====== BigObject PolyhedralComplex ====== //from application [[..:fan|fan]]//\\ \\ A polyhedral complex. The derivation from ''[[..:fan:PolyhedralFan |PolyhedralFan]]'' works like the derivation of ''[[..:polytope:Polytope |Polytope]]'' from ''[[..:polytope:Cone |Cone]]''. ? Type Parameters: :: ''Scalar'': numeric data type used for the coordinates, must be an ordered field. Default is ''[[..:common#Rational |Rational]]''. ? derived from: : ''[[..:fan:PolyhedralFan |PolyhedralFan]]'' ? Example: :: The following defines a subdivision of a square in the plane into two triangles. :: > $c=new PolyhedralComplex(VERTICES=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1]],MAXIMAL_POLYTOPES=>[[0,1,2],[1,2,3]]); ===== Properties ===== ==== Input property ==== These properties are for input only. They allow redundant information. ---- {{anchor:input_polytopes:}} ? **''INPUT_POLYTOPES''** :: Maybe redundant list of not necessarily maximal polytopes. Indices refer to ''[[..:fan:PolyhedralComplex#POINTS |POINTS]]''. Each polytope must list all vertices of ''[[..:fan:PolyhedralComplex#POINTS |POINTS]]'' it contains. The polytopes are allowed to contain lineality. An empty complex does not have any polytopes. Input section only. Ask for ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]'' if you want to know the maximal polytopes (indexed by ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''). Alias for property ''[[..:fan:PolyhedralFan#INPUT_CONES |INPUT_CONES]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ? Example: :: We can define a polyhderal complex consisting of two distinct triangles with the following (note that additionally stating one side of one of these triangles does not affect our resulting complex): :: > $c = new PolyhedralComplex(POINTS=>[[1,1,0],[1,1,1],[1,0,1],[1,-1,0],[1,-1,-1],[1,0,-1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5],[0,1]]); :: print $c->MAXIMAL_POLYTOPES; ---- {{anchor:points:}} ? **''POINTS''** :: Points in homogeneous coordinates from which the polytopes are formed. May be redundant. All vectors in the input must be non-zero. You also need to provide ''[[..:fan:PolyhedralComplex#INPUT_POLYTOPES |INPUT_POLYTOPES]]'' to define a complex completely. Input section only. Ask for ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]'' if you want a list of non-redundant points. Alias for property ''[[..:fan:PolyhedralFan#INPUT_RAYS |INPUT_RAYS]]''. ? Type: :''[[..:common#Matrix |Matrix]]'' ? Example: :: To obtain a complex consisting of two triangles we can do this (note that, contrary to a [[..:polytope:Polytope |polytope]], this complex is not convex): :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,3,-1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,3],[1,2,3]]); ---- ==== 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:compactification:}} ? **''COMPACTIFICATION''** :: The Hasse diagram of the compactification of the polyhedral complex. For a simplicial polyhedral complex, this is the cubical compactification (or cubical complex, see [Omid Amini: "The combinatorial Chow ring of products of graphs"]). For tropical varieties, this is the tropical compactification, as in [Brian Osserman and Joseph Rabinoff: "Lifting nonproper tropical intersections"]. The vertices of the compactification correspond to the faces of the original complex that have the same dimension as their recession cone. We call the face corresponding to a vertex the 'realisation' of the vertex. The decoration has four entries: 1. The face in the vertices of the compactification 2. The rank of the face 3. The realisation of the face. This is the union of the realisations of the new vertices. 4. The sedentarity of the face. This is the intersection of the sedentarities of the vertices. ? Type: :''[[..:graph:Lattice |Lattice]]<[[..:fan#SedentarityDecoration |SedentarityDecoration]],[[..:graph#Nonsequential |Nonsequential]]>'' ? Example: :: The compactification of the positive orthant in three dimensions has the same Hasse diagram as the three dimensional cube. :: > $pc1 = new PolyhedralComplex(POINTS=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], INPUT_POLYTOPES=>[[0,1,2,3]]); > print $pc1->COMPACTIFICATION->DECORATION; ({} 0 {} {}) ({0} 1 {0 1 2 3} {1 2 3}) ({1} 1 {0 2 3} {2 3}) ({2} 1 {0 1 3} {1 3}) ({3} 1 {0 1 2} {1 2}) ({4} 1 {0 3} {3}) ({5} 1 {0 2} {2}) ({6} 1 {0 1} {1}) ({7} 1 {0} {}) ({0 1} 2 {0 1 2 3} {2 3}) ({0 2} 2 {0 1 2 3} {1 3}) ({0 3} 2 {0 1 2 3} {1 2}) ({1 4} 2 {0 2 3} {3}) ({1 5} 2 {0 2 3} {2}) ({2 4} 2 {0 1 3} {3}) ({2 6} 2 {0 1 3} {1}) ({3 5} 2 {0 1 2} {2}) ({3 6} 2 {0 1 2} {1}) ({4 7} 2 {0 3} {}) ({5 7} 2 {0 2} {}) ({6 7} 2 {0 1} {}) ({0 1 2 4} 3 {0 1 2 3} {3}) ({0 1 3 5} 3 {0 1 2 3} {2}) ({0 2 3 6} 3 {0 1 2 3} {1}) ({1 4 5 7} 3 {0 2 3} {}) ({2 4 6 7} 3 {0 1 3} {}) ({3 5 6 7} 3 {0 1 2} {}) ({0 1 2 3 4 5 6 7} 4 {0 1 2 3} {}) ({-1} 5 {-1} {}) ---- {{anchor:maximal_polytopes:}} ? **''MAXIMAL_POLYTOPES''** :: Non redundant list of maximal polytopes. Indices refer to ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. An empty complex does not have any polytopes. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES |MAXIMAL_CONES]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ? Example: :: After creating a complex via the ''[[..:fan:PolyhedralComplex#INPUT_POLYTOPES |INPUT_POLYTOPES]]'' property, we can display all maximal polytopes rising from that definition: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2]]); > print $c->MAXIMAL_POLYTOPES; {0 1 2} {0 3} ---- {{anchor:maximal_polytopes_combinatorial_dims:}} ? **''MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS''** :: The combinatorial dimensions of the maximal polytopes. The i-th entry refers to the i-th entry of ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]''. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_COMBINATORIAL_DIMS |MAXIMAL_CONES_COMBINATORIAL_DIMS]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>'' ? Example: :: When connecting two vertices of a triangle to a vertex distinct from that triangle we receive a polyhedral complex with maximal polytopes of dimensions 2, 1 and 1, respectively: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2,3]]); > print $c->MAXIMAL_POLYTOPES; {0 1 2} {0 3} {2 3} :: > print $c->MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS; 2 1 1 ---- {{anchor:maximal_polytopes_incidences:}} ? **''MAXIMAL_POLYTOPES_INCIDENCES''** :: Array of incidence matrices of all [[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |maximal polytopes]]. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_INCIDENCES |MAXIMAL_CONES_INCIDENCES]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]%%>>%%'' ? Example: :: Here we construct a polyhedral complex made of two triangles which share a side; this fact can afterwards be read from this property: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[1,2,3]]); > print $c->MAXIMAL_POLYTOPES; {0 1 2} {1 2 3} :: > print $c->MAXIMAL_POLYTOPES_INCIDENCES; <{1 2} {0 2} {0 1} > <{1 2} {2 3} {1 3} > ---- {{anchor:maximal_polytopes_thru_vertices:}} ? **''MAXIMAL_POLYTOPES_THRU_VERTICES''** :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_THRU_RAYS |MAXIMAL_CONES_THRU_RAYS]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ---- {{anchor:n_maximal_polytopes:}} ? **''N_MAXIMAL_POLYTOPES''** :: Number of ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]''. Alias for property ''[[..:fan:PolyhedralFan#N_MAXIMAL_CONES |N_MAXIMAL_CONES]]''. ? Type: :''[[..:common#Int |Int]]'' ? Example: :: The number of maximal polytopes of a [[..:fan#planar_net |planar net]] of a polytope is the number of facets of that polytope; here we see this for the dodecahedron: :: > $c = planar_net(dodecahedron()); > print $c->N_MAXIMAL_POLYTOPES; 12 ---- {{anchor:n_polytopes:}} ? **''N_POLYTOPES''** :: Alias for property ''[[..:fan:PolyhedralFan#N_CONES |N_CONES]]''. ? Type: :''[[..:common#Int |Int]]'' ---- {{anchor:polytopes:}} ? **''POLYTOPES''** :: List of all polytopes of the complex of each dimension. Indices refer to ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. Alias for property ''[[..:fan:PolyhedralFan#CONES |CONES]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]%%>>%%'' ? Example: :: A complex whose only maximal polytope is a triangle also contains 3 line segments and 3 points: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2]]); > print $c->POLYTOPES; <{1} {2} {0} > <{1 2} {0 2} {0 1} > <{0 1 2} > ---- ==== 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''** :: # The possible linear span normals of all maximal polytopes. Empty if ''[[..:fan:PolyhedralFan#PURE |PURE]]'' and ''[[..:fan:PolyhedralFan#FULL_DIM |FULL_DIM]]'', i.e. each maximal polytope has the same dimension as the ambient space. Alias for property ''[[..:fan:PolyhedralFan#LINEAR_SPAN_NORMALS |LINEAR_SPAN_NORMALS]]''. ? Type: :''[[..:common#Matrix |Matrix]]'' ? Example: :: In the plane, when we construct a polyhedral complex with a 2-dimensional and two 1-dimensional maximal polytopes only the latter two will have a linear span with a normal in this ambient space: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,-1,0],[1,-1,-1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]); > print $c->AFFINE_HULL; 0 0 1 0 -1 1 ---- {{anchor:bounded:}} ? **''BOUNDED''** :: True if each object in ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]'' is [[..:polytope:Polytope#BOUNDED |bounded]]. ? Type: :''[[..:common#Bool |Bool]]'' ---- {{anchor:far_vertices:}} ? **''FAR_VERTICES''** :: Indices of vertices that are rays. ? Type: :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>'' ? Example: :: We construct a PolyhedralComplex consisting only of one unbounded ''[[..:polytope:Polytope |Polytope]]'' which is the Minkowski sum of an interval and a cone orthogonal to this line. Such a Minkowski sum always has a far vertex: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[0,0,1]],INPUT_POLYTOPES=>[[0,1,2]]); > print rows_numbered($c->VERTICES); 0:1 0 0 1:1 1 0 2:0 0 1 :: > print $c->FAR_VERTICES; {2} ---- {{anchor:maximal_polytopes_affine_hull_normals:}} ? **''MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS''** :: A basis of the normal space for each maximal polytope. This uniquely determines the affine hull of the corresponding maximal polytope. Indices refer to ''[[..:fan:PolyhedralComplex#AFFINE_HULL |AFFINE_HULL]]''. Rows correspond to ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]''. An empty row corresponds to a full-dimensional cone. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_LINEAR_SPAN_NORMALS |MAXIMAL_CONES_LINEAR_SPAN_NORMALS]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ? Example: :: In the plane, when we construct a polyhedral complex with a 2-dimensional and two 1-dimensional maximal polytopes only the latter will have a linear span with a normal in this ambient space: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,-1,0],[1,-1,-1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]); > print $c->MAXIMAL_POLYTOPES; {0 1 2} {0 3} {0 4} :: > print rows_numbered($c->AFFINE_HULL); 0:0 0 1 1:0 -1 1 :: > print $c->MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS; {} {0} {1} ---- {{anchor:maximal_polytopes_facets:}} ? **''MAXIMAL_POLYTOPES_FACETS''** :: Tells for each maximal polytope what are its facet normals, thus implying the facets. Each row corresponds to a maximal polytope and each column to the row with the same index of ''[[..:fan:PolyhedralComplex#AFFINE_HULL |AFFINE_HULL]]''. A negative number means that the corresponding row of ''[[..:fan:PolyhedralComplex#AFFINE_HULL |AFFINE_HULL]]'' has to be negated. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_FACETS |MAXIMAL_CONES_FACETS]]''. ? Type: :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>'' ? Example: :: Here we see the facet normals of the maximal polytopes of a complex made of two triangles (note that some facet normal appear to be redundant due to usage of homogeneous coordinates and the derivation from PolyhedralFan): :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,2,3]]); > print $c->MAXIMAL_POLYTOPES; {0 1 2} {0 2 3} :: > print rows_numbered($c->FACET_NORMALS); 0:1 -1 0 1:0 1 -1 2:0 0 1 3:1 0 -1 4:0 1 0 :: > print $c->MAXIMAL_POLYTOPES_FACETS; 1 1 1 0 0 0 -1 0 1 1 ---- {{anchor:n_points:}} ? **''N_POINTS''** :: Number of ''[[..:fan:PolyhedralComplex#POINTS |POINTS]]''. Alias for property ''[[..:fan:PolyhedralFan#N_INPUT_RAYS |N_INPUT_RAYS]]''. ? Type: :''[[..:common#Int |Int]]'' ? Example: :: In the plane, glueing two triangles together along one side gives us a complex with four vertices; nevertheless we can specify these two triangles using six points with redundancies: :: > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,0],[2,0,2],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5]]); > print $c->N_VERTICES; 4 :: > print $c->N_POINTS; 6 ---- {{anchor:n_vertices:}} ? **''N_VERTICES''** :: Number of ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. Alias for property ''[[..:fan:PolyhedralFan#N_RAYS |N_RAYS]]''. ? Type: :''[[..:common#Int |Int]]'' ---- {{anchor:vertices:}} ? **''VERTICES''** :: Vertices from which the polytopes are formed. Non-redundant. Co-exists with ''[[..:fan:PolyhedralFan#LINEALITY_SPACE |LINEALITY_SPACE]]''. Alias for property ''[[..:fan:PolyhedralFan#RAYS |RAYS]]''. ? Type: :''[[..:common#Matrix |Matrix]]'' ---- ==== Symmetry ==== These properties capture information of the object that is concerned with the action of permutation groups. ---- {{anchor:group:}} ? **''GROUP''** :: ? derived from: : ''[[..:fan:PolyhedralFan#GROUP |GROUP]]'' ? Type: :''[[..:group:Group |Group]]'' ? Properties of GROUP: : ? **''COORDINATE_ACTION''** :: ? Type: :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>'' ? Properties of COORDINATE_ACTION: : ? **''N_POINTS_GENERATORS''** :: Alias for property ''[[..:group:Action#N_INPUT_RAYS_GENERATORS |N_INPUT_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Int |Int]]'' ? **''N_VERTICES_GENERATORS''** :: Alias for property ''[[..:group:Action#N_RAYS_GENERATORS |N_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Int |Int]]'' ? **''POINTS_GENERATORS''** :: Alias for property ''[[..:group:Action#INPUT_RAYS_GENERATORS |INPUT_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''VERTICES_GENERATORS''** :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''. ? Type: :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''INPUT_POLYTOPES_ACTION''** :: Alias for property ''[[..:group:Group#INPUT_CONES_ACTION |INPUT_CONES_ACTION]]''. ? Type: :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>'' ? **''MATRIX_ACTION_ON_COMPLEX''** :: ? Type: :''[[..:group:MatrixActionOnVectors |MatrixActionOnVectors]]'' ? Properties of MATRIX_ACTION_ON_COMPLEX: : ? **''VERTICES_GENERATORS''** :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''. ? Type: :''[[..:common#Matrix |Matrix]]'' ? **''MAXIMAL_POLYTOPES_ACTION''** :: ? derived from: : ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_ACTION |MAXIMAL_CONES_ACTION]]'' ? Type: :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>'' ? Properties of MAXIMAL_POLYTOPES_ACTION: : ? **''MAXIMAL_POLYTOPES_GENERATORS''** :: Alias for property ''[[..:group:Action#MAXIMAL_CONES_GENERATORS |MAXIMAL_CONES_GENERATORS]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''N_POINTS_GENERATORS''** :: Alias for property ''[[..:group:Action#N_INPUT_RAYS_GENERATORS |N_INPUT_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Int |Int]]'' ? **''N_VERTICES_GENERATORS''** :: Alias for property ''[[..:group:Action#N_RAYS_GENERATORS |N_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Int |Int]]'' ? **''POINTS_GENERATORS''** :: Alias for property ''[[..:group:Action#INPUT_RAYS_GENERATORS |INPUT_RAYS_GENERATORS]]''. ? Type: :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''VERTICES_GENERATORS''** :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''. ? Type: :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''POINTS_ACTION''** :: Alias for property ''[[..:group:Group#INPUT_RAYS_ACTION |INPUT_RAYS_ACTION]]''. ? Type: :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>'' ? **''POLYTOPES_ORBIT_SIZES''** :: Alias for property ''[[..:fan:PolyhedralFan#CONES_ORBIT_SIZES |CONES_ORBIT_SIZES]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%'' ? **''REPRESENTATIVE_MAXIMAL_POLYTOPES''** :: Alias for property ''[[..:fan:PolyhedralFan#REPRESENTATIVE_MAXIMAL_CONES |REPRESENTATIVE_MAXIMAL_CONES]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ? **''REPRESENTATIVE_VERTICES''** :: Alias for property ''[[..:fan:PolyhedralFan#REPRESENTATIVE_RAYS |REPRESENTATIVE_RAYS]]''. ? Type: :''[[..:common#Matrix |Matrix]]'' ? **''VERTICES_ACTION''** :: Alias for property ''[[..:group:Group#RAYS_ACTION |RAYS_ACTION]]''. ? Type: :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>'' ---- ==== Visualization ==== These properties are for visualization. ---- {{anchor:point_labels:}} ? **''POINT_LABELS''** :: Unique names assigned to the ''[[..:fan:PolyhedralComplex#POINTS |POINTS]]''. Similar to ''[[..:fan:PolyhedralComplex#VERTEX_LABELS |VERTEX_LABELS]]'' for ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. Alias for property ''[[..:fan:PolyhedralFan#INPUT_RAY_LABELS |INPUT_RAY_LABELS]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#String |String]]>'' ---- {{anchor:vertex_labels:}} ? **''VERTEX_LABELS''** :: Unique names assigned to the ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. If specified, they are shown by visualization tools instead of vertex indices. For a polyhedral complex built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. Alias for property ''[[..:fan:PolyhedralFan#RAY_LABELS |RAY_LABELS]]''. ? Type: :''[[..:common#Array |Array]]<[[..:common#String |String]]>'' ---- ===== 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()''** :: Returns the dimension of the ambient space. ? Returns: :''[[..:common#Int |Int]]'' ? Example: :: The ambient dimension of a point in the line is 1: :: > $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]); > print $c->AMBIENT_DIM; 1 ---- {{anchor:dim:}} ? **''DIM()''** :: Returns the dimension of the linear space spanned by the complex. ? Returns: :''[[..:common#Int |Int]]'' ? Example: :: The dimension of a point in the line is 0: :: > $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]); > print $c->DIM; 0 ---- {{anchor:polytope:}} ? **''polytope([[..:common#Int |Int]] i)''** :: Returns the //i//-th facet of the complex as a ''[[..:polytope:Polytope |Polytope]]''. ? Parameters: :: ''[[..:common#Int |Int]]'' ''i'' ? Returns: :''[[..:polytope:Polytope |Polytope]]'' ? Example: :: The [[..:fan#planar_net |planar net]] of the 3-dimensional cross polytope consists only of triangles (and the according adjacent lines and vertices); asking for any of its polytopes thus gives us a triangle: :: > $c = planar_net(cross(3)); > $p = $c->polytope(5); > print rows_numbered($p->VERTICES); 0:1 0 0 1:1 0.707106781186547 -1.22474487139159 2:1 -0.707106781186549 -1.22474487139159 ---- ==== Visualization ==== These methods are for visualization. ---- {{anchor:visual_orbit_colored_graph:}} ? **''VISUAL_ORBIT_COLORED_GRAPH()''** :: Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color. ? Options: : option list ''[[..:graph#Visual_Graph_decorations |Visual::Graph::decorations]]'' ? Returns: :''[[..:polytope:Visual_PolytopeGraph |Visual::PolytopeGraph]]'' ---- ==== no category ==== {{anchor:visual:}} ? **''VISUAL()''** :: Visualizes the polyhedral complex. ? Options: : option list ''[[..:common#geometric_options |geometric_options]]'' ? Returns: :''[[..:fan:Visual_PolyhedralFan |Visual::PolyhedralFan]]'' ---- {{anchor:visual_metric_tight_span:}} ? **''VISUAL_METRIC_TIGHT_SPAN()''** ::UNDOCUMENTED ? Options: : option list ''[[..:fan#Visual_Graph_TightSpanDecorations |Visual::Graph::TightSpanDecorations]]'' ----