`‣ Cdd_PolyhedronByInequalities` ( ineq[, linearities_list] ) | ( function ) |

Returns: a CddPolyhedron

The function takes a list in which every entry represents an inequality (or equality). In case we want some entries to represent equalities we should refer in a second list to their indices.

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

`‣ Cdd_PolyhedronByGenerators` ( genes[, linearities_list] ) | ( function ) |

Returns: a CddPolyhedron

The function takes a list in which every entry represents a vertex in the ambient vector space. In case we want some vertices to be free (the vertex and its negative belong to the polyhedron) we should refer in a second list to their indices .

gap> A:= Cdd_PolyhedronByGenerators( [ [ 0, 1, 3 ], [ 1, 4, 5 ] ] ); <Polyhedron given by its V-representation> gap> Display( A ); V-representation begin 2 X 3 rational 0 1 3 1 4 5 end gap> B:= Cdd_PolyhedronByGenerators( [ [ 0, 1, 3 ] ], [ 1 ] ); <Polyhedron given by its V-representation> gap> Display( B ); V-representation linearity 1, [ 1 ] begin 1 X 3 rational 0 1 3 end

`‣ Cdd_FourierProjection` ( P, i ) | ( operation ) |

Returns: a CddPolyhedron

The function returns the Fourier projection of the polyhedron in the subspace (O,x_1,\dots,x_{i-1},x_{i+1},\dots,x_n) after applying the Fourier elemination algorithm to get rid of the variable x_{i}.

To illustrate this projection, Let P= \mathrm{conv}( (1,2), (4,5) ) in \mathbb{Q}^2. \newline To find its projection on the subspace (O, x_1), we apply the Fourier elemination to get rid of x_2

gap> P := Cdd_PolyhedronByGenerators( [ [ 1, 1, 2 ], [ 1, 4, 5 ] ] ); <Polyhedron given by its V-representation> gap> H := Cdd_H_Rep( P ); <Polyhedron given by its H-representation> gap> Display( H ); H-representation linearity 1, [ 3 ] begin 3 X 3 rational 4 -1 0 -1 1 0 -1 -1 1 end gap> P_x1 := Cdd_FourierProjection( H, 2); <Polyhedron given by its H-representation> gap> Display( P_x1 ); H-representation linearity 1, [ 3 ] begin 3 X 3 rational 4 -1 0 -1 1 0 0 0 1 end gap> Display( Cdd_V_Rep( P_x1 ) ); V-representation begin 2 X 3 rational 1 1 0 1 4 0 end

Let again Q= Conv( (2,3,4), (2,4,5) )+ nonneg( (1,1,1) ), and let us compute its projection on (O,x_2,x_3)

gap> Q := Cdd_PolyhedronByGenerators( [ [ 1, 2, 3, 4 ],[ 1, 2, 4, 5 ], [ 0, 1, 1, 1 ] ] ); <Polyhedron given by its V-representation> gap> R := Cdd_H_Rep( Q ); <Polyhedron given by its H-representation> gap> Display( R ); H-representation linearity 1, [ 4 ] begin 4 X 4 rational 2 1 -1 0 -2 1 0 0 -1 -1 1 0 -1 0 -1 1 end gap> P_x2_x3 := Cdd_FourierProjection( R, 1); <Polyhedron given by its H-representation> gap> Display( P_x2_x3 ); H-representation linearity 2, [ 1, 3 ] begin 3 X 4 rational -1 0 -1 1 -3 0 1 0 0 1 0 0 end gap> Display( Cdd_V_Rep( last ) ) ; V-representation begin 2 X 4 rational 0 0 1 1 1 0 3 4 end

`‣ Cdd_IsContained` ( P1, P2 ) | ( operation ) |

Returns: `true`

or `false`

The function returns `true`

if P_1 is contained in P_2, otherwise returns `false`

.

gap> A := Cdd_PolyhedronByInequalities( [ [ 10, -1, 1, 0 ], > [ -24, 9, 2, 0 ], [ 1, 1, -1, 0 ], [ -23, -12, 1, 11 ] ], [ 4 ] ); <Polyhedron given by its H-representation> gap> B := Cdd_PolyhedronByInequalities( [ [ 1, 0, 0, 0 ], > [ -4, 1, 0, 0 ], [ 10, -1, 1, 0 ], [ -3, -1, 0, 1 ] ], [ 3, 4 ] ); <Polyhedron given by its H-representation> gap> Cdd_IsContained( B, A ); true gap> Display( Cdd_V_Rep( A ) ); V-representation begin 3 X 4 rational 1 2 3 4 1 4 -6 7 0 1 1 1 end gap> Display( Cdd_V_Rep( B ) ); V-representation begin 2 X 4 rational 1 4 -6 7 0 1 1 1 end

`‣ Cdd_Intersection` ( P1, P2 ) | ( operation ) |

Returns: a CddPolyhedron

The function returns the intersection of P_1 and P_2

gap> A := Cdd_PolyhedronByInequalities( [ [ 3, 4, 5 ] ], [ 1 ] );; gap> B := Cdd_PolyhedronByInequalities( [ [ 9, 7, 2 ] ], [ 1 ] );; gap> C := Cdd_Intersection( A, B );; gap> Display( Cdd_V_Rep( A ) ); V-representation linearity 1, [ 2 ] begin 2 X 3 rational 1 -3/4 0 0 -5 4 end gap> Display( Cdd_V_Rep( B ) ); V-representation linearity 1, [ 2 ] begin 2 X 3 rational 1 -9/7 0 0 -2 7 end gap> Display( Cdd_V_Rep( C ) ); V-representation begin 1 X 3 rational 1 -13/9 5/9 end

`2.3-3 \+`

`‣ \+` ( P1, P2 ) | ( operation ) |

Returns: a CddPolyhedron

The function returns the Minkuwski sum of P_1 and P_2.

gap> P := Cdd_PolyhedronByGenerators( [ [ 1, 2, 5 ], [ 0, 1, 2 ] ] ); < Polyhedron given by its V-representation > gap> Q := Cdd_PolyhedronByGenerators( [ [ 1, 4, 6 ], [ 1, 3, 7 ], [ 0, 3, 1 ] ] ); < Polyhedron given by its V-representation > gap> S := P+Q; < Polyhedron given by its H-representation > gap> V := Cdd_V_Rep( S ); < Polyhedron given by its V-representation > gap> Display( V ); V-representation begin 4 X 3 rational 0 3 1 1 6 11 1 5 12 0 1 2 end gap> Cdd_GeneratingVertices( P ); [ [ 2, 5 ] ] gap> Cdd_GeneratingVertices( Q ); [ [ 3, 7 ], [ 4, 6 ] ] gap> Cdd_GeneratingVertices( S ); [ [ 5, 12 ], [ 6, 11 ] ] gap> Cdd_GeneratingRays( P ); [ [ 1, 2 ] ] gap> Cdd_GeneratingRays( Q ); [ [ 3, 1 ] ] gap> Cdd_GeneratingRays( S ); [ [ 1, 2 ], [ 3, 1 ] ]

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