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<head><title>Fan -- the class of all fans</title>
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<div><h1>Fan -- the class of all fans</h1>
<div class="single"><h2>Description</h2>
<div>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:<table class="examples"><tr><td><pre>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</pre>
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<p/>
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.<table class="examples"><tr><td><pre>i2 : C1 = posHull matrix {{2,2},{1,-1}};</pre>
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<tr><td><pre>i3 : C2 = posHull matrix {{2,-2},{1,1}};</pre>
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<tr><td><pre>i4 : C3 = posHull matrix {{-2,-2},{1,-1}};</pre>
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<tr><td><pre>i5 : C4 = posHull matrix {{-2,2},{-1,-1}};</pre>
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<tr><td><pre>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</pre>
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<p/>
This fan is for example the normal fan of a ''flattened'' crosspolytope in 2-space.<p/>
See also <a href="___Working_spwith_spfans.html" title="">Working with fans</a>.</div>
</div>
<div class="waystouse"><h2>Functions and methods returning an object of class Fan :</h2>
<ul><li><span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></li>
<li><span><a href="_cc__Refinement.html" title="computes the coarsest common refinement of a set of rays">ccRefinement</a> -- computes the coarsest common refinement of a set of rays</span></li>
<li><span><a href="_face__Fan.html" title=" computes the fan generated by the cones over the faces">faceFan</a> -- computes the fan generated by the cones over the faces</span></li>
<li><span><a href="_fan.html" title="generates a Fan">fan</a> -- generates a Fan</span></li>
<li><span><a href="_hirzebruch.html" title="computes the fan of the r-th Hirzebruch surface">hirzebruch</a> -- computes the fan of the r-th Hirzebruch surface</span></li>
<li><span><a href="_image__Fan.html" title=" computes the fan of the image">imageFan</a> -- computes the fan of the image</span></li>
<li><span><a href="_normal__Fan.html" title="computes the normalFan of a polyhedron">normalFan</a> -- computes the normalFan of a polyhedron</span></li>
<li><span><a href="_skeleton.html" title="computes the k-skeleton of a Fan">skeleton</a> -- computes the k-skeleton of a Fan</span></li>
<li><span><a href="_smooth__Subfan.html" title="computes the subfan of all smooth cones">smoothSubfan</a> -- computes the subfan of all smooth cones</span></li>
<li><span><a href="_stellar__Subdivision.html" title="computes the stellar subdivision of the fan by a ray">stellarSubdivision</a> -- computes the stellar subdivision of the fan by a ray</span></li>
</ul>
<h2>Methods that use an object of class Fan :</h2>
<ul><li><span>addCone(Cone,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>addCone(Fan,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>addCone(List,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>ambDim(Fan), see <span><a href="_amb__Dim.html" title="ambient dimension of a Polyhedron, Cone or Fan">ambDim</a> -- ambient dimension of a Polyhedron, Cone or Fan</span></span></li>
<li><span>commonFace(Cone,Fan), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Fan,Cone), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Fan,Fan), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>cones(ZZ,Fan), see <span><a href="_cones.html" title="computes all cones of a fan of a certain dimension">cones</a> -- computes all cones of a fan of a certain dimension</span></span></li>
<li><span>contains(Fan,Cone), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span><a href="_dim_lp__Fan_rp.html" title="computes the dimension of a fan">dim(Fan)</a> -- computes the dimension of a fan</span></li>
<li><span><a href="_direct__Product_lp__Fan_cm__Fan_rp.html" title="computes the direct product of two fans">directProduct(Fan,Fan)</a> -- computes the direct product of two fans</span></li>
<li><span><a href="___Fan_sp_st_sp__Fan.html" title="computes the direct product">Fan * Fan</a> -- computes the direct product</span></li>
<li><span><a href="___Fan_sp_eq_eq_sp__Fan.html" title="equality">Fan == Fan</a> -- equality</span></li>
<li><span>genCones(Fan), see <span><a href="_gen__Cones.html" title="displays the generating Cones of a Fan">genCones</a> -- displays the generating Cones of a Fan</span></span></li>
<li><span>incompCones(Cone,Fan), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>incompCones(Fan,Cone), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>incompCones(Fan,Fan), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>isComplete(Fan), see <span><a href="_is__Complete.html" title="checks completeness of a Fan">isComplete</a> -- checks completeness of a Fan</span></span></li>
<li><span>isPointed(Fan), see <span><a href="_is__Pointed.html" title="checks if a Cone or Fan is pointed">isPointed</a> -- checks if a Cone or Fan is pointed</span></span></li>
<li><span>isPolytopal(Fan), see <span><a href="_is__Polytopal.html" title="checks if a Fan is polytopal">isPolytopal</a> -- checks if a Fan is polytopal</span></span></li>
<li><span>isPure(Fan), see <span><a href="_is__Pure.html" title="checks if a Fan is of pure dimension">isPure</a> -- checks if a Fan is of pure dimension</span></span></li>
<li><span>isSmooth(Fan), see <span><a href="_is__Smooth.html" title="checks if a Cone or Fan is smooth">isSmooth</a> -- checks if a Cone or Fan is smooth</span></span></li>
<li><span>linSpace(Fan), see <span><a href="_lin__Space.html" title="computes a basis of the lineality space">linSpace</a> -- computes a basis of the lineality space</span></span></li>
<li><span><a href="_net_lp__Fan_rp.html" title="displays characteristics of a fan">net(Fan)</a> -- displays characteristics of a fan</span></li>
<li><span>polytope(Fan), see <span><a href="_polytope.html" title="returns a polytope of which the fan is the normal fan if it is polytopal">polytope</a> -- returns a polytope of which the fan is the normal fan if it is polytopal</span></span></li>
<li><span>rays(Fan), see <span><a href="_rays.html" title="displays all rays of a Cone, a Fan, or a Polyhedron">rays</a> -- displays all rays of a Cone, a Fan, or a Polyhedron</span></span></li>
<li><span>skeleton(ZZ,Fan), see <span><a href="_skeleton.html" title="computes the k-skeleton of a Fan">skeleton</a> -- computes the k-skeleton of a Fan</span></span></li>
<li><span>smoothSubfan(Fan), see <span><a href="_smooth__Subfan.html" title="computes the subfan of all smooth cones">smoothSubfan</a> -- computes the subfan of all smooth cones</span></span></li>
<li><span>stellarSubdivision(Fan,Matrix), see <span><a href="_stellar__Subdivision.html" title="computes the stellar subdivision of the fan by a ray">stellarSubdivision</a> -- computes the stellar subdivision of the fan by a ray</span></span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Fan.html" title="the class of all fans">Fan</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="___Polyhedral__Object.html" title="the class of all polyhedral objects in Polyhedra">PolyhedralObject</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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