testBoundedRationalPolytope.cpp

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#include <iostream>
#include <vector>
#include <algorithm>
#include "DGtal/base/Common.h"
#include "DGtal/kernel/SpaceND.h"
#include "DGtal/geometry/volumes/BoundedRationalPolytope.h"
#include "DGtalCatch.h"
 
using namespace std;
using namespace DGtal;
 
 
// Functions for testing class BoundedRationalPolytope.
 
SCENARIO( "BoundedRationalPolytope< Z2 > unit tests", "[rational_polytope][2d]" )
{
  typedef SpaceND<2,int>                   Space;
  typedef Space::Point                     Point;
  typedef Space::Vector                    Vector;
  typedef BoundedRationalPolytope< Space >  Polytope;
 
  GIVEN( "A triangle P at (0,0), (5/2,0), (0,7/2)" ) {
    Point a( 0, 0 );
    Point b( 5, 0 );
    Point c( 0, 7 );
    Polytope P { Point(2,2), a, b, c };
    THEN( "It contains more than 3 integer points" ) {
      REQUIRE( P.count() > 3 );
    }
    THEN( "It contains more points than its area" ) {
      REQUIRE( P.count() > (5*7/8) );
    }
    THEN( "It satisfies #In(P) <= #Int(P) + #Bd(P)" ) {
      auto     nb = P.count();
      auto nb_int = P.countInterior();
      auto  nb_bd = P.countBoundary();
      CAPTURE( nb );
      CAPTURE( nb_int );
      CAPTURE( nb_bd );
      REQUIRE( nb <= nb_int + nb_bd );
    }
    WHEN( "Cut by some half-space" ) {
      Polytope Q = P;
      Q.cut( Vector( -1, 1 ), 3 );
      THEN( "It contains less points" ) {
        REQUIRE( Q.count() < P.count() );
      }
    }
    WHEN( "Multiplied by 4 as Q = 4 * P, it becomes a lattice polytope" ) {
      Polytope Q = 4 * P;
      THEN( "It satisfies Pick's formula, ie 2*Area(P) = 2*#Int(P) + #Bd(P) - 2" ) {
        Polytope IntQ = Q.interiorPolytope();
        auto   nb_int = IntQ.count();
        auto    nb_bd = Q.count() - IntQ.count();
        auto    area2 = (5*2)*(7*2);
      CAPTURE( Q );
      CAPTURE( nb_int );
      CAPTURE( nb_bd );
      CAPTURE( area2 );
      REQUIRE( area2 == 2*nb_int + nb_bd - 2 );
      }
    }
  }
  GIVEN( "A closed segment S at (4/2,0/2), (-8/2,-4/2)" ) {
    Point a( 4, 0 );
    Point b( -8, -4 );
    Polytope P { Point(2,2), a, b };
    THEN( "Its interior is empty #Int(P) == 0" ) {
      auto nb_int = P.countInterior();
      REQUIRE( nb_int == 0 );
    }
    THEN( "It satisfies #In(P) == #Int(P) + #Bd(P) == #Bd(P) == 3" ) {
      auto     nb = P.count();
      auto nb_int = P.countInterior();
      auto  nb_bd = P.countBoundary();
      CAPTURE( nb );
      CAPTURE( nb_int );
      CAPTURE( nb_bd );
      std::vector<Point> Ppts;
      P.getPoints( Ppts );
      CAPTURE( P );
      CAPTURE( Ppts );
      REQUIRE( nb_bd == 3 );
      REQUIRE( nb == nb_int + nb_bd );
    }
  }
  GIVEN( "A thin triangle P at (4/4,2/4), (2/4,4/4), (9/4,9/4)" ) {
    Point a( 4, 2 );
    Point b( 2, 4 );
    Point c( 9, 9 );
    Polytope P { Point(4,4), a, b, c };
    THEN( "It contains 2 integer points" ) {
      REQUIRE( P.count() == 2 );
    }
    THEN( "Its boundary is empty" ) {
      REQUIRE( P.countBoundary() == 0 );
    }
    WHEN( "Multiplied by 4 as Q = 4 * P, it becomes a lattice polytope" ) {
      Polytope Q = 4 * P;
      THEN( "It satisfies Pick's formula, ie 2*Area(P) = 2*#Int(P) + #Bd(P) - 2" ) {
        Polytope IntQ = Q.interiorPolytope();
        auto   nb_int = IntQ.count();
        auto    nb_bd = Q.count() - IntQ.count();
        auto    area2 = 24;
        CAPTURE( Q );
        CAPTURE( nb_int );
        CAPTURE( nb_bd );
        CAPTURE( area2 );
        REQUIRE( area2 == 2*nb_int + nb_bd - 2 );
      }
    }
    WHEN( "Multiplied by 10/3 as Q = 10/3 * P, it is a rational polytope" ) {
      Polytope Q = Polytope::Rational( 10, 3 ) * P;
      THEN( "It has a denominator 3 * gcd(4,10) == 6" ) {
        REQUIRE( Q.denominator() == 6 );
      }
      THEN( "#( 3P Cap Z2 ) <= #( Q Cap Z2 ) < #( 4P Cap Z2 )" ) {
        Polytope R = 3 * P;
        Polytope S = 4 * P;
        auto   nbQ = Q.count(); 
        auto   nbR = R.count(); 
        auto   nbS = S.count();
        CAPTURE( nbR );
        CAPTURE( nbQ );
        CAPTURE( nbS );
        REQUIRE( nbR <= nbQ );
        REQUIRE( nbQ <= nbS );
      }
      THEN( "6/5*Q is a polytope equal to 4*P." ) {
        Polytope R = Polytope::Rational( 6, 5 ) * Q;
        Polytope S = 4 * P;
        std::vector<Point> Rpts, Spts;
        R.getPoints( Rpts );
        S.getPoints( Spts );
        CAPTURE( Rpts );
        CAPTURE( Spts );
        REQUIRE( std::equal( Rpts.cbegin(), Rpts.cend(), Spts.cbegin() ) );
      }
    }
  }
} // SCENARIO( "BoundedRationalPolytope< Z2 > unit tests", "[rational_polytope][2d]" )
 
SCENARIO( "BoundedRationalPolytope< Z3 > unit tests", "[rational_polytope][3d]" )
{
  typedef SpaceND<3,int>                   Space;
  typedef Space::Point                     Point;
  typedef BoundedRationalPolytope< Space >  Polytope;
 
  GIVEN( "A tetrahedron P at (0,0,0), (5/2,0,0), (0,7/2,0), (0,0,3/2)" ) {
    Point a( 0, 0, 0 );
    Point b( 5, 0, 0 );
    Point c( 0, 7, 0 );
    Point d( 0, 0, 3 );
    Polytope P { Point(2,2), a, b, c, d };
    THEN( "It contains more than 3 integer points" ) {
      REQUIRE( P.count() > 3 );
    }
    THEN( "It contains more points than its volume" ) {
      REQUIRE( P.count() > (5*7*3/8/6) );
    }
    THEN( "It satisfies #In(P) <= #Int(P) + #Bd(P)" ) {
      auto     nb = P.count();
      auto nb_int = P.countInterior();
      auto  nb_bd = P.countBoundary();
      CAPTURE( nb );
      CAPTURE( nb_int );
      CAPTURE( nb_bd );
      REQUIRE( nb <= nb_int + nb_bd );
    }
    WHEN( "Multiplied by 4 as Q = 2 * P, it becomes a lattice polytope" ) {
      Polytope Q = 2 * P;
      THEN( "Its denominator is 1" ) {
        REQUIRE( Q.denominator() == 1 );
      }
      THEN( "It has more interior and boundary points than P" ) {
        auto     nb = P.count();
        auto nb_int = P.countInterior();
        auto  nb_bd = P.countBoundary();
        Polytope IntQ = Q.interiorPolytope();
        auto  nb_intQ = IntQ.count();
        auto      nbQ = Q.count();
        auto   nb_bdQ = nbQ - nb_intQ;
        CAPTURE( Q );
        REQUIRE( nb_int < nb_intQ );
        REQUIRE( nb_bd  < nb_bdQ  );
        REQUIRE( nb     < nbQ     );
      }
    }
    WHEN( "Considering an increasing series f_i = 3/2, 2, 9/4, 3, 11/3, the number of inside points of f_i * P is increasing" ) {
      Polytope Q1 = Polytope::Rational( 3 , 2 ) * P;      
      Polytope Q2 = Polytope::Rational( 2 , 1 ) * P;      
      Polytope Q3 = Polytope::Rational( 9 , 4 ) * P;      
      Polytope Q4 = Polytope::Rational( 3 , 1 ) * P;      
      Polytope Q5 = Polytope::Rational( 11, 3 ) * P;
      auto    nb  = P .count();
      auto    nb1 = Q1.count();
      auto    nb2 = Q2.count();
      auto    nb3 = Q3.count();
      auto    nb4 = Q4.count();
      auto    nb5 = Q5.count();
      REQUIRE( nb  < nb1 );
      REQUIRE( nb1 < nb2 );
      REQUIRE( nb2 < nb3 );
      REQUIRE( nb3 < nb4 );
      REQUIRE( nb4 < nb5 );
    }
  }
}
 
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