BigObject Visual::Polytope

from application 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).

DIRECTED_GRAPH(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:

LinearProgram lp: a Linear Program object attached to the polytope

Returns:

Visual::Polytope

Example:

Attaches a linear program to the 3-dimensional cube and visualizes the directed graph, giving the cube a blue facet color

 > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,0,0,1]);
 > $p->VISUAL(FacetColor=>"blue")->DIRECTED_GRAPH;

LATTICE()

Visualize the LATTICE_POINTS of a polytope

Options:

option list Visual::PointSet::decorations

Returns:

Visual::Polytope

Example:

Visualizes the lattice points of the threedimensional cube.

 > cube(3)->VISUAL->LATTICE;

LATTICE_COLORED()

Visualize the LATTICE_POINTS of a polytope in different colors (interior / boundary / vertices)

Options:

option list Visual::PointSet::decorations

Returns:

Visual::Polytope

Example:

Creates the threedimensional unit cube scaled by 1.5 and displays the colored version of its lattice points

 > cube(3,(3/2),0)->VISUAL->LATTICE_COLORED;

MIN_MAX_FACE(LinearProgram lp)

Illustrate the behavior of a linear objective function on the polytope. Draw the facets contained in MAXIMAL_FACE and MINIMAL_FACE in distinct colors.

Parameters:

LinearProgram lp: a LinearProgram object attached to the polytope.

Options:

Color min: minimal face decoration (default: yellow vertices and/or facets)

Color max: maximal face decoration (default: red vertices and/or facets)

Returns:

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

 > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);
 > $p->VISUAL->MIN_MAX_FACE(max=>"purple");

STEINER()

Add the STEINER_POINTS to the 3-d visualization. The facets become transparent.

Options:

option list Visual::PointSet::decorations

Returns:

Visual::Polytope

Example:

Displays the Steiner points of a random threedimensional sphere with 20 vertices. The labels of the vertices are turned off.

 > rand_sphere(3,20)->VISUAL(VertexLabels=>"hidden")->STEINER;

TRIANGULATION(Array<Set<Int>> t)

Add the triangulation to the drawing. You may specify any triangulation of the current polytope. Per default, the TRIANGULATION property is taken. (Currently there is only one possible alternative triangulation: TRIANGULATION_INT). Hint: Use the method Method → Effect → Explode Group of Geometries of JavaView for better insight in the internal structure.

Parameters:

Array<Set<Int>> t: facets of the triangulation

Options:

option list Visual::Polygons::decorations

Returns:

Visual::Polytope

Example:

Displays a triangulation of the threedimensional cube. Facets are made transparent and vertices are hidden.

 > cube(3)->VISUAL->TRIANGULATION(FacetTransparency=>0.7,VertexStyle=>"hidden");

TRIANGULATION_BOUNDARY()

Draw the edges of the TRIANGULATION_BOUNDARY. The facets are made transparent.

Options:

option list Visual::Graph::decorations

Returns:

Visual::Polytope

Example:

Displays the boundary triangulation of the threedimensional cube.

 > cube(3)->VISUAL->TRIANGULATION_BOUNDARY;

Example:

For a slightly different visualization of essentially the same:

 > cube(3)->TRIANGULATION->BOUNDARY->VISUAL;

VERTEX_COLORS(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:

LinearProgram lp: a LinearProgram object attached to the polytope

Options:

Color min: minimal vertex color (default: yellow)

Color max: maximal vertex color (default: red)

Returns:

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

 > $p = cube(3);
 > $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);
 > $p->VISUAL->VERTEX_COLORS(max=>"purple");