GAP (CddInterface) - Chapter 4: Attributes and properties

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4 Attributes and properties

4.1 Attributes and properties of polyhedron

4.1-1 Cdd_Canonicalize

‣ Cdd_Canonicalize( P ) ( attribute )

Returns: a CddPolyhedron

The function takes a polyhedron and reduces its defining inequalities (generators set) by deleting all redundant inequalities (generators).

gap> A:= Cdd_PolyhedronByInequalities( [ [ 0, 2, 6 ], [ 0, 1, 3 ], [1, 4, 10 ] ] );
<Polyhedron given by its H-representation>
gap> B:= Cdd_Canonicalize( A );
<Polyhedron given by its H-representation>
gap> Display( B );
H-representation
begin
   2 X 3  rational

   0   1   3
   1   4  10
end

4.1-2 Cdd_V_Rep

‣ Cdd_V_Rep( P ) ( attribute )

Returns: a CddPolyhedron

The function takes a polyhedron and returns its reduced V-representation.

4.1-3 Cdd_H_Rep

‣ Cdd_H_Rep( P ) ( attribute )

Returns: a CddPolyhedron

The function takes a polyhedron and returns its reduced H-representation.

gap> A:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1 ], [ 0, 5, 5 ] ] );
<Polyhedron given by its H-representation>
gap> B:= Cdd_V_Rep( A );
<Polyhedron given by its V-representation>
gap> Display( B );
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   0   1   0
   0  -1   1
end
gap> C:= Cdd_H_Rep( B );
<Polyhedron given by its H-representation>
gap> Display( C );
H-representation
begin
   1 X 3  rational

   0  1  1
end
gap> D:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1, 34, 22, 43 ],
> [ 11, 2, 2, 54, 53, 221 ], [33, 23, 45, 2, 40, 11 ] ] );
<Polyhedron given by its H-representation>
gap> Cdd_V_Rep( D );
<Polyhedron given by its V-representation>
gap> Display( last );
V-representation
linearity 2, [ 5, 6 ]
begin
   6 X 6  rational

   1  -743/14   369/14    11/14        0        0
   0    -1213      619       22        0        0
   0       -1        1        0        0        0
   0      764     -390      -11        0        0
   0   -13526     6772       99      154        0
   0  -116608    59496     1485        0      154
end

4.1-4 Cdd_AmbientSpaceDimension

‣ Cdd_AmbientSpaceDimension( P ) ( attribute )

Returns: The dimension of the ambient space of the polyhedron(i.e., the space that contains P).

4.1-5 Cdd_Dimension

‣ Cdd_Dimension( P ) ( attribute )

Returns: The dimension of the polyhedron, where the dimension, \mathrm{dim}(P), of a polyhedron P is the maximum number of affinely independent points in P minus 1.

4.1-6 Cdd_GeneratingVertices

‣ Cdd_GeneratingVertices( P ) ( attribute )

Returns: The reduced generating vertices of the polyhedron

4.1-7 Cdd_GeneratingRays

‣ Cdd_GeneratingRays( P ) ( attribute )

Returns: list

The output is the reduced generating rays of the polyhedron

4.1-8 Cdd_Equalities

‣ Cdd_Equalities( P ) ( attribute )

Returns: a list

The output is the reduced equalities of the polyhedron.

4.1-9 Cdd_Inequalities

‣ Cdd_Inequalities( P ) ( attribute )

The output is the reduced inequalities of the polyhedron.

4.1-10 Cdd_InteriorPoint

‣ Cdd_InteriorPoint( P ) ( attribute )

Returns: a list

The output is an interior point in the polyhedron

4.1-11 Cdd_Faces

‣ Cdd_Faces( P ) ( attribute )

Returns: a list

This function takes a H-represented polyhedron P and returns a list. Every entry in this list is a again a list, contains the dimension and linearity of the face defined as a polyhedron over the same system of inequalities.

4.1-12 Cdd_FacesWithFixedDimension

‣ Cdd_FacesWithFixedDimension( P, d ) ( operation )

Returns: a list

This function takes a H-represented polyhedron P and a positive integer d. The output is a list. Every entry in this list is the linearity of an d- dimensional face of P defined as a polyhedron over the same system of inequalities.

4.1-13 Cdd_FacesWithInteriorPoints

‣ Cdd_FacesWithInteriorPoints( P ) ( attribute )

Returns: a list

This function takes a H-represented polyhedron P and returns a list. Every entry in this list is a again a list, contains the dimension, linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in the face.

4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints

‣ Cdd_FacesWithFixedDimensionAndInteriorPoints( P, d ) ( operation )

Returns: a list

This function takes a H-represented polyhedron P and a positive integer d. The output is a list. Every entry in this list is a again a list, contains the linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in this face.

4.1-15 Cdd_Facets

‣ Cdd_Facets( P ) ( attribute )

Returns: a list

This function takes a H-represented polyhedron P and returns a list. Every entry in this is the linearity of a facet defined as a polyhedron over the same system of inequalities.

4.1-16 Cdd_Lines

‣ Cdd_Lines( P ) ( attribute )

Returns: a list

This function takes a H-represented polyhedron P and returns a list. Every entry in this is the linearity of a ray (1-dimensional face) defined as a polyhedron over the same system of inequalities.

4.1-17 Cdd_Vertices

‣ Cdd_Vertices( P ) ( attribute )

Returns: a list

This function takes a H-represented polyhedron P and returns a list. Every entry in this list is the linearity of a vertex defined as a polyhedron over the same system of inequalities.

4.1-18 Cdd_IsEmpty

‣ Cdd_IsEmpty( P ) ( property )

Returns: true or false

The output is true if the polyhedron is empty and false otherwise

4.1-19 Cdd_IsCone

‣ Cdd_IsCone( P ) ( property )

Returns: true or false

The output is true if the polyhedron is cone and false otherwise

4.1-20 Cdd_IsPointed

‣ Cdd_IsPointed( P ) ( property )

Returns: true or false

The output is true if the polyhedron is pointed and false otherwise

gap> poly:= Cdd_PolyhedronByInequalities( [ [ 1, 3, 4, 5, 7 ], [ 1, 3, 5, 12, 34 ],
> [ 9, 3, 0, 2, 13 ]  ], [ 1 ] );
<Polyhedron given by its H-representation>
gap> Cdd_InteriorPoint( poly );
[ -194/75, 46/25, -3/25, 0 ]
gap> Cdd_FacesWithInteriorPoints( poly );
[ [ 3, [ 1 ], [ -194/75, 46/25, -3/25, 0 ] ], [ 2, [ 1, 2 ],
[ -62/25, 49/25, -7/25, 0 ] ], [ 1, [ 1, 2, 3 ],
[ -209/75, 56/25, -8/25, 0 ] ], [ 2, [ 1, 3 ], [ -217/75, 53/25, -4/25, 0 ] ] ]
gap> Cdd_Dimension( poly );
3
gap> Cdd_IsPointed( poly );
false
gap> Cdd_IsEmpty( poly );
false
gap> Cdd_Faces( poly );
[ [ 3, [ 1 ] ], [ 2, [ 1, 2 ] ], [ 1, [ 1, 2, 3 ] ], [ 2, [ 1, 3  ] ] ]
gap> poly1 := Cdd_ExtendLinearity( poly, [ 1, 2, 3 ] );
<Polyhedron given by its H-representation>
gap> Display( poly1 );
H-representation 
linearity 3, [ 1, 2, 3 ]
begin
   3 X 5  rational

   1   3   4   5   7 
   1   3   5  12  34 
   9   3   0   2  13 
end
gap> Cdd_Dimension( poly1 );
1
gap> Cdd_Facets( poly );
[ [ 1, 2 ], [ 1, 3 ] ]
gap> Cdd_GeneratingVertices( poly );
[ [ -209/75, 56/25, -8/25, 0 ] ]
gap> Cdd_GeneratingRays( poly );
[ [ -97, 369, -342, 75 ], [ -8, -9, 12, 0 ],
[ 23, -21, 3, 0 ], [ 97, -369, 342, -75 ] ]
gap> Cdd_Inequalities( poly );
[ [ 1, 3, 5, 12, 34 ], [ 9, 3, 0, 2, 13 ] ]
gap> Cdd_Equalities( poly );
[ [ 1, 3, 4, 5, 7 ] ]
gap> P := Cdd_FourierProjection( poly, 2);
<Polyhedron given by its H-representation>
gap> Display( P );
H-representation 
linearity 1, [ 3 ]
begin 
   3 X 5  rational

    9    3    0    2   13 
   -1   -3    0   23  101 
    0    0    1    0    0 
end

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