BigObject PlanarNet<Scalar>

from application fan

A special big object class devoted to planar unfoldings of 3-polytopes. Its main functionality is the visualization.

Type Parameters:

Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.

derived from:

PolyhedralComplex

Example:

To visualize a planar net of some Johnson solid (with flaps, such that you can print, cut and glue):

 > planar_net(polytope::johnson_solid(52))->VISUAL->FLAPS;

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

DUAL_TREE

The spanning tree in the DUAL_GRAPH which defines the layout. Encoded as a list of edges, directed away from the root facet. Indices correspond to MAXIMAL_POLYTOPES as well as FACETS.

Type:

Array<Pair<Int,Int>>

Example:

The planar net of the 3-cube consists of 6 squares. Looking at the dual tree we can see that the client planar_net gives us the classical cross-shaped planar net, as we have a vertex which is connected to four different vertices and a last one which has an edge to one of these four vertices:

 > print planar_net(cube(3))->DUAL_TREE;
 (0 2) (0 3) (0 4) (0 5) (2 1)

FLAPS

List of directed edges in the primal graph GRAPH which need flaps for gluing. Indices correspond to VERTICES. The orientation is chosen such that the facet in the layout is always on the left. In fact, the flaps define a spanning tree of GRAPH.

Type:

Array<Pair<Int,Int>>

VF_MAP

Map which tells which vertex/facet pair of the 3-polytope is mapped to which vertex of the planar net. The dependence on the facets comes from the fact that most VERTICES occur more than once in the planar net. The inverse is VF_MAP_INV.

Type:

Map<Pair<Int,Int>,Int>

VF_MAP_INV

Map which tells how the VERTICES of the planar net are mapped to the VERTICES of the 3-polytope. Inverse of VF_MAP.

Type:

Array<Int>

VIF_CYCLIC_NORMAL

Counter clockwise ordering of the VERTICES in each MAXIMAL_POLYTOPES. This is a copy of the property VIF_CYCLIC_NORMAL of POLYTOPE.

Type:

Array<Array<Int>>

These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

POLYTOPE

The corresponding 3-polytope.

Type:

Polytope<Scalar>

These methods are for visualization.

VISUAL()

This function is the main purpose of the entire class PlanarNet.

Options:

option list Visual::Polygons::decorations

Returns:

Visual::PlanarNet