====== BigObject Visual::Polytope ======
//from application [[..:polytope|polytope]]//\\
\\
 Visualization of a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).

===== Methods =====

==== no category ====
{{anchor:directed_graph:}}
  ?  **''DIRECTED_GRAPH([[..:polytope:LinearProgram |LinearProgram]] lp)''**
  :: Illustrate the behavior of a linear objective function on the polytope. Superpose the drawing with the directed graph induced by the objective function.
    ? Parameters:
    :: ''[[..:polytope:LinearProgram |LinearProgram]]'' ''lp'': a Linear Program object attached to the polytope
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Attaches a linear program to the 3-dimensional cube and visualizes the directed graph, giving the cube a blue facet color
    :: <code perl> > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,0,0,1]);
 > $p->VISUAL(FacetColor=>"blue")->DIRECTED_GRAPH;
</code>


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{{anchor:lattice:}}
  ?  **''LATTICE()''**
  :: Visualize the ''[[..:polytope:Polytope#LATTICE_POINTS |LATTICE_POINTS]]'' of a polytope
    ? Options:
    : option list ''[[..:common#Visual_PointSet_decorations |Visual::PointSet::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Visualizes the lattice points of the threedimensional cube.
    :: <code perl> > cube(3)->VISUAL->LATTICE;
</code>


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{{anchor:lattice_colored:}}
  ?  **''LATTICE_COLORED()''**
  :: Visualize the ''[[..:polytope:Polytope#LATTICE_POINTS |LATTICE_POINTS]]'' of a polytope in different colors (interior / boundary / vertices)
    ? Options:
    : option list ''[[..:common#Visual_PointSet_decorations |Visual::PointSet::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Creates the threedimensional unit cube scaled by 1.5 and displays the colored version of its lattice points
    :: <code perl> > cube(3,(3/2),0)->VISUAL->LATTICE_COLORED;
</code>


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{{anchor:min_max_face:}}
  ?  **''MIN_MAX_FACE([[..:polytope:LinearProgram |LinearProgram]] lp)''**
  :: Illustrate the behavior of a linear objective function on the polytope. Draw the facets contained in ''[[..:polytope:LinearProgram#MAXIMAL_FACE |MAXIMAL_FACE]]'' and ''[[..:polytope:LinearProgram#MINIMAL_FACE |MINIMAL_FACE]]'' in distinct colors.
    ? Parameters:
    :: ''[[..:polytope:LinearProgram |LinearProgram]]'' ''lp'': a LinearProgram object attached to the polytope.
    ? Options:
    : 
    :: ''[[..:common#Color |Color]]'' ''min'': minimal face decoration (default: yellow vertices and/or facets)
    :: ''[[..:common#Color |Color]]'' ''max'': maximal face decoration (default: red vertices and/or facets)
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Attaches a linear program to the threedimensional cube and displays the minimal/maximal faces in a different color, choosing purple instead of the default red for the maximal face
    :: <code perl> > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);
 > $p->VISUAL->MIN_MAX_FACE(max=>"purple");
</code>


----
{{anchor:steiner:}}
  ?  **''STEINER()''**
  :: Add the [[..:polytope:Polytope#STEINER_POINTS |STEINER_POINTS]] to the 3-d visualization. The facets become transparent.
    ? Options:
    : option list ''[[..:common#Visual_PointSet_decorations |Visual::PointSet::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Displays the Steiner points of a random threedimensional sphere with 20 vertices. The labels of the vertices are turned off.
    :: <code perl> > rand_sphere(3,20)->VISUAL(VertexLabels=>"hidden")->STEINER;
</code>


----
{{anchor:triangulation:}}
  ?  **''TRIANGULATION([[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%% t)''**
  :: Add the triangulation to the drawing.   You may specify any triangulation of the current polytope. Per default, the [[..:polytope:Cone#TRIANGULATION |TRIANGULATION]] property is taken. (Currently there is only one possible alternative triangulation: [[..:polytope:Cone#TRIANGULATION_INT |TRIANGULATION_INT]]). **Hint:** Use the method __Method -> Effect -> Explode Group of Geometries__ of [[:external_software#JavaView|JavaView]] for better insight in the internal structure.
    ? Parameters:
    :: ''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%'' ''t'': facets of the triangulation
    ? Options:
    : option list ''[[..:common#Visual_Polygons_decorations |Visual::Polygons::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Displays a triangulation of the threedimensional cube. Facets are made transparent and vertices are hidden.
    :: <code perl> > cube(3)->VISUAL->TRIANGULATION(FacetTransparency=>0.7,VertexStyle=>"hidden");
</code>


----
{{anchor:triangulation_boundary:}}
  ?  **''TRIANGULATION_BOUNDARY()''**
  :: Draw the edges of the [[..:polytope:Visual_Polytope#TRIANGULATION_BOUNDARY |TRIANGULATION_BOUNDARY]]. The facets are made transparent.
    ? Options:
    : option list ''[[..:graph#Visual_Graph_decorations |Visual::Graph::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Displays the boundary triangulation of the threedimensional cube.
    :: <code perl> > cube(3)->VISUAL->TRIANGULATION_BOUNDARY;
</code>
    ? Example:
    :: For a slightly different visualization of essentially the same:
    :: <code perl> > cube(3)->TRIANGULATION->BOUNDARY->VISUAL;
</code>


----
{{anchor:vertex_colors:}}
  ?  **''VERTEX_COLORS([[..:polytope:LinearProgram |LinearProgram]] lp)''**
  :: Illustrate the behavior of a linear objective function on the polytope. Color the vertices according to the values of the objective function.
    ? Parameters:
    :: ''[[..:polytope:LinearProgram |LinearProgram]]'' ''lp'': a LinearProgram object attached to the polytope
    ? Options:
    : 
    :: ''[[..:common#Color |Color]]'' ''min'': minimal vertex color (default: yellow)
    :: ''[[..:common#Color |Color]]'' ''max'': maximal vertex color (default: red)
    ? Returns:
    :''[[..:polytope:Visual_Polytope |Visual::Polytope]]''
    ? Example:
    :: Attaches a linear program to the threedimensional cube and displays the minimal/maximal vertices in a different color, choosing purple instead of the default red for the maximal vertices
    :: <code perl> > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);
 > $p->VISUAL->VERTEX_COLORS(max=>"purple");
</code>


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