====== BigObject HyperplaneArrangement<Scalar> ======
//from application [[..:fan|fan]]//\\
\\
 A hyperplane arrangement. The hyperplane arrangement is given by a matrix ''[[..:fan:HyperplaneArrangement#HYPERPLANES |HYPERPLANES]]'' whose rows are the linear equations of the hyperplanes and an optional support cone. The support cone defaults to being the whole space. Duplicate hyperplanes are ignored, as well as hyperplanes that intersect the support cone trivially. The support cone is subdivided by the hyperplanes resulting in a fan ''[[..:fan:HyperplaneArrangement#CELL_DECOMPOSITION |CELL_DECOMPOSITION]]''.

  ? Type Parameters:
  :: ''Scalar'': numeric data type used for the coordinates, must be an ordered field.  Default is ''[[..:common#Rational |Rational]]''.
  ? Example:
  :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
  :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1
</code>
  :: <code perl> > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>
  ? Permutations:
  :
    ? ConesPerm:
    :: permuting the ''[[..:fan:PolyhedralFan#RAYS |RAYS]]''
===== Properties =====

==== Input property ====
 These properties are for input only. They allow redundant information.
----
{{anchor:input_hyperplanes:}}
  ?  **''INPUT_HYPERPLANES''**
  :: A matrix containing the input hyperplanes of the arrangement as rows.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:support:}}
  ?  **''SUPPORT''**
  :: A cone being subdivided by the ''[[..:fan:HyperplaneArrangement#HYPERPLANES |HYPERPLANES]]'' defaults to the whole space.
    ? Type:
    :''[[..:polytope:Cone |Cone]]<Scalar>''
    ? Example:
    :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>
    ? Example:
    :: Subdivide the two-dimensional space along the axes
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 -1 0
 0 -1
 0 1
 1 0
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES;
 {0 1}
 {0 2}
 {1 3}
 {2 3}
</code>
    :: <code perl> > print $CD->COMPLETE;
 true
</code>


----

==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:cell_signatures:}}
  ?  **''CELL_SIGNATURES''**
  :: The i-th entry is the signature of the i-th maximal cone of the ''[[..:fan:HyperplaneArrangement#CELL_DECOMPOSITION |CELL_DECOMPOSITION]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
    ? Example:
    :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>
    :: <code perl> > print $HA->CELL_SIGNATURES;
 {}
 {0}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}
</code>
    :: <code perl> > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}
</code>


----

==== Geometry ====
 These properties capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:cell_decomposition:}}
  ?  **''CELL_DECOMPOSITION''**
  :: Slicing the ''[[..:fan:HyperplaneArrangement#SUPPORT |SUPPORT]]'' along every hyperplane of ''[[..:fan:HyperplaneArrangement#HYPERPLANES |HYPERPLANES]]'' one gets a polyhedral fan.
    ? Type:
    :''[[..:fan:PolyhedralFan |PolyhedralFan]]<Scalar>''
    ? Example:
    :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>


----
{{anchor:hyperplanes:}}
  ?  **''HYPERPLANES''**
  :: A matrix containing the hyperplanes of the arrangement as rows. This matrix is obtained from ''[[..:fan:HyperplaneArrangement#INPUT_HYPERPLANES |INPUT_HYPERPLANES]]'' by removing duplicates and also removing hyperplanes that are obsolete wrt the ''[[..:fan:HyperplaneArrangement#SUPPORT |SUPPORT]]'' cone.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: The same hyperplane with opposing directions.
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[1,-1],[-1,1],[1,1]]);
 > print $HA->HYPERPLANES;
 1 -1
 1 1
</code>
    ? Example:
    :: A hyperplane that does not cut through the ''[[..:fan:HyperplaneArrangement#SUPPORT |SUPPORT]]''
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[1,1]], "SUPPORT.INEQUALITIES"=>unit_matrix(2));
 > print $HA->HYPERPLANES;
 
</code>


----
===== Methods =====

==== Combinatorics ====
 These methods capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:cell_to_signature:}}
  ?  **''cell_to_signature''**
  :: Given a maximal cone of ''[[..:fan:HyperplaneArrangement#CELL_DECOMPOSITION |CELL_DECOMPOSITION]]'' as Set<Int> containing the indices of the rays spanning it, return the signature of the cone as Set<Int> of indices of the ''[[..:fan:HyperplaneArrangement#HYPERPLANES |HYPERPLANES]]'' that evaluate negatively on this cone.
    ? Example:
    :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>
    :: <code perl> > print $HA->CELL_SIGNATURES;
 {}
 {0}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}
</code>
    :: <code perl> > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}
</code>


----
{{anchor:signature_to_cell:}}
  ?  **''signature_to_cell''**
  :: Given a signature as a Set<Int> of indices that indicate which ''[[..:fan:HyperplaneArrangement#HYPERPLANES |HYPERPLANES]]'' should evaluate negatively (the remaining evaluate positively), return the maximal cone of ''[[..:fan:HyperplaneArrangement#CELL_DECOMPOSITION |CELL_DECOMPOSITION]]'' associated to this signature. The result the index of the maximal cone in the maximal cones of ''[[..:fan:HyperplaneArrangement#CELL_DECOMPOSITION |CELL_DECOMPOSITION]]''.
    ? Example:
    :: Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
    :: <code perl> > $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}
</code>
    :: <code perl> > print $HA->CELL_SIGNATURES;
 {}
 {0}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}
</code>
    :: <code perl> > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}
</code>
    :: <code perl> > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0
</code>
    :: <code perl> > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}
</code>


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