====== BigObject PropagatedPolytope<Scalar> ======
//from application [[..:polytope|polytope]]//\\
\\
 Polytope propagation means to define a polytope inductively by assigning vectors to arcs of a directed graph.  At each node of such a graph a polytope arises as the joint convex hull of the polytopes at the translated sources of the inward pointing arcs. For details see 
> Joswig: Polytope Propagation on Graphs.
> Chapter 6 in Pachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.

  ? Type Parameters:
  :: ''Scalar'': numeric data type used for the coordinates, must be an ordered field. Default is ''[[..:common#Rational |Rational]]''.
  ? derived from:
  : ''[[..:polytope:Polytope |Polytope]]''
  ? Example:
  ::  We define an acyclic digraph on 8 nodes with a unique source and a unique sink. Each arc is equipped with a vector in Q^2.
  :: <code perl> > $Gamma = new GraphAdjacency<Directed>(8);
 > $Vectors = new EdgeMap<Directed, Vector>($Gamma);
 > $Vectors->edge(0,1) = [1,0]; $Vectors->edge(0,2) = [0,0];
 > $Vectors->edge(1,3) = [1,0]; $Vectors->edge(1,4) = [0,1];
 > $Vectors->edge(2,3) = [0,0]; $Vectors->edge(2,4) = [0,2];
 > $Vectors->edge(3,5) = [1,0]; $Vectors->edge(3,6) = [0,1];
 > $Vectors->edge(4,5) = [0,0]; $Vectors->edge(4,6) = [0,2];
 > $Vectors->edge(5,7) = [0,0];
 > $Vectors->edge(6,7) = [0,0];
</code>
  ::  Then we can define the propagated polytope implicitly:
  :: <code perl> > $P = new PropagatedPolytope(CONE_AMBIENT_DIM=>3, "SUM_PRODUCT_GRAPH.ADJACENCY"=>$Gamma, "SUM_PRODUCT_GRAPH.TRANSLATIONS"=>$Vectors);
 > print $P->VERTICES;
 1 3 0
 1 1 0
 1 0 1
 1 1 3
 1 0 4
</code>
  ::  Note that it is necessary to specify the dimension of the (homogeneous) ambient space by setting ''[[..:polytope:Polytope#CONE_AMBIENT_DIM |CONE_AMBIENT_DIM]]''.
===== Properties =====

==== no category ====
{{anchor:sum_product_graph:}}
  ?  **''SUM_PRODUCT_GRAPH''**
  ::
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Directed |Directed]]>''
    ? Properties of SUM_PRODUCT_GRAPH:
    :
      ?  **''TRANSLATIONS''**
      :: The translation vectors of the arcs.
        ? Type:
        :''[[..:common#EdgeMap |EdgeMap]]<[[..:common#Directed |Directed]],[[..:common#Vector |Vector]]<Scalar%%>>%%''


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