GAP (Convex) - Chapter 5: Cone

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5 Cone
5.1 Cone: Category and Representations
5.1-1 IsCone
5.2 Cone: Properties
5.2-1 IsRay
5.3 Cone: Attributes
5.3-1 DualCone
5.3-2 HilbertBasis
5.3-3 RaysInFacets
5.3-4 Facets
5.3-5 GridGeneratedByCone
5.3-6 FactorGrid
5.3-7 GridGeneratedByOrthogonalCone
5.3-8 DefiningInequalities
5.3-9 IsContainedInFan
5.3-10 FactorGridMorphism
5.4 Cone: Methods
5.4-1 IntersectionOfCones
5.4-2 Contains
5.4-3 StarFan
5.4-4 StarFan
5.4-5 StarSubdivisionOfIthMaximalCone
5.5 Cone: Constructors
5.5-1 Cone
5.6 Cone: Examples
5.6-1 Cone example

### 5 Cone

#### 5.1 Cone: Category and Representations

##### 5.1-1 IsCone

`‣ IsCone`( M ) | ( category )

Returns: `true` or `false`

The GAP category of a cone.

Remember: Every cone is a convex object.

#### 5.2 Cone: Properties

##### 5.2-1 IsRay

`‣ IsRay`( cone ) | ( property )

Returns: `true` or `false`

Checks if the cone cone is a ray, i.e. if it has only one ray generator.

#### 5.3 Cone: Attributes

##### 5.3-1 DualCone

`‣ DualCone`( cone ) | ( attribute )

Returns: a cone

Returns the dual cone of the cone cone.

##### 5.3-2 HilbertBasis

`‣ HilbertBasis`( cone ) | ( attribute )

Returns: a list

Returns a Hilbert Basis of the cone cone.

##### 5.3-3 RaysInFacets

`‣ RaysInFacets`( cone ) | ( attribute )

Returns: a list

Returns an incidence matrix for the rays in the facets of the cone cone. The ith entry of the result corresponds to the ith facet, the jth entry of this is 1 if the jth ray is in th ith facet, 0 otherwise.

##### 5.3-4 Facets

`‣ Facets`( cone ) | ( attribute )

Returns: a list

Returns a list of the facets of the cone cone as homalg cones.

##### 5.3-5 GridGeneratedByCone

`‣ GridGeneratedByCone`( cone ) | ( attribute )

Returns: a homalg module

Returns the grid generated by the lattice points of the cone cone as a homalg module.

##### 5.3-6 FactorGrid

`‣ FactorGrid`( cone ) | ( attribute )

Returns: a homalg module

Returns the factor of the containing grid of the cone cone and the grid generated by cone.

##### 5.3-7 GridGeneratedByOrthogonalCone

`‣ GridGeneratedByOrthogonalCone`( cone ) | ( attribute )

Returns: a homalg module

Returns the grid generated by the lattice points of the orthogonal cone of the cone cone.

##### 5.3-8 DefiningInequalities

`‣ DefiningInequalities`( cone ) | ( attribute )

Returns: a list

Returns a list of the defining inequalities of the cone cone.

##### 5.3-9 IsContainedInFan

`‣ IsContainedInFan`( cone ) | ( attribute )

Returns: a fan

If the cone cone is constructed as part of a fan, this method returns the fan.

##### 5.3-10 FactorGridMorphism

`‣ FactorGridMorphism`( cone ) | ( attribute )

Returns: a morphism

Returns the morphism to the factor grid of the cone cone.

#### 5.4 Cone: Methods

##### 5.4-1 IntersectionOfCones

`‣ IntersectionOfCones`( cone1, cone2 ) | ( operation )

Returns: a cone

If the cones cone1 and cone2 share a face, the method returns their intersection,

##### 5.4-2 Contains

`‣ Contains`( cone1, cone2 ) | ( operation )

Returns: `true` or `false`

Returns `true` if the cone cone1 contains the cone cone2, `false` otherwise.

##### 5.4-3 StarFan

`‣ StarFan`( cone ) | ( operation )

Returns: a fan

Returns the star fan of the cone cone, as described in cox, 3.2.7

##### 5.4-4 StarFan

`‣ StarFan`( cone, fan ) | ( operation )

Returns: a fan

Returns the star fan of the fan fan along the cone cone, as described in cox, 3.2.7

##### 5.4-5 StarSubdivisionOfIthMaximalCone

`‣ StarSubdivisionOfIthMaximalCone`( fan, numb ) | ( operation )

Returns: a fan

Returns the star subdivision of the fan fan on the numbth maximal cone as in cox, 3.3.13.

#### 5.5 Cone: Constructors

##### 5.5-1 Cone

`‣ Cone`( cone ) | ( operation )

Returns: a cone

Returns a cone generated by the rays in cone.

#### 5.6 Cone: Examples

##### 5.6-1 Cone example

    gap> C := Cone([[1,2,3],[2,1,1],[1,0,0],[0,1,1]]);
    <A cone in |R^3>
    gap> Length( RayGenerators( C ) );
    3
    gap> IsSmooth( C );
    true
    gap> Length( HilbertBasis( C ) );
    3
    gap> IsSimplicial( C );
    true
    gap> DC := DualCone( C );
    <A cone in |R^3>
    gap> Length( HilbertBasis( DC ) );
    3

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