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<div><h1>Working with cones</h1>
<div>We start with a cone in 2-space which is the positive hull (<a href="_pos__Hull.html" title="computes the positive hull of rays, cones, and the cone over a polyhedron">posHull</a>) of a given set of rays.<table class="examples"><tr><td><pre>i1 : R = matrix {{1,1,2},{2,1,1}}
o1 = | 1 1 2 |
| 2 1 1 |
2 3
o1 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i2 : C = posHull R
o2 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of the cone => 2
number of facets => 2
number of rays => 2
o2 : Cone</pre>
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<tr><td><pre>i3 : ambDim C
o3 = 2</pre>
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<p/>
This gives an overview of the characteristics of the cone. If we want to know
more details, we can ask for them.<table class="examples"><tr><td><pre>i4 : rays C
o4 = | 2 1 |
| 1 2 |
2 2
o4 : Matrix QQ <--- QQ</pre>
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<p/>
Using <a href="_rays.html" title="displays all rays of a Cone, a Fan, or a Polyhedron">rays</a> we see that (1,1) is not an extremal ray of the cone.<table class="examples"><tr><td><pre>i5 : HS = halfspaces C
o5 = | -1 2 |
| 2 -1 |
2 2
o5 : Matrix QQ <--- QQ</pre>
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<p/>
The function <a href="_halfspaces.html" title="computes the defining half-spaces of a Cone or a Polyhedron">halfspaces</a> gives the defining linear half-spaces, i.e. <tt>C</tt> is given by all <tt>p</tt> in
the defining linear hyperplanes that satisfy <tt>HS*p >= 0</tt>. But in this case there are none, so the polyhedron is of full
dimension. Furthermore, we can construct the positive hull of a set of rays and a linear subspace.<table class="examples"><tr><td><pre>i6 : R1 = R || matrix {{0,0,0}}
o6 = | 1 1 2 |
| 2 1 1 |
| 0 0 0 |
3 3
o6 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i7 : LS = matrix {{1},{1},{1}}
o7 = | 1 |
| 1 |
| 1 |
3 1
o7 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i8 : C1 = posHull(R1,LS)
o8 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of the cone => 3
number of facets => 2
number of rays => 2
o8 : Cone</pre>
</td></tr>
<tr><td><pre>i9 : rays C1
o9 = | 0 0 |
| -1 1 |
| -2 -1 |
3 2
o9 : Matrix QQ <--- QQ</pre>
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<p/>
Note that the rays are given modulo the lineality space. On the other hand we can
construct cones as the <a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> of linear half-spaces and hyperplanes.<table class="examples"><tr><td><pre>i10 : HS = transpose R1
o10 = | 1 2 0 |
| 1 1 0 |
| 2 1 0 |
3 3
o10 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i11 : HP = matrix {{1,1,1}}
o11 = | 1 1 1 |
1 3
o11 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i12 : C2 = intersection(HS,HP)
o12 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 2
number of facets => 2
number of rays => 2
o12 : Cone</pre>
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<p/>
This is a two dimensional cone in 3-space with the following rays:<table class="examples"><tr><td><pre>i13 : rays C2
o13 = | 2 -1 |
| -1 2 |
| -1 -1 |
3 2
o13 : Matrix QQ <--- QQ</pre>
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<p/>
If we don't intersect with the hyperplane we get a full dimensional cone.<table class="examples"><tr><td><pre>i14 : C3 = intersection HS
o14 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of the cone => 3
number of facets => 2
number of rays => 2
o14 : Cone</pre>
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<tr><td><pre>i15 : rays C3
o15 = | 2 -1 |
| -1 2 |
| 0 0 |
3 2
o15 : Matrix QQ <--- QQ</pre>
</td></tr>
<tr><td><pre>i16 : linSpace C3
o16 = | 0 |
| 0 |
| 1 |
3 1
o16 : Matrix QQ <--- QQ</pre>
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<p/>
Again, the rays are given modulo the lineality space. Also, one can use
given cones, for example the positive orthant (<a href="_pos__Orthant.html" title="generates the positive orthant in n-space">posOrthant</a>):<table class="examples"><tr><td><pre>i17 : C4 = posOrthant 3
o17 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o17 : Cone</pre>
</td></tr>
<tr><td><pre>i18 : rays C4
o18 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o18 : Matrix QQ <--- QQ</pre>
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<p/>
Now that we can construct cones, we can turn to the functions
that can be applied to cones. First of all, we can apply the <a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a>
function also to a pair of cones in the same ambient space:<table class="examples"><tr><td><pre>i19 : C5 = intersection(C1,C2)
o19 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 2
number of facets => 2
number of rays => 2
o19 : Cone</pre>
</td></tr>
<tr><td><pre>i20 : rays C5
o20 = | 1 0 |
| 0 1 |
| -1 -1 |
3 2
o20 : Matrix QQ <--- QQ</pre>
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<p/>
On the other hand, we can take their positive hull by using <a href="_pos__Hull.html" title="computes the positive hull of rays, cones, and the cone over a polyhedron">posHull</a>:<table class="examples"><tr><td><pre>i21 : C6 = posHull(C1,C2)
o21 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of the cone => 3
number of facets => 2
number of rays => 2
o21 : Cone</pre>
</td></tr>
<tr><td><pre>i22 : rays C6
o22 = | 0 0 |
| 1 -1 |
| 0 -1 |
3 2
o22 : Matrix QQ <--- QQ</pre>
</td></tr>
<tr><td><pre>i23 : linSpace C6
o23 = | 1 |
| 1 |
| 1 |
3 1
o23 : Matrix QQ <--- QQ</pre>
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<p/>
Furthermore, both functions (<a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> and <a href="_pos__Hull.html" title="computes the positive hull of rays, cones, and the cone over a polyhedron">posHull</a>) can
be applied to a list containing any number of cones and matrices defining
rays and lineality space or linear half-spaces and hyperplanes. These must be in the
same ambient space. For example:<table class="examples"><tr><td><pre>i24 : R2 = matrix {{2,-1},{-1,2},{-1,-1}}
o24 = | 2 -1 |
| -1 2 |
| -1 -1 |
3 2
o24 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i25 : C7 = posHull {R2,C3,C4}
o25 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of the cone => 3
number of facets => 2
number of rays => 2
o25 : Cone</pre>
</td></tr>
<tr><td><pre>i26 : rays C7
o26 = | 2 -1 |
| -1 2 |
| 0 0 |
3 2
o26 : Matrix QQ <--- QQ</pre>
</td></tr>
<tr><td><pre>i27 : linSpace C7
o27 = | 0 |
| 0 |
| 1 |
3 1
o27 : Matrix QQ <--- QQ</pre>
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<p/>
Since they are all cones their positive hull is the same as their
Minkowski sum, so in fact:<table class="examples"><tr><td><pre>i28 : C6 == C1 + C2
o28 = true</pre>
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<p/>
But we can take the Minkowski sum of a cone and a polyhedron. For this,
both objects must lie in the same ambient space and the resulting object is then
a polyhedron:<table class="examples"><tr><td><pre>i29 : P = crossPolytope 3
o29 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 0
number of vertices => 6
o29 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i30 : P1 = C6 + P
o30 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of polyhedron => 3
number of facets => 2
number of rays => 2
number of vertices => 1
o30 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i31 : (vertices P1,rays P1)
o31 = (| 0 |, | 0 0 |)
| 0 | | 1 -1 |
| 1 | | 0 -1 |
o31 : Sequence</pre>
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<p/>
Furthermore, we can take the direct product (<a href="_direct__Product.html" title="computes the direct product of two convex objects">directProduct</a>) of
two cones.<table class="examples"><tr><td><pre>i32 : C8 = C * C1
o32 = {ambient dimension => 5 }
dimension of lineality space => 1
dimension of the cone => 5
number of facets => 4
number of rays => 4
o32 : Cone</pre>
</td></tr>
<tr><td><pre>i33 : rays C8
o33 = | 2 1 0 0 |
| 1 2 0 0 |
| 0 0 0 0 |
| 0 0 -1 1 |
| 0 0 -2 -1 |
5 4
o33 : Matrix QQ <--- QQ</pre>
</td></tr>
<tr><td><pre>i34 : linSpace C8
o34 = | 0 |
| 0 |
| 1 |
| 1 |
| 1 |
5 1
o34 : Matrix QQ <--- QQ</pre>
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<p/>
The result is in QQ^5.<table class="examples"><tr><td><pre>i35 : ambDim C8
o35 = 5</pre>
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<p/>
To find out more about this cone use for example <a href="_f__Vector.html" title="computes the f-vector of a Cone or Polyhedron">fVector</a>:<table class="examples"><tr><td><pre>i36 : fVector C8
o36 = {0, 1, 4, 6, 4, 1}
o36 : List</pre>
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<p/>
This function gives the number of faces of each dimension, so it has 1
vertex, the origin, 1 line, 4 two dimensional faces and so on. We can access the
faces of a certain codimension via <a href="_faces.html" title="computes all faces of a certain codimension of a Cone or Polyhedron">faces</a>:<table class="examples"><tr><td><pre>i37 : L = faces(1,C8)
o37 = {{ambient dimension => 5 }, {ambient dimension => 5
dimension of lineality space => 1 dimension of lineality space =>
dimension of the cone => 4 dimension of the cone => 4
number of facets => 3 number of facets => 3
number of rays => 3 number of rays => 3
-----------------------------------------------------------------------
}, {ambient dimension => 5 }, {ambient dimension => 5
1 dimension of lineality space => 1 dimension of lineality space
dimension of the cone => 4 dimension of the cone => 4
number of facets => 3 number of facets => 3
number of rays => 3 number of rays => 3
-----------------------------------------------------------------------
}}
=> 1
o37 : List</pre>
</td></tr>
<tr><td><pre>i38 : apply(L,rays)
o38 = {| 2 0 0 |, | 1 0 0 |, | 2 1 0 |, | 2 1 0 |}
| 1 0 0 | | 2 0 0 | | 1 2 0 | | 1 2 0 |
| 0 0 0 | | 0 0 0 | | 0 0 0 | | 0 0 0 |
| 0 -1 1 | | 0 -1 1 | | 0 0 -1 | | 0 0 1 |
| 0 -2 -1 | | 0 -2 -1 | | 0 0 -2 | | 0 0 -1 |
o38 : List</pre>
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<p/>
We can also check if the cone is smooth:<table class="examples"><tr><td><pre>i39 : isSmooth C8
o39 = false</pre>
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<p/>
Evenmore we can compute the Hilbert basis of the cone with <a href="_hilbert__Basis.html" title="computes the Hilbert basis of a Cone">hilbertBasis</a>.<table class="examples"><tr><td><pre>i40 : L = hilbertBasis C8
o40 = {| 0 |, | 0 |, | 0 |, | -1 |, | 0 |, | -1 |}
| 0 | | 0 | | 0 | | -1 | | -1 | | 0 |
| 0 | | 0 | | 0 | | -2 | | -2 | | -2 |
| 1 | | 2 | | 1 | | 0 | | 0 | | 0 |
| 1 | | 1 | | 2 | | 0 | | 0 | | 0 |
o40 : List</pre>
</td></tr>
<tr><td><pre>i41 : #L
o41 = 6</pre>
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</table>
<p/>
Finally, there is also a function to compute the dual cone, i.e.
the set of all points in the dual space that are positive on the cone.<table class="examples"><tr><td><pre>i42 : C9 = dualCone C8
o42 = {ambient dimension => 5 }
dimension of lineality space => 0
dimension of the cone => 4
number of facets => 4
number of rays => 4
o42 : Cone</pre>
</td></tr>
<tr><td><pre>i43 : rays C9
o43 = | 2 -1 0 0 |
| -1 2 0 0 |
| 0 0 2 -1 |
| 0 0 -1 2 |
| 0 0 -1 -1 |
5 4
o43 : Matrix QQ <--- QQ</pre>
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