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<div><h1>Working with fans</h1>
<div>We start by constructing a fan, which consists of a single cone and all of its
faces:<table class="examples"><tr><td><pre>i1 : C = posHull matrix {{1,0,0},{0,1,0},{0,0,1}}
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o1 : Cone</pre>
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<tr><td><pre>i2 : F = fan C
o2 = {ambient dimension => 3 }
number of generating cones => 1
number of rays => 3
top dimension of the cones => 3
o2 : Fan</pre>
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By this, we have already constructed the fan consisting of the
positive orthant and all of its faces. The package saves the generating cones
of the fan, which can be accessed by:<table class="examples"><tr><td><pre>i3 : genCones F
o3 = {{ambient dimension => 3 }}
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o3 : List</pre>
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Now we could expand the fan by adding more cones, for example the following:<table class="examples"><tr><td><pre>i4 : C1 = posHull matrix {{1,0,0},{1,1,0},{0,0,-1}}
o4 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o4 : Cone</pre>
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But in this case we can not, because the two cones are not compatible,
i.e. their intersection is not a face of each. So, when one tries to add a cone
to a fan that is not compatible with one of the generating cones of the fan, the
function <a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> gives an error. For two cones one can
check if their intersection is a common face by using <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>:<table class="examples"><tr><td><pre>i5 : commonFace(C,C1)
o5 = false</pre>
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Since the intersection of both is already computed in this function
there is a different function, which also returns the intersection, to save
computation time when one needs the intersection afterward anyway:<table class="examples"><tr><td><pre>i6 : (b,C2) = areCompatible(C,C1)
o6 = (false, {ambient dimension => 3 })
dimension of lineality space => 0
dimension of the cone => 2
number of facets => 2
number of rays => 2
o6 : Sequence</pre>
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<tr><td><pre>i7 : rays C2
o7 = | 0 1 |
| 1 1 |
| 0 0 |
3 2
o7 : Matrix QQ <--- QQ</pre>
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So we can make the cone compatible and add it to the fan.<table class="examples"><tr><td><pre>i8 : C1 = posHull matrix {{1,0,0},{0,1,0},{0,0,-1}}
o8 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o8 : Cone</pre>
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<tr><td><pre>i9 : F = addCone(C1,F)
o9 = {ambient dimension => 3 }
number of generating cones => 2
number of rays => 4
top dimension of the cones => 3
o9 : Fan</pre>
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<p/>
Instead of creating a fan with one cone and then adding more cones, we
can also make a fan out of a list of cones:<table class="examples"><tr><td><pre>i10 : C2 = posHull matrix {{-1,0,0},{0,1,0},{0,0,1}};</pre>
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<tr><td><pre>i11 : C3 = posHull matrix {{-1,0,0},{0,1,0},{0,0,-1}};</pre>
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<tr><td><pre>i12 : C4 = posHull matrix {{-1,0,0},{0,-1,0},{0,0,1}};</pre>
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<tr><td><pre>i13 : C5 = posHull matrix {{-1,0,0},{0,-1,0},{0,0,-1}};</pre>
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<tr><td><pre>i14 : F1 = fan {C2,C3,C4,C5}
o14 = {ambient dimension => 3 }
number of generating cones => 4
number of rays => 5
top dimension of the cones => 3
o14 : Fan</pre>
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Furthermore, we could add a list of cones to an existing fan:<table class="examples"><tr><td><pre>i15 : C6 = posHull matrix {{1,0,0},{0,-1,0},{0,0,1}};</pre>
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<tr><td><pre>i16 : C7 = posHull matrix {{1,0,0},{0,-1,0},{0,0,-1}};</pre>
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<tr><td><pre>i17 : F1 = addCone( {C6,C7}, F1)
o17 = {ambient dimension => 3 }
number of generating cones => 6
number of rays => 6
top dimension of the cones => 3
o17 : Fan</pre>
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Finally, we can add a whole fan to another fan:<table class="examples"><tr><td><pre>i18 : F1 = addCone(F,F1)
o18 = {ambient dimension => 3 }
number of generating cones => 8
number of rays => 6
top dimension of the cones => 3
o18 : Fan</pre>
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So, <a href="_fan.html" title="generates a Fan">fan</a> and <a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> are the methods to construct
fans ''from scratch'', but there are also methods to get fans directly, for example <a href="_normal__Fan.html" title="computes the normalFan of a polyhedron">normalFan</a>,
which constructs the inner normal fan of a polytope.<table class="examples"><tr><td><pre>i19 : P = hypercube 4
o19 = {ambient dimension => 4 }
dimension of lineality space => 0
dimension of polyhedron => 4
number of facets => 8
number of rays => 0
number of vertices => 16
o19 : Polyhedron</pre>
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<tr><td><pre>i20 : F2 = normalFan P
o20 = {ambient dimension => 4 }
number of generating cones => 16
number of rays => 8
top dimension of the cones => 4
o20 : Fan</pre>
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Now we have seen how to construct fans, so we turn to functions on fans,
for example the direct product (<a href="_direct__Product.html" title="computes the direct product of two convex objects">directProduct</a>:<table class="examples"><tr><td><pre>i21 : F3 = fan {posHull matrix {{1}},posHull matrix {{-1}}}
o21 = {ambient dimension => 1 }
number of generating cones => 2
number of rays => 2
top dimension of the cones => 1
o21 : Fan</pre>
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<tr><td><pre>i22 : F1 = F3 * F1
o22 = {ambient dimension => 4 }
number of generating cones => 16
number of rays => 8
top dimension of the cones => 4
o22 : Fan</pre>
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The result is in the direct product of the ambient spaces.<table class="examples"><tr><td><pre>i23 : ambDim F1
o23 = 4</pre>
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Of course, we can check if two fans are the same:<table class="examples"><tr><td><pre>i24 : F1 == F2
o24 = true</pre>
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A bit more on fans can be found in part 2: <a href="___Working_spwith_spfans_sp-_sp__Part_sp2.html" title="">Working with fans - Part 2</a>.</div>
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