BigObject PointConfiguration<Scalar> [polymake wiki]

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====== BigObject PointConfiguration<Scalar> ======
//from application [[..:polytope|polytope]]//\\
\\
 The ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' of an object of type PointConfiguration encode a not necessarily convex finite point set. The difference to a parent ''[[..:polytope:VectorConfiguration |VectorConfiguration]]'' is that the points have homogeneous coordinates, i.e. they will be normalized to have first coordinate 1 without warning.

  ? Type Parameters:
  :: ''Scalar'': default: ''[[..:common#Rational |Rational]]''
  ? derived from:
  : ''[[..:polytope:VectorConfiguration |VectorConfiguration]]''
  ? Specializations:
  :: ''PointConfiguration::ExactCoord'':  A point configuration with an exact coordinate type, like Rational.
===== Properties =====

==== Input property ====
 These properties are for input only. They allow redundant information.
----
{{anchor:points:}}
  ?  **''POINTS''**
  :: The points of the configuration.  Multiples allowed. Alias for property ''[[..:polytope:VectorConfiguration#VECTORS |VECTORS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----

==== 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:cocircuit_equations:}}
  ?  **''COCIRCUIT_EQUATIONS''**
  :: Tells the cocircuit equations that hold for the configuration, one for each interior ridge
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:graph:}}
  ?  **''GRAPH''**
  :: Graph of the point configuration. Two points are adjacent if they are neigbours in a edge of the ''[[..:polytope:PointConfiguration#CONVEX_HULL |CONVEX_HULL]]''.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Undirected |Undirected]]>''


----
{{anchor:interior_ridge_simplices:}}
  ?  **''INTERIOR_RIDGE_SIMPLICES''**
  :: Tells the number of codimension 1 simplices that are not on the boundary
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:max_boundary_simplices:}}
  ?  **''MAX_BOUNDARY_SIMPLICES''**
  :: Tells the full-dimensional simplices on the boundary that contain no points except for the vertices.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:max_interior_simplices:}}
  ?  **''MAX_INTERIOR_SIMPLICES''**
  :: Tells the full-dimensional simplices that contain no points except for the vertices.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:n_max_boundary_simplices:}}
  ?  **''N_MAX_BOUNDARY_SIMPLICES''**
  :: Tells the number of MAX_BOUNDARY_SIMPLICES
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_max_interior_simplices:}}
  ?  **''N_MAX_INTERIOR_SIMPLICES''**
  :: Tells the number of MAX_INTERIOR_SIMPLICES
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:simplexity_lower_bound:}}
  ?  **''SIMPLEXITY_LOWER_BOUND''**
  :: A lower bound for the minimal number of simplices in a triangulation
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:splits:}}
  ?  **''SPLITS''**
  :: The splits of the point configuration, i.e., hyperplanes cutting the configuration in two parts such that we have a regular subdivision.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:split_compatibility_graph:}}
  ?  **''SPLIT_COMPATIBILITY_GRAPH''**
  :: Two ''[[..:polytope:PointConfiguration#SPLITS |SPLITS]]'' are compatible if the defining hyperplanes do not intersect in the interior of the point configuration.  This defines a graph.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Undirected |Undirected]]>''


----

==== 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:affine_hull:}}
  ?  **''AFFINE_HULL''**
  :: Dual basis of the affine hull of the point configuration
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:barycenter:}}
  ?  **''BARYCENTER''**
  :: The center of gravity of the point configuration.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''


----
{{anchor:bounded:}}
  ?  **''BOUNDED''**
  :: True if the point configuration is bounded.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:centered:}}
  ?  **''CENTERED''**
  :: True if (1, 0, 0, ...) is in the relative interior.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:convex:}}
  ?  **''CONVEX''**
  :: True if the ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' are in convex position.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:convex_hull:}}
  ?  **''CONVEX_HULL''**
  ::
    ? Type:
    :''[[..:polytope:Polytope |Polytope]]<Scalar>''
    ? Properties of CONVEX_HULL:
    :
      ?  **''VERTEX_POINT_MAP''**
      :: Indices of ''[[..:polytope:Polytope#VERTICES |VERTICES]]'' of the ''[[..:polytope:PointConfiguration#CONVEX_HULL |CONVEX_HULL]]'' as ''[[..:polytope:Polytope#POINTS |POINTS]]''.
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:far_points:}}
  ?  **''FAR_POINTS''**
  :: Indices of ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' that are rays.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''


----
{{anchor:multiple_points:}}
  ?  **''MULTIPLE_POINTS''**
  :: Tells if multiple points exist. Alias for property ''[[..:polytope:VectorConfiguration#MULTIPLE_VECTORS |MULTIPLE_VECTORS]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:non_vertices:}}
  ?  **''NON_VERTICES''**
  :: ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' that are not ''[[..:polytope:Polytope#VERTICES |VERTICES]]'' of the ''[[..:polytope:PointConfiguration#CONVEX_HULL |CONVEX_HULL]]''
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''


----
{{anchor:n_points:}}
  ?  **''N_POINTS''**
  :: Number of ''[[..:polytope:PointConfiguration#POINTS |POINTS]]''. Alias for property ''[[..:polytope:VectorConfiguration#N_VECTORS |N_VECTORS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:vertex_point_map:}}
  ?  **''VERTEX_POINT_MAP''**
  :: Indices of ''[[..:polytope:Polytope#VERTICES |VERTICES]]'' of the ''[[..:polytope:PointConfiguration#CONVEX_HULL |CONVEX_HULL]]'' as ''[[..:polytope:PointConfiguration#POINTS |POINTS]]''
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----

==== Symmetry ====
 These properties capture information of the object that is concerned with the action of permutation groups.
----
{{anchor:group:}}
  ?  **''GROUP''**
  ::
    ? derived from:
    : ''[[..:polytope:VectorConfiguration#GROUP |GROUP]]''
    ? Type:
    :''[[..:group:Group |Group]]''
    ? Properties of GROUP:
    :
      ?  **''MATRIX_ACTION''**
      ::
        ? Type:
        :''[[..:group:MatrixActionOnVectors |MatrixActionOnVectors]]<Scalar>''
        ? Properties of MATRIX_ACTION:
        :
          ?  **''POINTS_ORBITS''**
          :: Alias for property ''[[..:group:MatrixActionOnVectors#VECTORS_ORBITS |VECTORS_ORBITS]]''.
            ? Type:
            :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''POINTS_ACTION''**
      ::
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''
        ? Properties of POINTS_ACTION:
        :
          ?  **''SYMMETRIZED_COCIRCUIT_EQUATIONS''**
          :: The cocircuit equations, projected to a certain direct sum of isotypic components
            ? Type:
            :''[[..:polytope:SymmetrizedCocircuitEquations |SymmetrizedCocircuitEquations]]''
      ?  **''REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES''**
      :: One representative for each orbit of boundary ridge simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES''**
      :: One representative for each orbit of interior ridge simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES''**
      :: One representative for each orbit of maximal-dimensional boundary simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_MAX_INTERIOR_SIMPLICES''**
      :: One representative for each orbit of maximal-dimensional interior simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''


----

==== Triangulation and volume ====
 These properties collect information about triangulations of the object and properties usually computed from such, as the volume.
----
{{anchor:polytopal_subdivision:}}
  ?  **''POLYTOPAL_SUBDIVISION''**
  ::
    ? Type:
    :''[[..:fan:SubdivisionOfPoints |SubdivisionOfPoints]]<Scalar>''
    ? Properties of POLYTOPAL_SUBDIVISION:
    :
      ?  **''REFINED_SPLITS''**
      :: The splits that are coarsenings of the subdivision. If the subdivision is regular these form the unique split decomposition of the corresponding weight function.
        ? Type:
        :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''


----
{{anchor:triangulation:}}
  ?  **''TRIANGULATION''**
  ::
    ? Type:
    :''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]<Scalar>''
    ? Properties of TRIANGULATION:
    :
      ?  **''BOUNDARY''**
      ::
        ? derived from:
        : ''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]''
        ? Type:
        :''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]''
        ? Properties of BOUNDARY:
        :
          ?  **''FACET_TRIANGULATIONS''**
          :: DOC_FIXME: Incomprehensible description! For each facet the set of simplex indices of BOUNDARY that triangulate it.
            ? Type:
            :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''GKZ_VECTOR''**
      :: GKZ-vector
      > See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:
      > Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994
        ? Type:
        :''[[..:common#Vector |Vector]]<Scalar>''
      ?  **''MASSIVE_GKZ_VECTOR''**
      :: Calculate the massive GKZ vectors of the triangulations of a integral PointConfiguration //A//. For a definition see Chapter 11 of Gelfand, Kapranov, and Zelevinsky: Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994.
        ? Type:
        :''[[..:common#Vector |Vector]]<Scalar>''
        ? Example:
        :: To calculate the massive GKZ vector of a triangulation of a point configuration. This example is from the book mentioned above (p. 369, top right example).
        :: <code perl> > $A=new PointConfiguration(POINTS=>[[1,0,0],[1,1,0],[1,2,0],[1,3,0],[1,0,1],[1,1,1],[1,0,2]]);
     > $A->add("TRIANGULATION", WEIGHTS=>[0,1,0,1,1,1,0]);
     > print $A->TRIANGULATION->MASSIVE_GKZ_VECTOR;
     1 0 3 1 0 0 4
    </code>
      ?  **''REFINED_SPLITS''**
      :: The splits that are coarsenings of the current ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''. If the triangulation is regular these form the unique split decomposition of the corresponding weight function.
        ? Type:
        :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
      ?  **''REGULAR''**
      :: Checks regularity of ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''.
        ? Type:
        :''[[..:common#Bool |Bool]]''
      ?  **''WEIGHTS''**
      :: Weight vector to construct a regular ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''. Must be generic.
        ? Type:
        :''[[..:common#Vector |Vector]]<Scalar>''


----

==== Visualization ====
 These properties are for visualization.
----
{{anchor:pif_cyclic_normal:}}
  ?  **''PIF_CYCLIC_NORMAL''**
  :: ''[[..:polytope:Polytope#VIF_CYCLIC_NORMAL |VIF_CYCLIC_NORMAL]]'' of the ''[[..:polytope:PointConfiguration#CONVEX_HULL |CONVEX_HULL]]'', but with the indices form ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' instead of ''[[..:polytope:Polytope#VERTICES |VERTICES]]''
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:point_labels:}}
  ?  **''POINT_LABELS''**
  :: Unique names assigned to the ''[[..:polytope:PointConfiguration#POINTS |POINTS]]''. Alias for property ''[[..:polytope:VectorConfiguration#LABELS |LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
===== 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:faces_of_dim:}}
  ?  **''faces_of_dim([[..:polytope:PointConfiguration |PointConfiguration]] p)''**
  :: Output the faces of a given dimension
    ? Parameters:
    :: ''[[..:polytope:PointConfiguration |PointConfiguration]]'' ''p'': the input point configuration
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----

==== Geometry ====
 These methods capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:ambient_dim:}}
  ?  **''AMBIENT_DIM()''**
  :: Ambient dimension of the point configuration (without the homogenization coordinate). Similar to ''[[..:polytope:Polytope#AMBIENT_DIM |AMBIENT_DIM]]''.
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dim:}}
  ?  **''DIM()''**
  :: Affine dimension of the point configuration. Similar to ''[[..:polytope:Polytope#DIM |DIM]]''.
    ? Returns:
    :''[[..:common#Int |Int]]''


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualize a point configuration.
    ? Options:
    : option list ''[[..:common#Visual_Polygons_decorations |Visual::Polygons::decorations]]''
    : option list ''[[..:common#geometric_options |geometric_options]]''
    ? Returns:
    :''[[..:polytope:Visual_PointConfiguration |Visual::PointConfiguration]]''


----
{{anchor:visual_points:}}
  ?  **''VISUAL_POINTS()''**
  :: Visualize the ''[[..:polytope:PointConfiguration#POINTS |POINTS]]'' of a point configuration.
    ? Options:
    : option list ''[[..:common#Visual_Polygons_decorations |Visual::Polygons::decorations]]''
    : option list ''[[..:common#geometric_options |geometric_options]]''
    ? Returns:
    :''[[..:common:Visual_Object |Visual::Object]]''


----