quadric.h

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/****************************************************************************
* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2004                                                \/)\/    *
* Visual Computing Lab                                            /\/|      *
* ISTI - Italian National Research Council                           |      *
*                                                                    \      *
* All rights reserved.                                                      *
*                                                                           *
* This program is free software; you can redistribute it and/or modify      *
* it under the terms of the GNU General Public License as published by      *
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* (at your option) any later version.                                       *
*                                                                           *
* This program is distributed in the hope that it will be useful,           *
* but WITHOUT ANY WARRANTY; without even the implied warranty of            *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the             *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt)          *
* for more details.                                                         *
*                                                                           *
****************************************************************************/
#ifndef __VCGLIB_QUADRIC
#define __VCGLIB_QUADRIC

#include <vcg/space/point3.h>
#include <vcg/space/plane3.h>
#include <vcg/math/matrix33.h>
#include <eigenlib/Eigen/Core>

namespace vcg {
namespace math {

/*
 *  This class encode a quadric function 
 *  f(x) = xAx +bx + c
 *  where A is a symmetric 3x3 matrix, b a vector and c a scalar constant.  
 */ 
template<typename  _ScalarType>
class Quadric
{
public:
  typedef _ScalarType ScalarType;
  ScalarType a[6];              // Symmetric Matrix 3x3 : a11 a12 a13 a22 a23 a33
  ScalarType b[3];              // Vector r3
  ScalarType c;                   // Scalar (-1 means null/un-initialized quadric)
  
  inline Quadric() { c = -1; }
  
  bool IsValid() const { return c>=0; }
  void SetInvalid() { c = -1.0; }
  
   // Initialize the quadric to keep the squared distance from a given Plane  
  template< class PlaneType >
  void ByPlane( const PlaneType & p )
  {
    a[0] =  (ScalarType)p.Direction()[0]*p.Direction()[0];      // a11
    a[1] =  (ScalarType)p.Direction()[1]*p.Direction()[0];      // a12 (=a21)
    a[2] =  (ScalarType)p.Direction()[2]*p.Direction()[0];      // a13 (=a31)
    a[3] =  (ScalarType)p.Direction()[1]*p.Direction()[1];      // a22
    a[4] =  (ScalarType)p.Direction()[2]*p.Direction()[1];      // a23 (=a32)
    a[5] =  (ScalarType)p.Direction()[2]*p.Direction()[2];      // a33
    b[0] =  (ScalarType)(-2.0)*p.Offset()*p.Direction()[0];
    b[1] =  (ScalarType)(-2.0)*p.Offset()*p.Direction()[1];
    b[2] =  (ScalarType)(-2.0)*p.Offset()*p.Direction()[2];
    c    =  (ScalarType)p.Offset()*p.Offset();
  }
  
  /* 
   * Initializes the quadric as the squared distance from a given line.
   * Note that this code also works for a vcg::Ray<T>, even though the (squared) distance
   * from a ray is different "before" its origin.
   */
  template< class LineType >
  void ByLine( const LineType & r ) // Init dato un raggio
  {
    ScalarType K = (ScalarType)(r.Origin()*r.Direction());
    a[0] = (ScalarType)1.0-r.Direction()[0]*r.Direction()[0]; // a11
    a[1] = (ScalarType)-r.Direction()[0]*r.Direction()[1]; // a12 (=a21)
    a[2] = (ScalarType)-r.Direction()[0]*r.Direction()[2]; // a13 (=a31)
    a[3] = (ScalarType)1.0-r.Direction()[1]*r.Direction()[1]; // a22
    a[4] = (ScalarType)-r.Direction()[1]*r.Direction()[2]; // a23 (=a32)
    a[5] = (ScalarType)1.0-r.Direction()[2]*r.Direction()[2]; // a33
    b[0] = (ScalarType)2.0*(r.Direction()[0]*K - r.Origin()[0]);
    b[1] = (ScalarType)2.0*(r.Direction()[1]*K - r.Origin()[1]);
    b[2] = (ScalarType)2.0*(r.Direction()[2]*K - r.Origin()[2]);
    c = -K*K + (ScalarType)(r.Origin()*r.Origin());
  }

  void SetZero()
  {
    a[0] = 0;
    a[1] = 0;
    a[2] = 0;
    a[3] = 0;
    a[4] = 0;
    a[5] = 0;
    b[0] = 0;
    b[1] = 0;
    b[2] = 0;
    c    = 0;
  }
  
  void operator = ( const Quadric & q )
  {
    assert( q.IsValid() );
    
    a[0] = q.a[0];
    a[1] = q.a[1];
    a[2] = q.a[2];
    a[3] = q.a[3];
    a[4] = q.a[4];
    a[5] = q.a[5];
    b[0] = q.b[0];
    b[1] = q.b[1];
    b[2] = q.b[2];
    c    = q.c;
  }
  
  void operator += ( const Quadric & q )
  {
    assert( IsValid() );
    assert( q.IsValid() );
    
    a[0] += q.a[0];
    a[1] += q.a[1];
    a[2] += q.a[2];
    a[3] += q.a[3];
    a[4] += q.a[4];
    a[5] += q.a[5];
    b[0] += q.b[0];
    b[1] += q.b[1];
    b[2] += q.b[2];
    c    += q.c;
  }
  
  void operator *= ( const ScalarType & w )                     // Amplifica una quadirca
  {
    assert( IsValid() );
    
    a[0] *= w;
    a[1] *= w;
    a[2] *= w;
    a[3] *= w;
    a[4] *= w;
    a[5] *= w;
    b[0] *= w;
    b[1] *= w;
    b[2] *= w;
    c    *= w;
  }
  
  
  
  /* Evaluate a quadric over a point p.
   */
  template <class ResultScalarType>
  ResultScalarType Apply( const Point3<ResultScalarType> & p ) const
  {
    assert( IsValid() );
    return ResultScalarType (
            p[0]*p[0]*a[0] + 2*p[0]*p[1]*a[1] + 2*p[0]*p[2]*a[2] + p[0]*b[0]
        +   p[1]*p[1]*a[3] + 2*p[1]*p[2]*a[4] + p[1]*b[1]
        +   p[2]*p[2]*a[5] + p[2]*b[2]  + c);
  }
  
  
  static double &RelativeErrorThr()
  {
    static double _err = 0.000001;
    return _err;
  }
  
  // Find the point minimizing the quadric xAx + bx + c 
  // by solving the first derivative 2 Ax + b = 0 
  // return true if the found solution fits the system. 
  
  template <class ReturnScalarType>
  bool Minimum(Point3<ReturnScalarType> &x)
  {
    Eigen::Matrix3d A;
    Eigen::Vector3d be;
    A << a[0], a[1], a[2],
         a[1], a[3], a[4],
         a[2], a[4], a[5];
    be << -b[0]/2, -b[1]/2, -b[2]/2;
  
  //  Eigen::Vector3d xe = A.colPivHouseholderQr().solve(bv);
  //  Eigen::Vector3d xe = A.partialPivLu().solve(bv);
    Eigen::Vector3d xe = A.fullPivLu().solve(be);
    double relative_error = (A*xe - be).norm() / be.norm();
    if(relative_error> Quadric<ScalarType>::RelativeErrorThr() ) 
      return false;
    
    x.FromEigenVector(xe);
    return true;
  }
  
  
  
  
  
// spostare..risolve un sistema 3x3
template<class FLTYPE>
bool Gauss33( FLTYPE x[], FLTYPE C[3][3+1] )
{
    const FLTYPE keps = (FLTYPE)1e-3;
    int i,j,k;

    FLTYPE eps;                                 // Determina valore cond.
                eps = math::Abs(C[0][0]);
    for(i=1;i<3;++i)
    {
                FLTYPE t = math::Abs(C[i][i]);
                if( eps<t ) eps = t;
    }
    eps *= keps;

    for (i = 0; i < 3 - 1; ++i)                 // Ciclo di riduzione
    {
        int ma = i;                             // Ricerca massimo pivot
        FLTYPE vma = math::Abs( C[i][i] );
        for (k = i + 1; k < 3; k++)
        {
            FLTYPE t = math::Abs( C[k][i] );
            if (t > vma)
            {
                vma = t;
                ma  = k;
            }
        }
        if (vma<eps)
            return false;                               // Matrice singolare
        if(i!=ma)                               // Swap del massimo pivot
            for(k=0;k<=3;k++)
            {
                FLTYPE t = C[i][k];
                C[i][k] = C[ma][k];
                C[ma][k] = t;
            }
        for (k = i + 1; k < 3; k++)             //  Riduzione
        {
            FLTYPE s;
            s = C[k][i] / C[i][i];
            for (j = i+1; j <= 3; j++)
                C[k][j] -= C[i][j] * s;
            C[k][i] = 0.0;
        }
    }

        // Controllo finale singolarita'
    if( math::Abs(C[3-1][3- 1])<eps)
        return false;

    for (i=3-1; i>=0; i--)                      // Sostituzione
    {
        FLTYPE t;
        for (t = 0.0, j = i + 1; j < 3; j++)
            t += C[i][j] * x[j];
        x[i] = (C[i][3] - t) / C[i][i];
    }

    return true;
}



template <class ReturnScalarType>
bool MinimumOld(Point3<ReturnScalarType> &x)
{
                ReturnScalarType C[3][4];
                C[0][0]=a[0]; C[0][1]=a[1]; C[0][2]=a[2];
                C[1][0]=a[1]; C[1][1]=a[3]; C[1][2]=a[4];
                C[2][0]=a[2]; C[2][1]=a[4]; C[2][2]=a[5];

                C[0][3]=-b[0]/2;
                C[1][3]=-b[1]/2;
                C[2][3]=-b[2]/2;
                return Gauss33(&(x[0]),C);
}

// determina il punto di errore minimo vincolato nel segmento (a,b)
bool Minimum(Point3<ScalarType> &x,Point3<ScalarType> &pa,Point3<ScalarType> &pb){
ScalarType      t1,t2, t4, t5, t8, t9,
        t11,t12,t14,t15,t17,t18,t25,t26,t30,t34,t35,
        t41,t42,t44,t45,t50,t52,t54,
        t56,t21,t23,t37,t64,lambda;

          t1 = a[4]*pb.z();
          t2 = t1*pa.y();
      t4 = a[1]*pb.y();
      t5 = t4*pa.x();
      t8 = a[1]*pa.y();
      t9 = t8*pa.x();
      t11 = a[4]*pa.z();
      t12 = t11*pa.y();
      t14 = pa.z()*pa.z();
      t15 = a[5]*t14;
      t17 = a[2]*pa.z();
      t18 = t17*pa.x();
      t21 = 2.0*t11*pb.y();
      t23 = a[5]*pb.z()*pa.z();
      t25 = a[2]*pb.z();
      t26 = t25*pa.x();
      t30 = a[0]*pb.x()*pa.x();
      t34 = 2.0*a[3]*pb.y()*pa.y();
      t35 = t17*pb.x();
      t37 = t8*pb.x();
      t41 = pa.x()*pa.x();
      t42 = a[0]*t41;
      t44 = pa.y()*pa.y();
      t45 = a[3]*t44;
      t50 = 2.0*t30+t34+2.0*t35+2.0*t37-(-b[2]/2)*pa.z()-(-b[0]/2)*pa.x()-2.0*t42-2.0*t45+(-b[1]/2)*pb.y()
+(-b[0]/2)*pb.x()-(-b[1]/2)*pa.y();
      t52 = pb.y()*pb.y();
      t54 = pb.z()*pb.z();
      t56 = pb.x()*pb.x();
      t64 = t5+t37-t9+t30-t18+t35+t26-t25*pb.x()+t2-t1*pb.y()+t23;
      lambda = (2.0*t2+2.0*t5+(-b[2]/2)*pb.z()-4.0*t9-4.0*t12-2.0*t15-4.0*t18+t21+2.0*t23+
2.0*t26+t50)/(-t45-a[3]*t52-a[5]*t54-a[0]*t56-t15-t42+t34-2.0*t12+t21-2.0*t4*pb.x()+
2.0*t64)/2.0;

          if(lambda<0)  lambda=0;  else   if(lambda>1)   lambda = 1;

                 x = pa*(1.0-lambda)+pb*lambda;
                 return true;
        }

};

typedef Quadric<short>  Quadrics;
typedef Quadric<int>      Quadrici;
typedef Quadric<float>  Quadricf;
typedef Quadric<double> Quadricd;



        } // end namespace math
} // end namespace vcg

#endif