();
  }

  template <typename T>
  TMatrix4x4<T>::TMatrix4x4
    (const T& m11, const T& m12, const T& m13, const T& m14, 
     const T& m21, const T& m22, const T& m23, const T& m24, 
     const T& m31, const T& m32, const T& m33, const T& m34, 
     const T& m41, const T& m42, const T& m43, const T& m44)
    
    : m11(m11), m12(m12), m13(m13), m14(m14), 
      m21(m21), m22(m22), m23(m23), m24(m24), 
      m31(m31), m32(m32), m33(m33), m34(m34), 
      m41(m41), m42(m42), m43(m43), m44(m44)
  {
    initializeComponentPointers_();
  }

  template <typename T>
  void TMatrix4x4<T>::clear()
    
  {
    set((T)0);
  }

  template <typename T>
  void TMatrix4x4<T>::set(const T* ptr)
    throw(Exception::NullPointer)
  {
    if (ptr == 0) 
    {
      throw Exception::NullPointer(__FILE__, __LINE__);
    }

    m11 = *ptr++; m12 = *ptr++; m13 = *ptr++; m14 = *ptr++; 
    m21 = *ptr++; m22 = *ptr++; m23 = *ptr++; m24 = *ptr++; 
    m31 = *ptr++; m32 = *ptr++; m33 = *ptr++; m34 = *ptr++; 
    m41 = *ptr++; m42 = *ptr++; m43 = *ptr++; m44 = *ptr; 
  }

  template <typename T>
  void TMatrix4x4<T>::set(const T array_ptr[4][4])
    throw(Exception::NullPointer)
  {
    if (array_ptr == 0)
    {
      throw Exception::NullPointer(__FILE__, __LINE__);
    }
    
    const T *ptr = *array_ptr;

    m11 = *ptr++; m12 = *ptr++; m13 = *ptr++; m14 = *ptr++; 
    m21 = *ptr++; m22 = *ptr++; m23 = *ptr++; m24 = *ptr++; 
    m31 = *ptr++; m32 = *ptr++; m33 = *ptr++; m34 = *ptr++; 
    m41 = *ptr++; m42 = *ptr++; m43 = *ptr++; m44 = *ptr; 
  }

  template <typename T>
  void TMatrix4x4<T>::set(const TMatrix4x4<T>& m)
    
  {
    m11 = m.m11; m12 = m.m12; m13 = m.m13; m14 = m.m14; 
    m21 = m.m21; m22 = m.m22; m23 = m.m23; m24 = m.m24; 
    m31 = m.m31; m32 = m.m32; m33 = m.m33; m34 = m.m34; 
    m41 = m.m41; m42 = m.m42; m43 = m.m43; m44 = m.m44;
  }

  template <typename T>
  void TMatrix4x4<T>::set
    (const TVector4<T>& col1, const TVector4<T>& col2,
     const TVector4<T>& col3, const TVector4<T>& col4)
    
  {
    m11 = col1.x; m12 = col1.y; m13 = col1.z; m14 = col1.h; 
    m21 = col2.x; m22 = col2.y; m23 = col2.z; m24 = col2.h; 
    m31 = col3.x; m32 = col3.y; m33 = col3.z; m34 = col3.h; 
    m41 = col4.x; m42 = col4.y; m43 = col4.z; m44 = col4.h;
  }

  template <typename T>
  void TMatrix4x4<T>::set
    (const T& c11, const T& c12, const T& c13, const T& c14, 
     const T& c21, const T& c22, const T& c23, const T& c24, 
     const T& c31, const T& c32, const T& c33, const T& c34, 
     const T& c41, const T& c42, const T& c43, const T& c44)
    
  {
    m11 = c11; m12 = c12; m13 = c13; m14 = c14;
    m21 = c21; m22 = c22; m23 = c23; m24 = c24;
    m31 = c31; m32 = c32; m33 = c33; m34 = c34;
    m41 = c41; m42 = c42; m43 = c43; m44 = c44;
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T>& TMatrix4x4<T>::operator = (const T* ptr)
    throw(Exception::NullPointer)
  {
    set(ptr);
    return *this;
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T>& TMatrix4x4<T>::operator = (const T array_ptr[4][4])
    throw(Exception::NullPointer)
  {
    set(array_ptr);
    return *this;
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T>& TMatrix4x4<T>::operator = (const TMatrix4x4<T>& m)
    
  {
    set(m);
    return *this;
  }

  template <typename T>
  void TMatrix4x4<T>::get(T* ptr) const
    throw(Exception::NullPointer)
  {
    if (ptr == 0)
    {
      throw Exception::NullPointer(__FILE__, __LINE__);
    }

    *ptr++ = m11; *ptr++ = m12; *ptr++ = m13; *ptr++ = m14; 
    *ptr++ = m21; *ptr++ = m22; *ptr++ = m23; *ptr++ = m24; 
    *ptr++ = m31; *ptr++ = m32; *ptr++ = m33; *ptr++ = m34; 
    *ptr++ = m41; *ptr++ = m42; *ptr++ = m43; *ptr   = m44; 
  }

  template <typename T>
  void TMatrix4x4<T>::get(T array_ptr[4][4]) const
    throw(Exception::NullPointer)
  {
    if (array_ptr == 0)
    {
       throw Exception::NullPointer(__FILE__, __LINE__);
    }
 
    T *ptr = *array_ptr;

    *ptr++ = m11; *ptr++ = m12; *ptr++ = m13; *ptr++ = m14; 
    *ptr++ = m21; *ptr++ = m22; *ptr++ = m23; *ptr++ = m24; 
    *ptr++ = m31; *ptr++ = m32; *ptr++ = m33; *ptr++ = m34; 
    *ptr++ = m41; *ptr++ = m42; *ptr++ = m43; *ptr   = m44; 
  }

  template <typename T>
  void TMatrix4x4<T>::get(TMatrix4x4<T>& m) const
    
  {
    m.set(*this);
  }

  template <typename T>
  void TMatrix4x4<T>::get
    (TVector4<T>& col1, TVector4<T>& col2,
     TVector4<T>& col3, TVector4<T>& col4) const
    
  {
    col1.x = m11; col1.y = m12; col1.z = m13; col1.h = m14; 
    col2.x = m21; col2.y = m22; col2.z = m23; col2.h = m24; 
    col3.x = m31; col3.y = m32; col3.z = m33; col3.h = m34; 
    col4.x = m41; col4.y = m42; col4.z = m43; col4.h = m44;
  }

  template <typename T>
  void TMatrix4x4<T>::get
    (T& c11, T& c12, T& c13, T& c14, 
     T& c21, T& c22, T& c23, T& c24, 
     T& c31, T& c32, T& c33, T& c34, 
     T& c41, T& c42, T& c43, T& c44) const
    
  {
    c11 = m11; c12 = m12; c13 = m13; c14 = m14;
    c21 = m21; c22 = m22; c23 = m23; c24 = m24;
    c31 = m31; c32 = m32; c33 = m33; c34 = m34;
    c41 = m41; c42 = m42; c43 = m43; c44 = m44;
  }

  template <typename T>
  BALL_INLINE 
  T TMatrix4x4<T>::getTrace() const
    
  {
    return (m11 + m22 + m33 + m44);
  }

  template <typename T>
  BALL_INLINE 
  const TMatrix4x4<T>& TMatrix4x4<T>::getZero()
    
  {
    static TMatrix4x4<T> null_matrix
      (0, 0, 0, 0,
       0, 0, 0, 0,
       0, 0, 0, 0,
       0, 0, 0, 0);
    
    return null_matrix;
  }


  template <typename T>
  BALL_INLINE
  void TMatrix4x4<T>::setIdentity()
    
  {
    m12 = m13 = m14 = m21 = m23 = m24 = m31 = m32 = m34 = m41 = m42 = m43 = 0;
    m11 = m22 = m33 = m44 = (T)1;
  }
  template <typename T>
  BALL_INLINE 
  const TMatrix4x4<T>& TMatrix4x4<T>::getIdentity()
    
  {
    static TMatrix4x4<T> identity
      (1, 0, 0, 0,
       0, 1, 0, 0,
       0, 0, 1, 0,
       0, 0, 0, 1);

    return identity;
  }

  template <typename T>
  void TMatrix4x4<T>::set(const T& t)
    
  {
      m11 = m12 = m13 = m14 
    = m21 = m22 = m23 = m24 
    = m31 = m32 = m33 = m34
    = m41 = m42 = m43 = m44
    = t;
  }

  template <typename T>
  void TMatrix4x4<T>::swap(TMatrix4x4<T>& m)
    
  {
    T tmp = m11; m11 = m.m11; m.m11 = tmp;
      tmp = m12; m12 = m.m12; m.m12 = tmp;
      tmp = m13; m13 = m.m13; m.m13 = tmp;
      tmp = m14; m14 = m.m14; m.m14 = tmp;
      tmp = m21; m21 = m.m21; m.m21 = tmp;
      tmp = m22; m22 = m.m22; m.m22 = tmp;
      tmp = m23; m23 = m.m23; m.m23 = tmp;
      tmp = m24; m24 = m.m24; m.m24 = tmp;
      tmp = m31; m31 = m.m31; m.m31 = tmp;
      tmp = m32; m32 = m.m32; m.m32 = tmp;
      tmp = m33; m33 = m.m33; m.m33 = tmp;
      tmp = m34; m34 = m.m34; m.m34 = tmp;
      tmp = m41; m41 = m.m41; m.m41 = tmp;
      tmp = m42; m42 = m.m42; m.m42 = tmp;
      tmp = m43; m43 = m.m43; m.m43 = tmp;
      tmp = m44; m44 = m.m44; m.m44 = tmp;
  }

  template <typename T>
  void TMatrix4x4<T>::transpose()
    
  {
    T tmp = m12;
    m12 = m21;
    m21 = tmp;

    tmp  = m13;
    m13 = m31;
    m31 = tmp;

    tmp  = m14;
    m14 = m41;
    m41 = tmp;

    tmp  = m23;
    m23 = m32;
    m32 = tmp;

    tmp  = m24;
    m24 = m42;
    m42 = tmp;

    tmp  = m34;
    m34 = m43;
    m43 = tmp;
  }

  template <typename T>
  TVector4<T> TMatrix4x4<T>::getRow(Position row) const
    throw(Exception::IndexOverflow)
  {
    if (row > 3)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, row, 3);
    }

    // calculate the start of the row in the array
    const T* ptr = comp_ptr_[4 * row];
    return TVector4<T> (ptr[0], ptr[1], ptr[2], ptr[3]);
  }

  template <typename T>
  TVector4<T> TMatrix4x4<T>::getColumn(Position col) const
    throw(Exception::IndexOverflow)
  {
    if (col > 3)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, col, 3);
    }
    
    const T* ptr = comp_ptr_[col];

    return TVector4<T> (ptr[0], ptr[4], ptr[8], ptr[12]);
  }


  template <typename T>
  void TMatrix4x4<T>::setRow(Position row, const TVector4<T>& row_value)
    throw(Exception::IndexOverflow)
  {
    if (row > 3)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, row, 3);
    }

    // calculate a pointer to the start of the row
    T* ptr = comp_ptr_[4 * row];

    ptr[0] = row_value.x;
    ptr[1] = row_value.y;
    ptr[2] = row_value.z;
    ptr[3] = row_value.h;
  }

  template <typename T>
  void TMatrix4x4<T>::setColumn(Position col, const TVector4<T>& col_value)
    throw(Exception::IndexOverflow)
  {
    if (col > 3)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, col, 3);
    }

    // calculate a pointer to the start of the column
    T* ptr = comp_ptr_[col];

    ptr[0] = col_value.x;
    ptr[4] = col_value.y;
    ptr[8] = col_value.z;
    ptr[12] = col_value.h;
  }

  template <typename T>
  bool TMatrix4x4<T>::isEqual(const TMatrix4x4<T>& m) const
    
  {
    // iterate over all component pointers
    // and compare the elements for approximate equality
    for (Position i = 0; i < 16; i++)
    {
      if (Maths::isEqual(*comp_ptr_[i], *m.comp_ptr_[i]) == false)
      {
        return false;
      } 
    }

    return true;
  }

  template <typename T>
  TVector4<T>TMatrix4x4<T>::getDiagonal() const
    
  {
    return TVector4<T>(m11, m22, m33, m44);
  }

  template <typename T>
  BALL_INLINE  
  T& TMatrix4x4<T>::operator () (Position row, Position col)
    throw(Exception::IndexOverflow)
  {
    if ((row > 3) || (col > 3))
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, row + col, 3);
    }

    return *comp_ptr_[4 * row + col];
  }

  template <typename T>
  BALL_INLINE 
  const T& TMatrix4x4<T>::operator () (Position row, Position col) const
    throw(Exception::IndexOverflow)
  {
    if ((row > 3) || (col > 3))
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, row + col, 3);
    }

    return *comp_ptr_[4 * row + col];
  }

  template <typename T>
  BALL_INLINE
  const T& TMatrix4x4<T>::operator [] (Position position) const
    throw(Exception::IndexOverflow)
  {
    if (position > 15)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, position, 15);
    }
    return *comp_ptr_[position];
  }

  template <typename T>
  BALL_INLINE
  T& TMatrix4x4<T>::operator [] (Position position)
    throw(Exception::IndexOverflow)
  {
    if (position > 15)
    {
      throw Exception::IndexOverflow(__FILE__, __LINE__, position, 15);
    }
    return *comp_ptr_[position];
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T> TMatrix4x4<T>::operator + () const
    
  {
    return *this;
  }

  template <typename T>
  BALL_INLINE TMatrix4x4<T>
  TMatrix4x4<T>::operator - () const
    
  {
    return TMatrix4x4<T>
      (-m11, -m12, -m13, -m14,
       -m21, -m22, -m23, -m24,
       -m31, -m32, -m33, -m34,
       -m41, -m42, -m43, -m44);
  }

  template <typename T>
  TMatrix4x4<T> TMatrix4x4<T>::operator + (const TMatrix4x4<T>& m) const
    
  {
    return TMatrix4x4<T>
      (m11 + m.m11, m12 + m.m12, m13 + m.m13, m14 + m.m14,
       m21 + m.m21, m22 + m.m22, m23 + m.m23, m24 + m.m24,
       m31 + m.m31, m32 + m.m32, m33 + m.m33, m34 + m.m34,
       m41 + m.m41, m42 + m.m42, m43 + m.m43, m44 + m.m44);
  }

  template <typename T>
  TMatrix4x4<T>& TMatrix4x4<T>::operator += (const TMatrix4x4<T>& m)
    
  {
    m11 += m.m11;
    m12 += m.m12;
    m13 += m.m13;
    m14 += m.m14;
    m21 += m.m21;
    m22 += m.m22;
    m23 += m.m23;
    m24 += m.m24;
    m31 += m.m31;
    m32 += m.m32;
    m33 += m.m33;
    m34 += m.m34;
    m41 += m.m41;
    m42 += m.m42;
    m43 += m.m43;
    m44 += m.m44;

    return *this;
  }

  template <typename T>
  TMatrix4x4<T> TMatrix4x4<T>::operator - (const TMatrix4x4<T>& m) const
    
  {
    return TMatrix4x4<T>
      (m11 - m.m11, m12 - m.m12, m13 - m.m13, m14 - m.m14,
       m21 - m.m21, m22 - m.m22, m23 - m.m23, m24 - m.m24,
       m31 - m.m31, m32 - m.m32, m33 - m.m33, m34 - m.m34,
       m41 - m.m41, m42 - m.m42, m43 - m.m43, m44 - m.m44);
  }

  template <typename T>
  TMatrix4x4<T>& TMatrix4x4<T>::operator -= (const TMatrix4x4<T>& m)
    
  {
    m11 -= m.m11;
    m12 -= m.m12;
    m13 -= m.m13;
    m14 -= m.m14;
    m21 -= m.m21;
    m22 -= m.m22;
    m23 -= m.m23;
    m24 -= m.m24;
    m31 -= m.m31;
    m32 -= m.m32;
    m33 -= m.m33;
    m34 -= m.m34;
    m41 -= m.m41;
    m42 -= m.m42;
    m43 -= m.m43;
    m44 -= m.m44;

    return *this;
  }

  template <typename T>
  TMatrix4x4<T> TMatrix4x4<T>::operator * (const T& scalar) const
    
  {
    return TMatrix4x4<T>
      (m11 * scalar, m12 * scalar, m13 * scalar, m14 * scalar,
       m21 * scalar, m22 * scalar, m23 * scalar, m24 * scalar,
       m31 * scalar, m32 * scalar, m33 * scalar, m34 * scalar,
       m41 * scalar, m42 * scalar, m43 * scalar, m44 * scalar);
  }

  template <typename T>
  TMatrix4x4<T> operator * (const T& scalar, const TMatrix4x4<T>& m)
    
  {
    return TMatrix4x4<T>
      (scalar * m.m11, scalar * m.m12, scalar * m.m13, scalar * m.m14,
       scalar * m.m21, scalar * m.m22, scalar * m.m23, scalar * m.m24,
       scalar * m.m31, scalar * m.m32, scalar * m.m33, scalar * m.m34,
       scalar * m.m41, scalar * m.m42, scalar * m.m43, scalar * m.m44);
  }

  template <typename T>
  TMatrix4x4<T>& TMatrix4x4<T>::operator *= (const T& scalar)
    
  {
    m11 *= scalar;
    m12 *= scalar;
    m13 *= scalar;
    m14 *= scalar;
    m21 *= scalar;
    m22 *= scalar;
    m23 *= scalar;
    m24 *= scalar;
    m31 *= scalar;
    m32 *= scalar;
    m33 *= scalar;
    m34 *= scalar;
    m41 *= scalar;
    m42 *= scalar;
    m43 *= scalar;
    m44 *= scalar;

    return *this;
  }

  template <typename T>
  TVector3<T> operator *(const TMatrix4x4<T>& matrix, const TVector3<T>& vector)
    
  {
    return TVector3<T>
      (matrix.m11 * vector.x + matrix.m12 * vector.y + matrix.m13 * vector.z + matrix.m14,
       matrix.m21 * vector.x + matrix.m22 * vector.y + matrix.m23 * vector.z + matrix.m24,
       matrix.m31 * vector.x + matrix.m32 * vector.y + matrix.m33 * vector.z + matrix.m34);
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T>TMatrix4x4<T>::operator / (const T& scalar) const
    throw(Exception::DivisionByZero)
  {
    if (scalar == (T)0)
    {
      throw Exception::DivisionByZero(__FILE__, __LINE__);
    }
    
    return (*this * ((T)1 / scalar));
  }

  template <typename T>
  BALL_INLINE 
  TMatrix4x4<T>& TMatrix4x4<T>::operator /= (const T& scalar)
    throw(Exception::DivisionByZero)
  {
    if (scalar == (T)0)
    {
      throw Exception::DivisionByZero(__FILE__, __LINE__);
    }
    
    return (*this *= (T)1 / scalar);
  }

  template <typename T>
  TMatrix4x4<T> TMatrix4x4<T>::operator * (const TMatrix4x4<T>& m) const
    
  {
    return TMatrix4x4<T>
        (m11 * m.m11 + m12 * m.m21 + m13 * m.m31 + m14 * m.m41,
         m11 * m.m12 + m12 * m.m22 + m13 * m.m32 + m14 * m.m42,
         m11 * m.m13 + m12 * m.m23 + m13 * m.m33 + m14 * m.m43,
         m11 * m.m14 + m12 * m.m24 + m13 * m.m34 + m14 * m.m44,

         m21 * m.m11 + m22 * m.m21 + m23 * m.m31 + m24 * m.m41,
         m21 * m.m12 + m22 * m.m22 + m23 * m.m32 + m24 * m.m42,
         m21 * m.m13 + m22 * m.m23 + m23 * m.m33 + m24 * m.m43,
         m21 * m.m14 + m22 * m.m24 + m23 * m.m34 + m24 * m.m44,
     
         m31 * m.m11 + m32 * m.m21 + m33 * m.m31 + m34 * m.m41,
         m31 * m.m12 + m32 * m.m22 + m33 * m.m32 + m34 * m.m42,
         m31 * m.m13 + m32 * m.m23 + m33 * m.m33 + m34 * m.m43,
         m31 * m.m14 + m32 * m.m24 + m33 * m.m34 + m34 * m.m44,
     
         m41 * m.m11 + m42 * m.m21 + m43 * m.m31 + m44 * m.m41,
         m41 * m.m12 + m42 * m.m22 + m43 * m.m32 + m44 * m.m42,
         m41 * m.m13 + m42 * m.m23 + m43 * m.m33 + m44 * m.m43,
         m41 * m.m14 + m42 * m.m24 + m43 * m.m34 + m44 * m.m44);
  }

  template <typename T>
  TMatrix4x4<T>& TMatrix4x4<T>::operator *= (const TMatrix4x4<T>& m)
    
  {
    set(m11 * m.m11 + m12 * m.m21 + m13 * m.m31 + m14 * m.m41,
        m11 * m.m12 + m12 * m.m22 + m13 * m.m32 + m14 * m.m42,
        m11 * m.m13 + m12 * m.m23 + m13 * m.m33 + m14 * m.m43,
        m11 * m.m14 + m12 * m.m24 + m13 * m.m34 + m14 * m.m44,
 
        m21 * m.m11 + m22 * m.m21 + m23 * m.m31 + m24 * m.m41,
        m21 * m.m12 + m22 * m.m22 + m23 * m.m32 + m24 * m.m42,
        m21 * m.m13 + m22 * m.m23 + m23 * m.m33 + m24 * m.m43,
        m21 * m.m14 + m22 * m.m24 + m23 * m.m34 + m24 * m.m44,

        m31 * m.m11 + m32 * m.m21 + m33 * m.m31 + m34 * m.m41,
        m31 * m.m12 + m32 * m.m22 + m33 * m.m32 + m34 * m.m42,
        m31 * m.m13 + m32 * m.m23 + m33 * m.m33 + m34 * m.m43,
        m31 * m.m14 + m32 * m.m24 + m33 * m.m34 + m34 * m.m44,

        m41 * m.m11 + m42 * m.m21 + m43 * m.m31 + m44 * m.m41,
        m41 * m.m12 + m42 * m.m22 + m43 * m.m32 + m44 * m.m42,
        m41 * m.m13 + m42 * m.m23 + m43 * m.m33 + m44 * m.m43,
        m41 * m.m14 + m42 * m.m24 + m43 * m.m34 + m44 * m.m44);

    return *this;
  }

  template <typename T>
  TVector4<T> TMatrix4x4<T>::operator * (const TVector4<T>& v) const
    
  {
    return TVector4<T>
      (m11 * v.x + m12 * v.y + m13 * v.z + m14 * v.h,
       m21 * v.x + m22 * v.y + m23 * v.z + m24 * v.h,
       m31 * v.x + m32 * v.y + m33 * v.z + m34 * v.h,
       m41 * v.x + m42 * v.y + m43 * v.z + m44 * v.h);
  }

  template <typename T>
  bool TMatrix4x4<T>::invert(TMatrix4x4<T>& inverse) const
    
  {
    Index i, j, k;

    T a[4][4] = // holds the matrix we want to invert
    {
      { m11, m12, m13, m14 },
      { m21, m22, m23, m24 },
      { m31, m32, m33, m34 },
      { m41, m42, m43, m44 }
    };
  
    // holds the maximum in the part of A we still have to work with
    T scale, sum_of_squares, sigma, tau;
    T c[4], d[4];
    for (k=0; k<3; k++)
    {
      scale = (T)0;
      // find the maximum in a
      for (i=k; i<4; i++)
        scale = Maths::max((T)fabs(a[i][k]), scale);

      // is the matrix singular?
      if (scale == (T)0)
        return false;

      // nope. we can normalize the remaining rows
      for (i=k; i<4; i++)
        a[i][k] /= scale;
      
      sum_of_squares = (T)0;
      for (i=k; i<4; i++)
        sum_of_squares += a[i][k]*a[i][k];

      // shift the diagonal element
      sigma = (a[k][k] >= 0) ? sqrt(sum_of_squares) : -sqrt(sum_of_squares);
      a[k][k] += sigma;

      c[k] =  sigma*a[k][k];
      d[k] = -scale*sigma;

      for (j = k+1; j<4; j++)
      {
        // store the scalar product of a_[k] and a_[j]
        sum_of_squares = (T)0;
        for (i = k; i<4; i++)
          sum_of_squares += a[i][k] * a[i][j];

        tau = sum_of_squares / c[k];

        // prepare the matrix
        for (i=k; i<4; i++)
          a[i][j] -= tau*a[i][k];
      }
    }
    d[3] = a[3][3];
    
    // is the matrix singular?
    if (d[3] == (T)0)
      return 1;

    // now we have the QR decomposition. The upper triangle of A contains
    // R, except for the diagonal elements, which are stored in d. c contains
    // the values needed to compute the Householder matrices Q, and the vectors
    // u needed for the determination of the Qs are stored in the lower triangle
    // of A
    //
    // now we need to solve four linear systems of equations, one for each column
    // of the resulting matrix
    T result[4][4];
    result[0][0] = 1; result[0][1] = 0; result[0][2] = 0; result[0][3] = 0;
    result[1][0] = 0; result[1][1] = 1; result[1][2] = 0; result[1][3] = 0;
    result[2][0] = 0; result[2][1] = 0; result[2][2] = 1; result[2][3] = 0;
    result[3][0] = 0; result[3][1] = 0; result[3][2] = 0; result[3][3] = 1;

    for (k=0; k<4; k++) // k generates the k-th column of the inverse
    {
      // form the vector Q^t * b, which is simple, since b = e_k
      for (j=0; j<3; j++)
      {
        sum_of_squares = (T)0;
        for (i=j; i<4; i++)
          sum_of_squares += a[i][j]*result[i][k];
        
        tau = sum_of_squares / c[j];
        
        for (i=j; i<4; i++)
          result[i][k] -= tau*a[i][j];
      }

      // and solve the resulting system
      result[3][k] /= d[3];
      for (i=2; i>=0; i--)
      {
        sum_of_squares = (T)0;
        for (j=i+1; j<4; j++)
          sum_of_squares += a[i][j] * result[j][k];

        result[i][k] = (result[i][k] - sum_of_squares) / d[i];
      }
    }
    
    T* k_ptr = *result;
    inverse.m11 = *k_ptr++;
    inverse.m12 = *k_ptr++;
    inverse.m13 = *k_ptr++;
    inverse.m14 = *k_ptr++;
    inverse.m21 = *k_ptr++;
    inverse.m22 = *k_ptr++;
    inverse.m23 = *k_ptr++;
    inverse.m24 = *k_ptr++;
    inverse.m31 = *k_ptr++;
    inverse.m32 = *k_ptr++;
    inverse.m33 = *k_ptr++;
    inverse.m34 = *k_ptr++;
    inverse.m41 = *k_ptr++;
    inverse.m42 = *k_ptr++;
    inverse.m43 = *k_ptr++;
    inverse.m44 = *k_ptr;

    return true;
  }

  template <typename T>
  BALL_INLINE bool TMatrix4x4<T>::invert()
    
  {
    return invert(*this);
  }

  template <typename T>
  T TMatrix4x4<T>::getDeterminant() const
    
  {
    Position i;
    Position j;
    Position k;
    T submatrix[3][3];
    T matrix[4][4] =
    {
      { m11, m12, m13, m14 },
      { m21, m22, m23, m24 },
      { m31, m32, m33, m34 },
      { m41, m42, m43, m44 }
    };
    T determinant = 0;
      
    for (i = 0; i < 4; i++)
    {
      for (j = 0; j < 3; j++)
      {
        for (k = 0; k < 3; k++)
        {
          submatrix[j][k] =
          matrix[j + 1][(k < i) ? k : k + 1];
        }
      }
      
      determinant += matrix[0][i] * (T)(i / 2.0 == (i >> 1) ? 1 : -1)
                  * (submatrix[0][0] * submatrix[1][1] * submatrix[2][2] 
                     + submatrix[0][1] * submatrix[1][2] * submatrix[2][0] 
                     + submatrix[0][2] * submatrix[1][0] * submatrix[2][1] 
                     - submatrix[0][2] * submatrix[1][1] * submatrix[2][0] 
                     - submatrix[0][0] * submatrix[1][2] * submatrix[2][1] 
                     - submatrix[0][1] * submatrix[1][0] * submatrix[2][2]);
    }

    return determinant;
  }

  template <typename T>
  void TMatrix4x4<T>::translate(const T& x, const T& y, const T& z)
    
  {
    m14 += m11 * x + m12 * y + m13 * z;
    m24 += m21 * x + m22 * y + m23 * z;
    m34 += m31 * x + m32 * y + m33 * z;
    m44 += m41 * x + m42 * y + m43 * z;
  }

  template <typename T>
  void TMatrix4x4<T>::translate(const TVector3<T>& v)
    
  {
    m14 += m11 * v.x + m12 * v.y + m13 * v.z;
    m24 += m21 * v.x + m22 * v.y + m23 * v.z;
    m34 += m31 * v.x + m32 * v.y + m33 * v.z;
    m44 += m41 * v.x + m42 * v.y + m43 * v.z;
  }

  template <typename T>
  void TMatrix4x4<T>::setTranslation(const T& x, const T& y, const T& z)
    
  {
    m11 = m22 = m33 = m44 = 1;

    m12 = m13 = 
    m21 = m23 = 
    m31 = m32 =  
    m41 = m42 = m43 = 0;

    m14 = x;
    m24 = y;
    m34 = z;
  }

  template <typename T>
  void TMatrix4x4<T>::setTranslation(const TVector3<T>& v)
    
  {
    m11 = m22 = m33 = m44 = 1;

    m12 = m13 = 
    m21 = m23 = 
    m31 = m32 =  
    m41 = m42 = m43 = 0;

    m14 = v.x;
    m24 = v.y;
    m34 = v.z;
  }

  template <typename T>
  void TMatrix4x4<T>::scale(const T& x_scale, const T& y_scale, const T& z_scale)
    
  {
    m11 *= x_scale;
    m21 *= x_scale;
    m31 *= x_scale;
    m41 *= x_scale;

    m12 *= y_scale;
    m22 *= y_scale;
    m32 *= y_scale;
    m42 *= y_scale;

    m13 *= z_scale;
    m23 *= z_scale;
    m33 *= z_scale;
    m43 *= z_scale;
  }

  template <typename T>
  void TMatrix4x4<T>::scale(const T& scale)
    
  {
    m11 *= scale;
    m21 *= scale;
    m31 *= scale;
    m41 *= scale;

    m12 *= scale;
    m22 *= scale;
    m32 *= scale;
    m42 *= scale;

    m13 *= scale;
    m23 *= scale;
    m33 *= scale;
    m43 *= scale;
  }

  template <typename T>
  void TMatrix4x4<T>::scale(const TVector3<T>& v)
    
  {
    m11 *= v.x;
    m21 *= v.x;
    m31 *= v.x;
    m41 *= v.x;

    m12 *= v.y;
    m22 *= v.y;
    m32 *= v.y;
    m42 *= v.y;

    m13 *= v.z;
    m23 *= v.z;
    m33 *= v.z;
    m43 *= v.z;
  }

  template <typename T>
  void TMatrix4x4<T>::setScale(const T& x_scale, const T& y_scale, const T& z_scale)
    
  {
    m11 = x_scale;
    m22 = y_scale;
    m33 = z_scale;
    m44 = 1;

    m12 = m13 = m14 =
    m21 = m23 = m24 =
    m31 = m32 = m34 = 
    m41 = m42 = m43 = 0;
  }

  template <typename T>
  void TMatrix4x4<T>::setScale(const T& scale)
    
  {
    m11 = scale;
    m22 = scale;
    m33 = scale;
    m44 = 1;

    m12 = m13 = m14 =
    m21 = m23 = m24 =
    m31 = m32 = m34 = 
    m41 = m42 = m43 = 0;
  }

  template <typename T>
  void TMatrix4x4<T>::setScale(const TVector3<T>& v)
    
  {
    m11 = v.x;
    m22 = v.y;
    m33 = v.z;
    m44 = 1;

    m12 = m13 = m14 =
    m21 = m23 = m24 =
    m31 = m32 = m34 = 
    m41 = m42 = m43 = 0;
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::rotateX(const TAngle<T>& phi)
    
  {
    TMatrix4x4<T> rotation;

    rotation.setRotationX(phi);
    *this *= rotation;
  }

  template <typename T>
  void TMatrix4x4<T>::setRotationX(const TAngle<T>& phi)
    
  {
    m11 = m44 = 1;

      m12 = m13 = m14 
    = m21 = m24 
    = m31 = m34  
    = m41 = m42 = m43 
    = 0;

    m22 = m33 = cos(phi);
    m23 = -(m32 = sin(phi));
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::rotateY(const TAngle<T>& phi)
    
  {
    TMatrix4x4<T> rotation;

    rotation.setRotationY(phi);
    *this *= rotation;
  }

  template <typename T>
  void TMatrix4x4<T>::setRotationY(const TAngle<T>& phi)
    
  {
    m22 = m44 = 1;

      m12 = m14 
    = m21 = m23 = m24 
    = m32 = m34 
    = m41 = m42 = m43 
    = 0;

    m11 = m33 = cos(phi);
    m31 = -(m13 = sin(phi));
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::rotateZ(const TAngle<T>& phi)
    
  {
    TMatrix4x4<T> rotation;

    rotation.setRotationZ(phi);
    *this *= rotation;
  }

  template <typename T>
  void TMatrix4x4<T>::setRotationZ(const TAngle<T>& phi)
    
  {
    m33 = m44 = 1;

    m13 = m14 = m23 = m24 = m31 = 
    m32 = m34 = m41 = m42 = m43 = 0;

    m11 =  m22 = cos(phi);
    m12 = -(m21 = sin(phi));
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::rotate(const TAngle<T>& phi, const TVector3<T>& v)
    
  {
    rotate(phi, v.x, v.y, v.z);
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::rotate(const TAngle<T>& phi, const TVector4<T>& v)
    
  {
    rotate(phi, v.x, v.y, v.z);
  }

  //
  //     Arbitrary axis rotation matrix.
  //
  //  [Taken from the MESA-Library. But modified for additional Speed-Up.]
  //
  //  This function was contributed by Erich Boleyn (erich@uruk.org).
  //
  //  This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied
  //  like so:  Rz * Ry * T * Ry' * Rz'.  T is the final rotation
  //  (which is about the X-axis), and the two composite transforms
  //  Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary
  //  from the arbitrary axis to the X-axis then back.  They are
  //  all elementary rotations.
  //
  //  Rz' is a rotation about the Z-axis, to bring the axis vector
  //  into the x-z plane.  Then Ry' is applied, rotating about the
  //  Y-axis to bring the axis vector parallel with the X-axis.  The
  //  rotation about the X-axis is then performed.  Ry and Rz are
  //  simply the respective inverse transforms to bring the arbitrary
  //  axis back to it's original orientation.  The first transforms
  //  Rz' and Ry' are considered inverses, since the data from the
  //  arbitrary axis gives you info on how to get to it, not how
  //  to get away from it, and an inverse must be applied.
  //
  //  The basic calculation used is to recognize that the arbitrary
  //  axis vector (x, y, z), since it is of unit length, actually
  //  represents the sines and cosines of the angles to rotate the
  //  X-axis to the same orientation, with theta being the angle about
  //  Z and phi the angle about Y (in the order described above)
  //  as follows:
  //
  //  cos ( theta ) = x / sqrt ( 1 - z^2 )
  //  sin ( theta ) = y / sqrt ( 1 - z^2 )
  //
  //  cos ( phi ) = sqrt ( 1 - z^2 )
  //  sin ( phi ) = z
  //
  //  Note that cos ( phi ) can further be inserted to the above
  //  formulas:
  //
  //  cos ( theta ) = x / cos ( phi )
  //  sin ( theta ) = y / sin ( phi )
  //
  //  ...etc.  Because of those relations and the standard trigonometric
  //  relations, it is pssible to reduce the transforms down to what
  //  is used below.  It may be that any primary axis chosen will give the
  //  same results (modulo a sign convention) using thie method.
  //
  //  Particularly nice is to notice that all divisions that might
  //  have caused trouble when parallel to certain planes or
  //  axis go away with care paid to reducing the expressions.
  //  After checking, it does perform correctly under all cases, since
  //  in all the cases of division where the denominator would have
  //  been zero, the numerator would have been zero as well, giving
  //  the expected result.

  template <typename T>
  void TMatrix4x4<T>::rotate(const TAngle<T>& phi, const T& ax, const T& ay, const T& az)
    
  {
    T xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c;
    T x = ax;
    T y = ay;
    T z = az;

    double sin_angle = sin(phi);
    double cos_angle = cos(phi);

    xx = x * x;
    yy = y * y;
    zz = z * z;

    T mag = sqrt(xx + yy + zz);
    
    if (mag == (T)0) 
    {
      m12 = m13 = m14 = m21 = m23 = m24 = m31 = m32 = m34 = m41 = m42 = m43 = 0;
      m11 = m22 = m33 = m44 = (T)1;
    }

    x /= mag;
    y /= mag;
    z /= mag;

    // we need to recalculate xx, yy, zz due to the
    // normalization. recalculation is probably faster
    // than normalizing xx, yy, zz
    xx = x*x;
    yy = y*y;
    zz = z*z;

    xy = x * y;
    yz = y * z;
    zx = z * x;
    xs = (T) (x * sin_angle);
    ys = (T) (y * sin_angle);
    zs = (T) (z * sin_angle);
    one_c = (T) (1 - cos_angle);

    m11 = (T)( (one_c * xx) + cos_angle );
    m12 = (one_c * xy) - zs;
    m13 = (one_c * zx) + ys;
    m14 = 0;
    
    m21 = (one_c * xy) + zs;
    m22 = (T) ((one_c * yy) + cos_angle);
    m23 = (one_c * yz) - xs;
    m24 = 0;
    
    m31 = (one_c * zx) - ys;
    m32 = (one_c * yz) + xs;
    m33 = (T) ((one_c * zz) + cos_angle);
    m34 = 0;
     
    m41 = 0;
    m42 = 0;
    m43 = 0;
    m44 = 1;
  }

  template <typename T>
  void TMatrix4x4<T>::setRotation(const TAngle<T>& phi, const T& x, const T& y, const T& z)
    
  {
    m12 = m13 = m14 = m21 = m23 = m24 = m31 = m32 = m34 = m41 = m42 = m43 = 0;
    m11 = m22 = m33 = m44 = (T)1;
    rotate(phi, x, y, z);
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::setRotation(const TAngle<T>& phi, const TVector3<T>& v)
    
  {
    m12 = m13 = m14 = m21 = m23 = m24 = m31 = m32 = m34 = m41 = m42 = m43 = 0;
    m11 = m22 = m33 = m44 = (T)1;
    rotate(phi, v.x, v.y, v.z);
  }

  template <typename T>
  BALL_INLINE 
  void TMatrix4x4<T>::setRotation(const TAngle<T>& phi, const TVector4<T>& v)
    
  {
    m12 = m13 = m14 = m21 = m23 = m24 = m31 = m32 = m34 = m41 = m42 = m43 = 0;
    m11 = m22 = m33 = m44 = (T)1;
    rotate(phi, v.x, v.y, v.z);
  }

  template <typename T>
  bool TMatrix4x4<T>::operator == (const TMatrix4x4<T>& m) const
    
  {
    return 
      (   m11 == m.m11
       && m12 == m.m12
       && m13 == m.m13
       && m14 == m.m14
       && m21 == m.m21
       && m22 == m.m22
       && m23 == m.m23
       && m24 == m.m24
       && m31 == m.m31
       && m32 == m.m32
       && m33 == m.m33
       && m34 == m.m34
       && m41 == m.m41
       && m42 == m.m42
       && m43 == m.m43
       && m44 == m.m44);
  }

  template <typename T>
  bool TMatrix4x4<T>::operator != (const TMatrix4x4<T>& m) const
    
  {
    return 
      (   m11 != m.m11
       || m12 != m.m12
       || m13 != m.m13
       || m14 != m.m14
       || m21 != m.m21
       || m22 != m.m22
       || m23 != m.m23
       || m24 != m.m24
       || m31 != m.m31
       || m32 != m.m32
       || m33 != m.m33
       || m34 != m.m34
       || m41 != m.m41
       || m42 != m.m42
       || m43 != m.m43
       || m44 != m.m44);
  }

  template <typename T>
  bool TMatrix4x4<T>::isIdentity() const
    
  {
    return 
      (   m11 == (T)1
       && m12 == (T)0
       && m13 == (T)0
       && m14 == (T)0
       && m21 == (T)0
       && m22 == (T)1
       && m23 == (T)0
       && m24 == (T)0
       && m31 == (T)0
       && m32 == (T)0
       && m33 == (T)1
       && m34 == (T)0
       && m41 == (T)0
       && m42 == (T)0
       && m43 == (T)0
       && m44 == (T)1);
  }

  template <typename T>
  BALL_INLINE 
  bool TMatrix4x4<T>::isRegular() const
    
  {
    return (getDeterminant() != (T)0);
  }

  template <typename T>
  BALL_INLINE
  bool TMatrix4x4<T>::isSingular() const
    
  {
    return (getDeterminant() == (T)0);
  }

  template <typename T>
  bool TMatrix4x4<T>::isSymmetric() const
    
  {
    return (   m12 == m21 && m13 == m31
            && m14 == m41 && m23 == m32
            && m24 == m42 && m34 == m43);
  }

  template <typename T>
  bool TMatrix4x4<T>::isLowerTriangular() const
    
  {
    return (   m12 == (T)0
            && m13 == (T)0
            && m14 == (T)0
            && m23 == (T)0
            && m24 == (T)0
            && m34 == (T)0);
  }

  template <typename T>
  bool TMatrix4x4<T>::isUpperTriangular() const
    
  {
    return (   m21 == (T)0
            && m31 == (T)0
            && m32 == (T)0
            && m41 == (T)0
            && m42 == (T)0
            && m43 == (T)0);
  }

  template <typename T>
  BALL_INLINE 
  bool TMatrix4x4<T>::isDiagonal() const
    
  {
    return (   m12 == (T)0
            && m13 == (T)0
            && m14 == (T)0
            && m21 == (T)0
            && m23 == (T)0
            && m24 == (T)0
            && m31 == (T)0
            && m32 == (T)0
            && m34 == (T)0
            && m41 == (T)0
            && m42 == (T)0
            && m43 == (T)0);
  }

  template <typename T>
  bool TMatrix4x4<T>::isValid() const
    
  {
    T **ptr = (T **)comp_ptr_;
    
    return (   *ptr++ == &m11
            && *ptr++ == &m12
            && *ptr++ == &m13
            && *ptr++ == &m14
            && *ptr++ == &m21
            && *ptr++ == &m22
            && *ptr++ == &m23
            && *ptr++ == &m24
            && *ptr++ == &m31
            && *ptr++ == &m32
            && *ptr++ == &m33
            && *ptr++ == &m34
            && *ptr++ == &m41
            && *ptr++ == &m42
            && *ptr++ == &m43
            && *ptr   == &m44);
  }

  template <typename T>
  std::istream& operator >> (std::istream& s, TMatrix4x4<T>& m)
    
  {   
    char c;
    s >> c
      >> m.m11 >> m.m12 >> m.m13 >> m.m14 >> c >> c
      >> m.m21 >> m.m22 >> m.m23 >> m.m24 >> c >> c
      >> m.m31 >> m.m32 >> m.m33 >> m.m34 >> c >> c
      >> m.m41 >> m.m42 >> m.m43 >> m.m44 >> c;
    
    return s;
  }

  template <typename T>
  std::ostream& operator << (std::ostream& s, const TMatrix4x4<T>& m)
    
  { 
    s << '/'  <<  std::setw(14) << m.m11 << ' ' << std::setw(14) << m.m12 << ' ' << std::setw(14) << m.m13 << ' ' << std::setw(14) << m.m14 << " \\" << std::endl
      << '|'  <<  std::setw(14) << m.m21 << ' ' << std::setw(14) << m.m22 << ' ' << std::setw(14) << m.m23 << ' ' << std::setw(14) << m.m24 << " |"  << std::endl
      << '|'  <<  std::setw(14) << m.m31 << ' ' << std::setw(14) << m.m32 << ' ' << std::setw(14) << m.m33 << ' ' << std::setw(14) << m.m34 << " |"  << std::endl
      << '\\' <<  std::setw(14) << m.m41 << ' ' << std::setw(14) << m.m42 << ' ' << std::setw(14) << m.m43 << ' ' << std::setw(14) << m.m44 << " /" << std::endl;

    return s;
  }

  template <typename T>
  void TMatrix4x4<T>::dump(std::ostream& s, Size depth) const
    
  {
    BALL_DUMP_STREAM_PREFIX(s);

    BALL_DUMP_HEADER(s, this, this);

    BALL_DUMP_DEPTH(s, depth);
    s << m11 << " " << m12 << " " << m13 << " " << m14 << std::endl;

    BALL_DUMP_DEPTH(s, depth);
    s << m21 << " " << m22 << " " << m23 << " " << m24 << std::endl;

    BALL_DUMP_DEPTH(s, depth);
    s << m31 << " " << m32 << " " << m33 << " " << m34 << std::endl;

    BALL_DUMP_DEPTH(s, depth);
    s << m41 << " " << m42 << " " << m43 << " " << m44 << std::endl;

    BALL_DUMP_STREAM_SUFFIX(s);
  }

  template <typename T>
  TMatrix4x4<T> operator * (const T& scalar, const TMatrix4x4<T>& m)
    ;

  template <typename T>
  TVector3<T> operator * (const TMatrix4x4<T>& matrix, const TVector3<T>& vector)
    ;

  typedef TMatrix4x4<float> Matrix4x4;

} // namespace BALL

#endif // BALL_MATHS_MATRIX44_H