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
4.1-2 Cdd_V_Rep
4.1-3 Cdd_H_Rep
4.1-4 Cdd_AmbientSpaceDimension
4.1-5 Cdd_Dimension
4.1-6 Cdd_GeneratingVertices
4.1-7 Cdd_GeneratingRays
4.1-8 Cdd_Equalities
4.1-9 Cdd_Inequalities
4.1-10 Cdd_InteriorPoint
4.1-11 Cdd_Faces
4.1-12 Cdd_FacesWithFixedDimension
4.1-13 Cdd_FacesWithInteriorPoints
4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints
4.1-15 Cdd_Facets
4.1-16 Cdd_Lines
4.1-17 Cdd_Vertices
4.1-18 Cdd_IsEmpty
4.1-19 Cdd_IsCone
4.1-20 Cdd_IsPointed

### 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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