  
  [1X5 [33X[0;0YCone[133X[101X
  
  
  [1X5.1 [33X[0;0YCone: Category and Representations[133X[101X
  
  [1X5.1-1 IsCone[101X
  
  [29X[2XIsCone[102X( [3XM[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of a cone.[133X
  
  [33X[0;0YRemember: Every cone is a convex object.[133X
  
  
  [1X5.2 [33X[0;0YCone: Properties[133X[101X
  
  [1X5.2-1 IsRay[101X
  
  [29X[2XIsRay[102X( [3Xcone[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the cone [3Xcone[103X is a ray, i.e. if it has only one ray generator.[133X
  
  
  [1X5.3 [33X[0;0YCone: Attributes[133X[101X
  
  [1X5.3-1 DualCone[101X
  
  [29X[2XDualCone[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya cone[133X
  
  [33X[0;0YReturns the dual cone of the cone [3Xcone[103X.[133X
  
  [1X5.3-2 HilbertBasis[101X
  
  [29X[2XHilbertBasis[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a Hilbert Basis of the cone [3Xcone[103X.[133X
  
  [1X5.3-3 RaysInFacets[101X
  
  [29X[2XRaysInFacets[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns an incidence matrix for the rays in the facets of the cone [3Xcone[103X. 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.[133X
  
  [1X5.3-4 Facets[101X
  
  [29X[2XFacets[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of the facets of the cone [3Xcone[103X as homalg cones.[133X
  
  [1X5.3-5 GridGeneratedByCone[101X
  
  [29X[2XGridGeneratedByCone[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya homalg module[133X
  
  [33X[0;0YReturns  the  grid  generated  by  the  lattice points of the cone [3Xcone[103X as a
  homalg module.[133X
  
  [1X5.3-6 FactorGrid[101X
  
  [29X[2XFactorGrid[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya homalg module[133X
  
  [33X[0;0YReturns  the  factor  of  the  containing grid of the cone [3Xcone[103X and the grid
  generated by [3Xcone[103X.[133X
  
  [1X5.3-7 GridGeneratedByOrthogonalCone[101X
  
  [29X[2XGridGeneratedByOrthogonalCone[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya homalg module[133X
  
  [33X[0;0YReturns  the  grid generated by the lattice points of the orthogonal cone of
  the cone [3Xcone[103X.[133X
  
  [1X5.3-8 DefiningInequalities[101X
  
  [29X[2XDefiningInequalities[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of the defining inequalities of the cone [3Xcone[103X.[133X
  
  [1X5.3-9 IsContainedInFan[101X
  
  [29X[2XIsContainedInFan[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YIf  the  cone  [3Xcone[103X is constructed as part of a fan, this method returns the
  fan.[133X
  
  [1X5.3-10 FactorGridMorphism[101X
  
  [29X[2XFactorGridMorphism[102X( [3Xcone[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns the morphism to the factor grid of the cone [3Xcone[103X.[133X
  
  
  [1X5.4 [33X[0;0YCone: Methods[133X[101X
  
  [1X5.4-1 IntersectionOfCones[101X
  
  [29X[2XIntersectionOfCones[102X( [3Xcone1[103X, [3Xcone2[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya cone[133X
  
  [33X[0;0YIf  the  cones  [3Xcone1[103X  and  [3Xcone2[103X  share  a  face,  the method returns their
  intersection,[133X
  
  [1X5.4-2 Contains[101X
  
  [29X[2XContains[102X( [3Xcone1[103X, [3Xcone2[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YReturns [10Xtrue[110X if the cone [3Xcone1[103X contains the cone [3Xcone2[103X, [10Xfalse[110X otherwise.[133X
  
  [1X5.4-3 StarFan[101X
  
  [29X[2XStarFan[102X( [3Xcone[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the star fan of the cone [3Xcone[103X, as described in cox, 3.2.7[133X
  
  [1X5.4-4 StarFan[101X
  
  [29X[2XStarFan[102X( [3Xcone[103X, [3Xfan[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns  the  star  fan  of the fan [3Xfan[103X along the cone [3Xcone[103X, as described in
  cox, 3.2.7[133X
  
  [1X5.4-5 StarSubdivisionOfIthMaximalCone[101X
  
  [29X[2XStarSubdivisionOfIthMaximalCone[102X( [3Xfan[103X, [3Xnumb[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the star subdivision of the fan [3Xfan[103X on the [3Xnumb[103Xth maximal cone as in
  cox, 3.3.13.[133X
  
  
  [1X5.5 [33X[0;0YCone: Constructors[133X[101X
  
  [1X5.5-1 Cone[101X
  
  [29X[2XCone[102X( [3Xcone[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya cone[133X
  
  [33X[0;0YReturns a cone generated by the rays in [3Xcone[103X.[133X
  
  
  [1X5.6 [33X[0;0YCone: Examples[133X[101X
  
  
  [1X5.6-1 [33X[0;0YCone example[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC := Cone([[1,2,3],[2,1,1],[1,0,0],[0,1,1]]);[127X[104X
    [4X[28X<A cone in |R^3>[128X[104X
    [4X[25Xgap>[125X [27XLength( RayGenerators( C ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XIsSmooth( C );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLength( HilbertBasis( C ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XIsSimplicial( C );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDC := DualCone( C );[127X[104X
    [4X[28X<A cone in |R^3>[128X[104X
    [4X[25Xgap>[125X [27XLength( HilbertBasis( DC ) );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
