geometry.inl

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/*
Copyright (c) 2019, Michael Kazhdan
All rights reserved.
 
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are permitted provided that the following conditions are met:
 
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#include <algorithm>
#include <SVD/SVDFit.h>
#include <SVD/MatrixMNTC.h>
#include <Util/exceptions.h>
 
namespace Util
{
    // Point //
    template< unsigned int Dim >
    void Point< Dim >::_init( const double *values , unsigned int sz )
    {
        if     ( sz==0   ) memset( _p , 0 , sizeof(_p) );
        else if( sz==Dim ) memcpy( _p , values , sizeof(_p) );
        else ERROR_OUT( "Should never be called" );
    }
 
    template< unsigned int Dim >
    Point< Dim >::Point( void ){ memset( _p , 0 , sizeof(_p) ); }
 
    template< unsigned int Dim >
    Point< Dim >::Point( const Point &p ){ memcpy( _p , p._p , sizeof(_p) ); }
 
    template< unsigned int Dim >
    template< typename ... Doubles >
    Point< Dim >::Point( Doubles ... values )
    {
        static_assert( sizeof...(values)==Dim || sizeof...(values)==0 , "[ERROR] Point< Dim >::Point: Invalid number of coefficients" );
        const double _values[] = { static_cast< double >( values )... };
        _init( _values , sizeof...(values) );
    }
 
    template< unsigned int Dim >
    Point< Dim > Point< Dim >::operator * ( double s ) const { Point p ; for( int i=0 ; i<Dim ; i++ ) p._p[i] = _p[i] * s ; return p; }
 
    template< unsigned int Dim >
    Point< Dim > Point< Dim >::operator + ( const Point &p ) const { Point q ; for( int i=0 ; i<Dim ; i++ ) q._p[i] += _p[i] + p._p[i] ; return q; }
 
    template< unsigned int Dim >
    double Point< Dim >::dot( const Point &q ) const
    {
        double dot = 0;
        for( int i=0 ; i<Dim ; i++ ) dot += _p[i] * q._p[i];
        return dot;
    }
 
    template< unsigned int Dim >
    double& Point< Dim >::operator[] ( int i ){ return _p[i]; }
 
    template< unsigned int Dim >
    const double &Point< Dim >::operator[] ( int i ) const { return _p[i]; }
 
    template< unsigned int Dim >
    Point< Dim >  Point< Dim >::operator * ( const Point &q ) const
    {
        Point p;
        for( int i=0 ; i<Dim ; i++ ) p[i] = _p[i]*q._p[i];
        return p;
    }
 
    template< unsigned int Dim >
    Point< Dim >  Point< Dim >::operator / ( const Point &q ) const
    {
        Point p;
        for( int i=0 ; i<Dim ; i++ ) p[i] = _p[i]/q._p[i];
        return p;
    }
 
    template< unsigned int Dim >
    Point< Dim > &Point< Dim >::operator *= ( const Point &q ){  return (*this) = (*this) * q; }
 
    template< unsigned int Dim >
    Point< Dim > &Point< Dim >::operator /= ( const Point &q ){  return (*this) = (*this) / q; }
 
    template< unsigned int Dim >
    template< typename ... Points >
    Point< Dim > Point< Dim >::CrossProduct( Points ... points )
    {
        static_assert( sizeof ... ( points )==Dim-1 , "[ERROR] Number of points in cross-product must be one less than the dimension" );
        const Point< Dim > _points[] = { points ... };
        return CrossProduct( _points );
    }
 
    template< unsigned int Dim >
    Point< Dim > Point< Dim >::CrossProduct( Point *points ){ return CrossProduct( (const Point *)points );}
 
    template< unsigned int Dim >
    Point< Dim > Point< Dim >::CrossProduct( const Point *points )
    {
        Matrix< Dim , Dim > M;
        for( int d=0 ; d<Dim ; d++ ) for( int c=0 ; c<Dim-1 ; c++ ) M(d,c) = points[c][d];
        Point p;
        for( int d=0 ; d<Dim ; d++ ) p[d] = ( d&1 ) ? -M.subDeterminant( d , Dim-1 ) : M.subDeterminant( d , Dim-1 );
        return p;
    }
 
    template< unsigned int Dim >
    std::ostream &operator << ( std::ostream &stream , const Point< Dim > &p )
    {
        for( int i=0 ; i<Dim-1 ; i++ ) stream << p[i] << " ";
        stream << p[Dim-1];
        return stream;
    }
    template< unsigned int Dim >
    std::istream &operator >> ( std::istream &stream , Point< Dim > &p )
    {
        for( int i=0 ; i<Dim ; i++ ) stream >> p[i];
        return stream;
    }

    /////////////////
    // _BaseMatrix //
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    double _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::dot( const _BaseMatrix &m ) const
    {
        double dot = 0;
        for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<Cols ; c++ ) dot += operator()(r,c) * m(r,c);
        return dot;
    }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::_BaseMatrix( void ){ memset( _m , 0 , sizeof(_m) ); }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    double& _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::operator() ( int r , int c ) { return _m[r][c]; }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    const double &_BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::operator() ( int r , int c ) const { return _m[r][c]; }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    MatrixTransposeType _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::transpose( void ) const
    {
        MatrixTransposeType n;
        for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<Cols ; c++ ) n(c,r) = operator()(r,c);
        return n;
    }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    Point< Rows > _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::operator * ( const Point< Cols > &p ) const
    {
        Point< Rows > q;
        for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<Cols ; c++ ) q[r] += operator()(r,c) * p[c];
        return q;
    }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    MatrixType _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::operator * ( double s ) const { MatrixType n ; for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<Cols ; c++ ) n(r,c) = operator()(r,c) * s ; return n; }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    MatrixType _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType >::operator + ( const MatrixType &m ) const { MatrixType n ; for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<Cols ; c++ ) n(r,c) = operator()(r,c) + m(r,c) ; return n; }

    // Matrix //
    template< unsigned int Rows , unsigned int Cols > Matrix< Rows , Cols >::Matrix( void ) : _BaseMatrix< Rows , Cols , Matrix< Rows , Cols > , Matrix< Cols , Rows > >() {}
 
    template< unsigned int Rows , unsigned int Cols >
    template< unsigned int _Cols >
    Matrix< Rows , _Cols > Matrix< Rows , Cols >::operator * ( const Matrix< Cols , _Cols > &m ) const
    {
        Matrix< Rows , _Cols > n;
        for( int r=0 ; r<Rows ; r++ ) for( int c=0 ; c<_Cols ; c++ ) for( int i=0 ; i<Cols ; i++ ) n(r,c) += operator()(r,i) * m(i,c);
        return n;
    }

    // SquareMatrix //
    template< unsigned int Dim > SquareMatrix< Dim >::Matrix( void ) : _BaseMatrix< Dim , Dim , SquareMatrix< Dim > , SquareMatrix< Dim > >() {}
    template< unsigned int Dim >
    SquareMatrix< Dim >::Matrix( const SquareMatrix< Dim+1 > &n ){ for( int i=0 ; i<Dim ; i++ ) for( int j=0 ; j<Dim ; j++ ) operator()(i,j) = n(i,j); }
 
    template< unsigned int Dim >
    SquareMatrix< Dim >::Matrix( const SquareMatrix< Dim-1 > &n , Point< Dim-1 > p ) : Matrix()
    {
        for( int i=0 ; i<Dim-1 ; i++ ) for( int j=0 ; j<Dim-1 ; j++ ) operator()(i,j) = n(i,j);
        operator()(Dim-1,Dim-1) = 1.;
        for( int i=0 ; i<Dim-1 ; i++ ) operator()(i,Dim-1) = p[i];
    }
 
    template< unsigned int Dim >
    template< unsigned int Cols >
    Matrix< Dim , Cols > SquareMatrix< Dim >::operator * ( const Matrix< Dim , Cols > &m ) const
    {
        Matrix< Dim , Cols > n;
        for( int r=0 ; r<Dim ; r++ ) for( int c=0 ; c<Cols ; c++ ) for( int i=0 ; i<Dim ; i++ ) n(r,c) += operator()(r,i) * m(i,c);
        return n;
    }
 
    template< unsigned int Dim >
    double SquareMatrix< Dim >::subDeterminant( int r , int c ) const
    {
        SquareMatrix< Dim-1 > m;
        int rr[Dim-1] , cc[Dim-1];
        for( int a=0 , _r=0 , _c=0 ; a<Dim ; a++ )
        {
            if( a!=r ) rr[_r++] = a;
            if( a!=c ) cc[_c++] = a;
        }
        for( int _c=0 ; _c<Dim-1 ; _c++ ) for( int _r=0 ; _r<Dim-1 ; _r++ ) m(_r,_c) = operator()( rr[_r] , cc[_c] );
        return m.determinant();
    }
 
    template< unsigned int Dim >
    double SquareMatrix< Dim >::determinant( void ) const
    {
        double det = 0.;
        for( int d=0 ; d<Dim ; d++ ) 
            if( d&1 ) det -= operator()(0,d) * subDeterminant( 0 , d );
            else      det += operator()(0,d) * subDeterminant( 0 , d );
        return det;
    }
 
    template<>
    inline double SquareMatrix< 3 >::determinant( void ) const
    {
        return
            operator()(0,0)*( operator()(1,1)*operator()(2,2) - operator()(1,2)*operator()(2,1) ) -
            operator()(0,1)*( operator()(1,0)*operator()(2,2) - operator()(1,2)*operator()(2,0) ) +
            operator()(0,2)*( operator()(1,0)*operator()(2,1) - operator()(1,1)*operator()(2,0) );
    }
 
    template<>
    inline double SquareMatrix< 2 >::determinant( void ) const { return operator()(0,0)*operator()(1,1) - operator()(0,1)*operator()(1,0); }
 
    template<>
    inline double SquareMatrix< 1 >::determinant( void ) const { return operator()(0,0); }
 
    template< unsigned int Dim >
    double SquareMatrix< Dim >::trace( void ) const
    {
        double tr = 0;
        for( int i=0 ; i<Dim ; i++ ) tr += operator()(i,i);
        return tr;
    }
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::inverse( void ) const
    {
        Matrix inv;
        if( !setInverse( inv ) ) THROW( " singular matrix" );
        return inv;
    }
 
    template< unsigned int Dim >
    bool SquareMatrix< Dim >::setInverse( Matrix &inv ) const
    {
        double d = determinant();
        if( !d ) return false;
        for( int i=0 ; i<Dim ; i++ ) for( int j=0 ; j<Dim ; j++ )
            if( (i+j)%2==0 ) inv(i,j) =  subDeterminant( j , i ) / d;
            else             inv(i,j) = -subDeterminant( j , i ) / d;
        return true;
    }
 
    template<>
    inline bool SquareMatrix< 3 >::setInverse( Matrix &inv ) const
    {
        double det = determinant();
        if( !det ) return false;
        inv(0,0) =  ( operator()(1,1)*operator()(2,2) - operator()(1,2)*operator()(2,1) ) / det;
        inv(0,1) = -( operator()(0,1)*operator()(2,2) - operator()(2,1)*operator()(0,2) ) / det;
        inv(0,2) =  ( operator()(0,1)*operator()(1,2) - operator()(0,2)*operator()(1,1) ) / det;
        inv(1,0) = -( operator()(1,0)*operator()(2,2) - operator()(1,2)*operator()(2,0) ) / det;
        inv(1,1) =  ( operator()(0,0)*operator()(2,2) - operator()(0,2)*operator()(2,0) ) / det;
        inv(1,2) = -( operator()(0,0)*operator()(1,2) - operator()(0,2)*operator()(1,0) ) / det;
        inv(2,0) =  ( operator()(1,0)*operator()(2,1) - operator()(1,1)*operator()(2,0) ) / det;
        inv(2,1) = -( operator()(0,0)*operator()(2,1) - operator()(0,1)*operator()(2,0) ) / det;
        inv(2,2) =  ( operator()(0,0)*operator()(1,1) - operator()(0,1)*operator()(1,0) ) / det;
 
        return true;
    }
 
    template<>
    inline bool SquareMatrix< 2 >::setInverse( Matrix &inv ) const
    {
        double det = determinant();
        if( !det ) return false;
        inv(0,0) =  operator()(1,1) / det;
        inv(1,1) =  operator()(0,0) / det;
        inv(1,0) = -operator()(1,0) / det;
        inv(0,1) = -operator()(0,1) / det;
 
        return true;
    }
 
    template<>
    inline bool SquareMatrix< 1 >::setInverse( Matrix &inv ) const
    {
        double det = operator()(0,0);
        if( !det ) return false;
        inv(0,0) = 1./det;
        return true;
    }
 
    template< unsigned int Dim >
    Point< Dim-1 > SquareMatrix< Dim >::operator * ( const Point< Dim-1 > &p ) const
    {
        Point< Dim > q;
        for( int i=0 ; i<Dim-1 ; i++ ) q[i] = p[i];
        q[Dim-1] = 1;
        q = (*this) * q;
        Point< Dim-1 > _q;
        for( int i=0 ; i<Dim-1 ; i++ ) _q[i] = q[i] / q[Dim-1];
        return _q;
    }
 
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::Identity( void )
    {
        Matrix m;
        for( int i=0 ; i<Dim ; i++ ) m(i,i) = 1;
        return m;
    }
 
    template< unsigned int Dim >
    void SquareMatrix< Dim >::SVD( Matrix& r1 , Matrix& d , Matrix& r2 ) const
    {
        GXMatrixMNd M( Dim , Dim );
        GXMatrixMNd U, W, Vt;
 
        d = r1 = r2 = Identity();
 
        for( int i=0 ; i<Dim ; i++ ) for(  int j=0 ; j<Dim ; j++ ) M(i,j) = operator()(i,j);
        SVDMat( M , U , W , Vt );  // M == U . DiagonalMatrix(W) . Vt
        for( int i=0 ; i<Dim ; i++ )
        {
            for( int j=0 ; j<Dim ; j++ ) r1(j,i) = U(j,i) , r2(j,i) = Vt(j,i);
            d(i,i) = W(i,0);
        }
    }
 
    // Code borrowed from:
    // Linear Combination of Transformations
    // Marc Alexa
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::SquareRoot( const Matrix& m , double eps )
    {
        Matrix X,Y;
        X = m;
        Y = Identity();
        while( (X*X-m).squareNorm()>eps*eps )
        {
            Matrix iX = X.inverse();
            Matrix iY = Y.inverse();
            X = (X+iY) / 2;
            Y = (Y+iX) / 2;
        }
        return X;
    }
 
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::Log( const Matrix& m , double eps )
    {
        Matrix I = Identity();
        Matrix X , Z , A=m;
        int k=0;
        while( (A-I).squareNorm()>0.25 )
        {
            A = SquareRoot( A , eps );
            k++;
        }
        A = I-A;
        X = Z = A;
        int i = 1;
        while( Z.squareNorm()>eps*eps )
        {
            Z = Z*A;
            i++;
            X += Z*( 1.0/i );
        }
        return X * ( -pow( 2.0 , (double)k ) );
    }
 
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::symmetrize( void ) const { return ( (*this)+transpose() ) / 2; }
 
    template< unsigned int Dim >
    SquareMatrix< Dim > SquareMatrix< Dim >::skewSymmetrize( void ) const { return ( (*this)-transpose() ) / 2; }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    std::ostream &operator << ( std::ostream &stream , const _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType > &m )
    {
        for( unsigned int c=0 ; c<Cols ; c++ ) for( unsigned int r=0 ; r<Rows ; r++ )
        {
            stream << m(r,c);
            if( r!=Rows-1 || c!=Cols-1 ) stream << " ";
        }
        return stream;
    }
 
    template< unsigned int Rows , unsigned int Cols , typename MatrixType , typename MatrixTransposeType >
    std::istream &operator >> ( std::istream &stream , _BaseMatrix< Rows , Cols , MatrixType , MatrixTransposeType > &m )
    {
        for( unsigned int c=0 ; c<Cols ; c++ ) for( unsigned int r=0 ; r<Rows ; r++ ) stream >> m(r,c); 
        return stream;
    }

    ///////////
    // Plane //
    template< unsigned int Dim >
    Plane< Dim >::Plane( void ) : distance(0) {}
 
    template< unsigned int Dim >
    Plane< Dim >::Plane( const Point< Dim > &n , const Point< Dim > &p )
    {
        normal = n.unit();
        distance = Point< Dim >::Dot( normal , p );
    }
 
    template< unsigned int Dim >
    template< typename ... Points >
    Plane< Dim >::Plane( Points ... points )
    {
        static_assert( sizeof ... ( points )==Dim , "[ERROR] Number of points in plane constructor must equal the dimension" );
        const Point< Dim > _points[] = { points ... };
        (*this) = Plane( _points );
    }
 
    template< unsigned int Dim >
    Plane< Dim >::Plane( Point< Dim > *points ) : Plane( (const Point< Dim > *)points ) {}
 
    template< unsigned int Dim >
    Plane< Dim >::Plane( const Point< Dim > *points )
    {
        Point< Dim > _points[Dim-1];
        for( int i=1 ; i<Dim ; i++ ) _points[i-1] = points[i] - points[0];
        normal = Point< Dim >::CrossProduct( _points ).unit();
        distance = Point< Dim >::Dot( normal , points[0] );
    }
 
    template< unsigned int Dim >
    double Plane< Dim >::operator() ( const Point< Dim > &p ) const
    {
        return Point< Dim >::Dot( normal , p ) - distance;
    }

    // Ray //
    template< unsigned int Dim >
    Ray< Dim >::Ray( void ){}
 
    template< unsigned int Dim >
    Ray< Dim >::Ray( const Point< Dim > &p , const Point< Dim > &d ) : position(p) , direction(d) {}
 
    template< unsigned int Dim >
    Point< Dim > Ray< Dim >::operator() ( double s ) const { return position+direction*s; }
 
    template< unsigned int Dim >
    Ray< Dim >  Ray< Dim >::operator +  ( const Point< Dim > &p ) const { return Ray( position+p , direction );}
 
    template< unsigned int Dim >
    Ray< Dim > &Ray< Dim >::operator += ( const Point< Dim > &p ){ position += p ; return *this; }
 
    template< unsigned int Dim >
    Ray< Dim >  Ray< Dim >::operator -  ( const Point< Dim > &p ) const { return Ray( position-p , direction );}
 
    template< unsigned int Dim >
    Ray< Dim > &Ray< Dim >::operator -= ( const Point< Dim > &p ){ position -= p ; return *this; }
 
    template< unsigned int Dim >
    Ray< Dim > operator * ( const Matrix< Dim+1 , Dim+1 > &m , const Ray< Dim >& r )
    {
        return Ray< Dim >( m * r.position , Matrix< Dim , Dim >(m) * r.direction );
    }
 

    // BoundingBox //
    template< unsigned int Dim >
    BoundingBox< Dim >::BoundingBox( void ){}
 
    template< unsigned int Dim >
    BoundingBox< Dim >::BoundingBox( const Point< Dim > &p1 , const Point< Dim > &p2 )
    {
        for( int d=0 ; d<Dim ; d++ ) _p[0][d] = std::min< double >( p1[d] , p2[d] ) , _p[1][d] = std::max< double >( p1[d] , p2[d] );
    }
 
    template< unsigned int Dim >
    BoundingBox< Dim >::BoundingBox( const Point< Dim > *pList , int pSize )
    {
        if( pSize>0 )
        {
            _p[0] = _p[1] = pList[0];
            for( int i=1 ; i<pSize ; i++ ) for( int j=0 ; j<Dim ; j++ ) _p[0][j] = std::min< double >( _p[0][j] , pList[i][j] ) , _p[1][j] = std::max< double >( _p[1][j] , pList[i][j] );
        }
    }
 
    template< unsigned int Dim >
    Point< Dim > &BoundingBox< Dim >::operator[] ( int idx ){ return _p[idx]; }
 
    template< unsigned int Dim >
    const Point< Dim > &BoundingBox< Dim >::operator[] ( int idx ) const { return _p[idx]; }
 
    template< unsigned int Dim >
    BoundingBox< Dim > BoundingBox< Dim >::operator + ( const BoundingBox &b ) const
    {
        Point< Dim > pList[4];
        Point< Dim > q;
 
        if( b.isEmpty() ) return *this;
        if(   isEmpty() ) return b;
        pList[0] = _p[0];
        pList[1] = _p[1];
        pList[2] = b._p[0];
        pList[3] = b._p[1];
        return BoundingBox( pList , 4 );
    }
 
    template< unsigned int Dim >
    BoundingBox< Dim > &BoundingBox< Dim >::operator += ( const BoundingBox &b ){ return (*this) = (*this) + b; }
 
    template< unsigned int Dim >
    BoundingBox< Dim > BoundingBox< Dim >::operator ^ ( const BoundingBox &b ) const
    {
 
        if( isEmpty() || b.isEmpty() ) return BoundingBox();
        BoundingBox _b;
        for( int j=0 ; j<Dim ; j++ ) _b._p[0][j] = std::max< double >( _p[0][j] , b._p[0][j] ) , _b._p[1][j] = std::min< double >( _p[1][j] , b._p[1][j] );
        if( _b.isEmpty() ) _b._p[0] = _b._p[1] = ( _b._p[0] + _b._p[1] ) / 2;
        return _b;
    }
 
    template< unsigned int Dim >
    BoundingBox< Dim > &BoundingBox< Dim >::operator ^= ( const BoundingBox &b ){ return (*this) = (*this) ^ b; }
 
    template< unsigned int Dim >
    BoundingBox< Dim > operator * ( const Matrix< Dim+1 , Dim+1 > &m , const BoundingBox< Dim > &b )
    {
        Point< Dim > v[1<<Dim];
        for( int idx=0 ; idx<(1<<Dim) ; idx++ )
        {
            Point< Dim > p;
            for( int d=0 ; d<Dim ; d++ ) p[d] = b[(idx>>d)&1][d];
            v[idx] = m * p;
        }
        return BoundingBox< Dim >( v , 1<<Dim );
    }
 
    template< unsigned int Dim >
    bool BoundingBox< Dim >::isInside( const Point< Dim > &p ) const
    {
        for( int d=0 ; d<Dim ; d++ ) if( p[d]<=_p[0][d] || p[d]>=_p[1][d] ) return false;
        return true;
    }
 
    template< unsigned int Dim >
    bool BoundingBox< Dim >::isEmpty( void ) const
    {
        for( int d=0 ; d<Dim ; d++ ) if( _p[0][d]>=_p[1][d] ) return true;
        return false;
    }
 
    template< unsigned int Dim >
    std::ostream &operator << ( std::ostream &stream , const BoundingBox< Dim > &b )
    {
        stream << "[ " << b[0] << " ] [ " << b[1] << " ]";
        return stream;
    }

    // Quadric //
    template< unsigned int Dim >
    Quadric< Dim >::Quadric( void ) : _C(0) {}
 
    template< unsigned int Dim >
    Quadric< Dim >::Quadric( const Matrix< Dim , Dim > &Q , const Point< Dim > &L , const double &C ) :_Q(Q) , _L(L) , _C(C) {}
 
    template< unsigned int Dim >
    Quadric< Dim >::Quadric( const Matrix< Dim+1 , Dim+1 > &Q )
    {
        _C = Q(Dim,Dim);
        for( unsigned int i=0 ; i<Dim ; i++ )
        {
            _L[i] = ( Q(i,Dim) + Q(Dim,i) ) / 2.;
            for( unsigned int j=0 ; j<Dim ; j++ ) _Q(i,j) = ( Q(i,j) + Q(j,i) ) / 2.;
        }
    }
 
    template< unsigned int Dim >
    SquareMatrix< Dim+1 > Quadric< Dim >::operator()( void ) const
    {
        SquareMatrix< Dim+1 > Q;
        Q(Dim,Dim) = _C;
        for( unsigned int i=0 ; i<Dim ; i++ )
        {
            Q(Dim,i) = Q(i,Dim) = _L[i];
            for( unsigned int j=0 ; j<Dim ; j++ ) Q(i,j) = _Q(i,j);
        }
        return Q;
    }
 
    template< unsigned int Dim >
    Matrix< Dim , Dim > Quadric< Dim >::getQuadratic( void ) const { return _Q; }
 
    template< unsigned int Dim >
    Point< Dim > Quadric< Dim >::getLinear( void ) const { return _L; }
 
    template< unsigned int Dim >
    double Quadric< Dim >::getConstant( void ) const { return _C; }
 
    template< unsigned int Dim >
    void Quadric< Dim >::setQuadratic( Matrix< Dim , Dim > Q ){ _Q = ( Q + Q.transpose() )/2.; }
 
    template< unsigned int Dim >
    void Quadric< Dim >::setLinear( Point< Dim > L ){ _L = L; }
 
    template< unsigned int Dim >
    void Quadric< Dim >::setConstant( double C ){ _C = C; }
 
    template< unsigned int Dim >
    double Quadric< Dim >::operator()( const Point< Dim > &p ) const { return Point< Dim >::Dot( p , _Q*p ) + 2. * Point< Dim >::Dot( p , _L ) + _C; }
 
    template< unsigned int Dim >
    template< unsigned int _Dim >
    Quadric< _Dim > Quadric< Dim >::operator * ( const Matrix< Dim , _Dim > &T ) const
    {
        // Q(p) = _Q( T(p) )
        //      = [ T * p ]^t * _Q._Q * [ T * p ] + 2 * _L^t * T * p + _C
        //      = p^t * [ T^t *_Q._Q * T ] *p + 2 ( [T^t*_L]^t * p ) + _C
        Quadric< _Dim > Q;
        Q._Q = T.transpose() * _Q * T;
        Q._L = T.transpose() * _L;
        Q._C = _C;
        return Q;
    }
 
    template< unsigned int Dim >
    Quadric< Dim > Quadric< Dim >::operator + ( const Point< Dim > &t ) const
    {
        // Q(p) = _Q( p+t )
        //      = [ p + t ]^t * _Q._Q * [ p + t ] + 2 * _Q._L^t * [ p + t ] + _Q._C
        //      = p^t * _Q._Q *p + 2 ( [ _Q._Q^t * t + _Q._L ]^t * p ) + _Q._C + t^t * _Q._Q * t + 2 * _Q._L^t * t
        Quadric< Dim > Q;
        Q._Q = _Q;
        Q._L = _Q*t + _L;
        Q._C = _C + Point< Dim >::Dot( t , _Q*t ) + 2. * Point< Dim >::Dot( _L , t );
        return Q;
    }
 
    template< unsigned int Dim >
    bool Quadric< Dim >::setExtremum( Point< Dim > &extremum ) const
    {
        // The equation for the quardic is given by
        //      Q(p) = p^t * _Q *p + 2 * < _L , p > + c
        // The gradient is:
        //      \nabla Q|_p = 2 * _Q * p + 2 * _L
        // Setting the gradient to zero gives:
        //      p = - _Q^{-1} * _L
        try{ extremum = -_Q.inverse() * _L; }
        catch( std::exception & ){ return false; }
        return true;
    }

    // QuadricBoundingBoxOverlap //
    template< unsigned int Dim >
    QuadricBoundingBoxOverlap< Dim >::QuadricBoundingBoxOverlap( void ){}
 
    template< unsigned int Dim >
    QuadricBoundingBoxOverlap< Dim >::QuadricBoundingBoxOverlap( const Quadric< Dim > &Q ){ _set( Q ); }
 
    template< unsigned int Dim >
    bool QuadricBoundingBoxOverlap< Dim >::operator()( const BoundingBox< Dim > & bBox ) const { return _intersect( _quadric , bBox ); }
 
    template< unsigned int Dim >
    void QuadricBoundingBoxOverlap< Dim >::_set( const Quadric< Dim > &Q )
    {
        _quadric = Q;
        _Q = Q._Q;
        try{ _Qinv = _Q.inverse(); }
        catch(...){}
        for( unsigned int d=0 ; d<Dim ; d++ )
        {
            unsigned idx = 0;
            for( unsigned int dd=0 ; dd<Dim ; dd++ ) if( dd!=d ) _boundaryInfo[d]._T(dd,idx++) = 1.;
            _boundaryInfo[d]._Tt = _boundaryInfo[d]._T.transpose();
            _boundaryInfo[d]._set( Q * _boundaryInfo[d]._T );
            _boundaryInfo[d]._Tt_Q = _boundaryInfo[d]._Tt * _Q;
        }
    }
 
    template< unsigned int Dim >
    bool QuadricBoundingBoxOverlap< Dim >::_intersect( const Quadric< Dim > &Q , const BoundingBox< Dim > &bBox ) const
    {
        // Intersection can be found by finding the minimizer of the quadric within the cube and testing if it evaluates to a positive
        // or negative value.
        // The minimizer will be:
        // 1. The point at which the gradient vanishes, if that point is inside the cube. Or,
        // 2. A point on the face at which the gradient vanishes
 
        // Case 2:
        {
            for( unsigned int d=0 ; d<Dim ; d++ )
            {
                BoundingBox< Dim-1 > _bBox;
 
                // Compute the bounding box and the transformation mapping the (Dim-1)-dimensional plane containing the face into Dim-dimensional space
 
                // Example:
                // We have (x,z) -> (x,c,z)
                // Setting
                //     | 1 0 |
                // T = | 0 0 |
                //     | 0 1 |
                // and
                //     | 0 |
                // v = | c |
                //     | 0 |
                // We get:
                //      _Q(_p) = Q( T*_p + v )
                //             = _p^t * [ T^t * Q._Q * T ] * _p + 2 * < T*_p + v , Q._L > + 2 * < Q._Q*T*_p , v > + _C + < v , Q._Q * v >
                //             = _p^t * [ T^t * Q._Q * T ] * _p + 2 * < _p , T^t*Q._L > + 2 * < _p , T^t*Q._Q*v > + _C + < v , Q._Q * v > + 2 * < v , Q._L >
 
                {
                    unsigned idx = 0;
                    for( unsigned int dd=0 ; dd<Dim ; dd++ ) if( dd!=d )
                    {
                        _bBox[0][idx] = bBox[0][dd];
                        _bBox[1][idx] = bBox[1][dd];
                        idx++;
                    }
                }
 
                Quadric< Dim-1 > _Q;
                _Q._Q = _boundaryInfo[d]._Q;
 
                Point< Dim-1 > _L = _boundaryInfo[d]._Tt * Q._L;
                Point< Dim-1 > _v;
                for( unsigned int i=0 ; i<Dim-1 ; i++ ) _v[i] = _boundaryInfo[d]._Tt_Q(i,d);
 
                for( unsigned int o=0 ; o<2 ; o++ )
                {
                    double v = bBox[o][d];
                    _Q._L = _L + _v * v;
                    _Q._C =  Q._C + v * v * Q._Q(d,d) + 2. * v * Q._L[d];
                    if( _boundaryInfo[d]._intersect( _Q , _bBox ) ) return true;
                }
            }
        }
 
        // Case 1:
        Point< Dim > extremum = -_Qinv * Q._L;
        return bBox.isInside( extremum ) && Q( extremum )<0;
    }
 
    inline void QuadricBoundingBoxOverlap< 1 >::_set( const Quadric< 1 > &Q )
    {
        _quadric = Q;
        _Q = Q._Q;
        try{ _Qinv = _Q.inverse(); } catch(...){}
    }
 
    inline bool QuadricBoundingBoxOverlap< 1 >::_intersect( const Quadric< 1 > &Q , const BoundingBox< 1 > &bBox ) const
    {
        if( Q(bBox[0])<0 || Q(bBox[1])<0 ) return true;
        Point< 1 > extremum = -_Qinv * Q._L;
        return extremum[0]>bBox[0][0] && extremum[0]<bBox[1][0] && Q(extremum)<0;
    }

    // RotationParameter //
    template< typename RotationParameterType , typename ParameterType >
    RotationParameterType RotationParameter< RotationParameterType , ParameterType >::operator * ( double scale ) const
    {
        RotationParameterType p;
        p.parameter = parameter * scale;
        return p;
    }
 
    template< typename RotationParameterType , typename ParameterType >
    RotationParameterType RotationParameter< RotationParameterType , ParameterType >::operator + ( const RotationParameterType &p ) const
    {
        RotationParameterType _p;
        _p.parameter = parameter + p.parameter;
        return _p;
    }

    // TransformationParameter //
    template< typename RotationParameterType >
    TransformationParameter< RotationParameterType > TransformationParameter< RotationParameterType >::operator * ( double scale ) const
    {
        TransformationParameter tp = *this;
        tp.rotationParameter *= scale;
        tp.translation *= scale;
        return tp;
    }
 
    template< typename RotationParameterType >
    TransformationParameter< RotationParameterType > TransformationParameter< RotationParameterType >::operator + ( const TransformationParameter &tp ) const
    {
        TransformationParameter _tp = *this;
        _tp.rotationParameter += tp.rotationParameter;
        _tp.translation += tp.translation;
        return _tp;
    }
 
    template< typename RotationParameterType >
    TransformationParameter< RotationParameterType >::TransformationParameter( void ) {}
 
    template< typename RotationParameterType >
    TransformationParameter< RotationParameterType >::TransformationParameter( const Matrix4D &m ) : rotationParameter( Matrix3D( m ) )
    {
        for( int i=0 ; i<3 ; i++ ) translation[i] = m(i,3);
    }
 
    template< typename RotationParameterType >
    TransformationParameter< RotationParameterType >::TransformationParameter( const Matrix4D &m , const TransformationParameter &p ) : rotationParameter( Matrix3D( m ) , p.rotationParameter )
    {
        for( int i=0 ; i<3 ; i++ ) translation[i] = m(i,3);
    }
 
    template< typename RotationParameterType >
    Matrix4D TransformationParameter< RotationParameterType >::operator () ( void ) const
    {
        return Matrix4D( rotationParameter() , translation );
    }
}