transitionsdf {zonohedra}

summarize the number of transitions and associated data, over all parallelograms in the boundary of a zonohedron

Description

The 2-transition surface is a union of parallelograms. For this function, the surface is required to be strictly starshaped at the center. For the definition of strictly starshaped see The 2-Transition Subcomplex and the 2-Transition Surface.

Each parallelogram has a unit normal that defines a linear functional.
If the 2-transition parallelogram is in the interior of the zonohedron then the functional is not maximized on the parallelogram, and there is a corresponding similar parallelogram on the boundary of the zonohedron where the functional *is* maximized. The first parallelogram (in the surface) is called deficient because the functional is not maximized, and the second parallelogram (in the boundary) is called abundant because the number of corresponding transitions across this parallelogram is more than 2. The difference between the functional values is called the deficit.
If the 2-transition parallelogram is on the boundary, then it is called coincident. It is also called non-deficient and the deficit is 0.

Usage

transitionsdf( x, trans2=TRUE )

Arguments

x

a zonohedron object as returned by the constructor zonohedron(). The 2-transition surface must be strictly starshaped.

trans2

if TRUE, then include metrics on the non-deficient (coincident) parallelograms, with 2 transitions. This is always the first row of the returned data frame.
if FALSE, then data on the non-deficient parallelograms is not included, and the returned data frame only has data on the deficient parallelograms, with more than 2 transitions.

Value

transitionsdf() returns a data.frame with a row for each number of transitions found, plus a final row with totals on appropriate columns. The columns are:

transitions

the number of transitions, a positive even integer, in increasing order.

parallelograms

the number of parallelograms with the given number of transitions

area

the min and max of the area of the parallelograms with the given number of transitions

area.sum

the total area of the parallelograms with the given number of transitions

deficit

the min and max of the deficit of the parallelograms with the given number of transitions. When there are 2 transitions the deficit should be exactly 0, but is usually slightly non-0 due to truncation. When there are more than 2 transitions the deficit is positive.

example

the 2 generators (from the ground set of the simplified matroid) of the parallelogram with the maximum area

In case of error, the function returns NULL.

Note

Because of the 1-1 correspondence between similar parallelograms, the surface areas of the 2-transition surface and the boundary of the zonohedron are equal.

See Also

zonohedron(), plot.zonohedron()