====== BigObject SubdivisionOfPoints<Scalar> ======
//from application [[..:fan|fan]]//\\
\\
 The inhomogeneous variant of ''[[..:fan:SubdivisionOfVectors |SubdivisionOfVectors]]'', similar to the derivation of ''[[..:polytope:PointConfiguration |PointConfiguration]]'' from ''[[..:polytope:VectorConfiguration |VectorConfiguration]]''.

  ? Type Parameters:
  :: ''Scalar'': default: ''[[..:common#Rational |Rational]]''
  ? derived from:
  : ''[[..:fan:SubdivisionOfVectors |SubdivisionOfVectors]]''
  ? Example:
  :: To produce a regular subdivision of the vertices of a square:
  :: <code perl> > $c=new SubdivisionOfPoints(POINTS=>polytope::cube(2)->VERTICES,WEIGHTS=>[0,0,0,1]);
 > print $c->MAXIMAL_CELLS;
 {0 1 2}
 {1 2 3}
</code>
===== Properties =====

==== 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:n_points:}}
  ?  **''N_POINTS''**
  :: The number of ''[[..:fan:SubdivisionOfPoints#POINTS |POINTS]]'' in the configuration. Alias for property ''[[..:fan:SubdivisionOfVectors#N_VECTORS |N_VECTORS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:points:}}
  ?  **''POINTS''**
  :: The points of the configuration.  Multiples allowed. Alias for property ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:polyhedral_complex:}}
  ?  **''POLYHEDRAL_COMPLEX''**
  :: The polyhedral complex induced by the cells of the subdivision.
    ? Type:
    :''[[..:fan:PolyhedralComplex |PolyhedralComplex]]<Scalar>''


----
{{anchor:regular:}}
  ?  **''REGULAR''**
  :: Whether the subdivision is regular, i.e. induced by a weight vector.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:tight_span:}}
  ?  **''TIGHT_SPAN''**
  :: The tight span of the subdivision.
    ? Type:
    :''[[..:fan:PolyhedralComplex |PolyhedralComplex]]<Scalar>''


----
{{anchor:unimodular:}}
  ?  **''UNIMODULAR''**
  :: A subdivision is unimodular if it is a triangulation such that each maximal simplex has unit normalized volume.
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: Unit square, triangulated.
    :: <code perl> > $S = new SubdivisionOfPoints(POINTS=>cube(2,0)->VERTICES, WEIGHTS=>[0,0,0,1]);
 > print $S->UNIMODULAR
 true
</code>
    ? Example:
    :: Unit 3-cube, triangulation induced by four compatible vertex splits.
    :: <code perl> > $S = new SubdivisionOfPoints(POINTS=>cube(3,0)->VERTICES, WEIGHTS=>[0,1,1,0,1,0,1,0]);
 > print $S->UNIMODULAR
 false
</code>


----
{{anchor:weights:}}
  ?  **''WEIGHTS''**
  :: Vector assigning a weight to each point to get a regular subdivision.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''


----

==== Visualization ====
 These properties are for visualization.
----
{{anchor:point_labels:}}
  ?  **''POINT_LABELS''**
  :: Unique names assigned to the ''[[..:fan:SubdivisionOfPoints#POINTS |POINTS]]''. If specified, they are shown by visualization tools instead of point indices. Alias for property ''[[..:fan:SubdivisionOfVectors#LABELS |LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


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===== Methods =====

==== 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:PointConfiguration#AMBIENT_DIM |AMBIENT_DIM]]''.


----
{{anchor:dim:}}
  ?  **''DIM''**
  :: Affine dimension of the point configuration. Similar to ''[[..:polytope:PointConfiguration#DIM |DIM]]''.


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{{anchor:cell:}}
  ?  **''cell([[..:common#Int |Int]] i)''**
  :: Returns the //i//-th cell of the complex as a ''[[..:polytope:PointConfiguration |PointConfiguration]]''
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''i''
    ? Returns:
    :''[[..:polytope:PointConfiguration |PointConfiguration]]''


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualizes the ''[[..:fan:SubdivisionOfPoints |SubdivisionOfPoints]]''.
    ? Options:
    : option list ''[[..:common#geometric_options |geometric_options]]''
    ? Returns:
    :''[[..:fan:Visual_PolyhedralFan |Visual::PolyhedralFan]]''


----