Fan -- the class of all fans

Description

A Fan represents a fan of rational convex polyhedral cones, i.e. a collection of cones, such that for every cone in the fan all faces are in the fan and for every two cones in the fan their intersection is a face of each (intersection condition). It need not be full dimensional or pure, and the cones need not be pointed. It is saved as a hash table which contains a list of the generating cones of the fan starting with those of maximal dimension. So for every cone in this list all faces are considered to be in the fan. The output of a Fan looks like this:

i1 : normalFan crossPolytope 3

o1 = {ambient dimension => 3         }
      number of generating cones => 6
      number of rays => 8
      top dimension of the cones => 3

o1 : Fan

This table displays a short summary of the properties of the Fan. However, one can not access the above information directly, because this is just a virtual hash table generated for the output. The data defining a Fan is extracted by the functions included in this package. A Fan can be constructed by collecting Cones that satisfy the intersection condition. Every cone that is added to a Fan is always considered as the collection of the Cone and all of its faces.

i2 : C1 = posHull matrix {{2,2},{1,-1}};

i3 : C2 = posHull matrix {{2,-2},{1,1}};

i4 : C3 = posHull matrix {{-2,-2},{1,-1}};

i5 : C4 = posHull matrix {{-2,2},{-1,-1}};

i6 : F = fan {C1,C2,C3,C4}

o6 = {ambient dimension => 2         }
      number of generating cones => 4
      number of rays => 4
      top dimension of the cones => 2

o6 : Fan

This fan is for example the normal fan of a ''flattened'' crosspolytope in 2-space.

Functions and methods returning an object of class Fan:

addCone -- adds cones to a Fan
ccRefinement -- computes the coarsest common refinement of a set of rays
faceFan -- computes the fan generated by the cones over the faces
fan -- generates a Fan
hirzebruch -- computes the fan of the r-th Hirzebruch surface
imageFan -- computes the fan of the image
normalFan -- computes the normalFan of a polyhedron
skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex
smoothSubfan -- computes the subfan of all smooth cones
stellarSubdivision(Fan,Matrix) -- see stellarSubdivision -- computes the stellar subdivision of the fan by a ray

Methods that use an object of class Fan:

addCone(Cone,Fan) -- see addCone -- adds cones to a Fan
addCone(Fan,Fan) -- see addCone -- adds cones to a Fan
addCone(List,Fan) -- see addCone -- adds cones to a Fan
commonFace(Cone,Fan) -- see commonFace -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans
commonFace(Fan,Cone) -- see commonFace -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans
commonFace(Fan,Fan) -- see commonFace -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans
cones(ZZ,Fan) -- see cones -- computes all cones of a fan of a certain dimension
contains(Fan,Cone) -- see contains -- checks if the first argument contains the second argument
dim(Fan) -- computes the dimension of a fan
directProduct(Fan,Fan) -- computes the direct product of two fans
Fan * Fan -- computes the direct product
Fan == Fan -- equality
incompCones(Cone,Fan) -- see incompCones -- returns the pairs of incompatible cones
incompCones(Fan,Cone) -- see incompCones -- returns the pairs of incompatible cones
incompCones(Fan,Fan) -- see incompCones -- returns the pairs of incompatible cones
isComplete(Fan) -- see isComplete -- checks completeness of a Fan or PolyhedralComplex
isPointed(Fan) -- see isPointed -- checks if a Cone or Fan is pointed
isPolytopal(Fan) -- see isPolytopal -- checks if a Fan is polytopal
isPure(Fan) -- see isPure -- checks if a Fan or PolyhedralComplex is of pure dimension
isSmooth(Fan) -- see isSmooth(Cone) -- checks if a Cone or Fan is smooth
linSpace(Fan) -- see linSpace -- computes a basis of the lineality space
maxCones(Fan) -- see maxCones -- displays the generating Cones of a Fan
net(Fan) -- displays characteristics of a fan
polytope(Fan) -- see polytope -- returns a polytope of which the fan is the normal fan if it is polytopal
rays(Fan) -- see rays -- displays all rays of a Cone, a Fan, or a Polyhedron
skeleton(ZZ,Fan) -- see skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex
smoothSubfan(Fan) -- see smoothSubfan -- computes the subfan of all smooth cones

For the programmer

The object Fan is a type, with ancestor classes PolyhedralObject < HashTable < Thing.

The source of this document is in OldPolyhedra.m2:4658:0.