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.
transitionsdf( x, trans2=TRUE )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:
the number of transitions, a positive even integer, in increasing order.
the number of parallelograms with the given number of transitions
the min and max of the area of the parallelograms with the given number of transitions
the total area of the parallelograms with the given number of transitions
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.
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.
a zonohedron object as returned by the constructor zonohedron().
The 2-transition surface must be strictly starshaped.
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.
zonohedron(),
plot.zonohedron()