zonogon-invert {zonohedra}

invert points in a zonogon

Description

A zonogon Z is the image of a linear map [0,1]^n \to Z \subset \bold{R}^2, from the n-cube to the plane. For a point in a zonogon, this function finds a point in the unit cube that maps to it. There are infinitely many such points in general (unless n=2), but this function picks a specific point using the standard tiling, see Details.

Usage

## S3 method for class 'zonogon'
invert( x, z, tol=0, plot=FALSE, ... )

Arguments

x

a zonogon object as returned by the constructor zonogon()

z

a numeric Mx2 matrix, with M points in the rows. z can also be a numeric vector that can be converted to such a matrix, by row.

tol

points that are within tol of the boundary are still processed, see Details

plot

if TRUE then the points in z are added to an existing plot of the zonogon x, using a red X symbol, see plot.zonogon(). If there is no plot open, a warning is issued.

...

not used

Details

The given points are first tested for being inside the zonogon, using inside() and the given tol. If any are outside, a warning is issued. When the corresponding point pcube is computed, it is clamped to the unit cube, so the inversion error may be as large as tol.

Inversion is not unique in general. For this function, the standard tiling of the zonogon by parallelograms is computed; it is an example of a zonotopal tiling. It is a regular zonotopal tiling because it arises from the projection of a zonohedron onto the plane, see Ziegler. The function plot.zonogon() has an option to plot this tiling. Given the point z, the function determines a parallelogram that contains the point. The pcube coordinates of the base of this parallelogram are all 0 or 1, and the coordinates of z within the parallelogram are in [0,1]. Thus, all coordinates of pcube are 0 or 1, except possibly for 2 of them.

Value

invert.zonogon() returns a data.frame with M rows and these columns:

z

the given point

pcube

a point in the unit cube that maps to z. Every pcube has all coordinates 0 or 1, except possibly for the 2 given by hyper, see Details.

hyper

the 2 indexes of the generators of the parallelogram that contains z, in the simplified matroid. These 2 coordinates in pcube are not 0 or 1 in general.

hyperidx

the index of the parallelogram that contains z

If a point z cannot be inverted, the other columns are all NA, and a warning message is printed.

If the row names of z are unique, they are copied to the row names of the output. The column names of pcube are copied from the ground set of the associated matroid.

In case of error, the function returns NULL.

References

Ziegler, G.M. Lectures on Polytopes. Graduate Texts in Mathematics. Springer New York. 2007.

See Also

zonogon(), inside(), plot.zonogon()

Examples

#   make a zonogon with 5 generators
pz20 = polarzonogon( 20, 5 )

#   make 7 random points in the zonogon
set.seed(0)
pcube = matrix( runif(5*7), 5, 7 )
z = t( getmatrix(pz20) %*% pcube )

#  invert these 7 points back to the cube
invert( pz20, z )
#         z.1       z.2    pcube.1    pcube.2    pcube.3    pcube.4    pcube.5
# 1 2.0676319 1.6279807 0.00000000 0.70030526 1.00000000 1.00000000 0.01553241
# 2 2.4031738 1.9658035 0.00000000 0.96572153 1.00000000 1.00000000 0.28450140
# 3 0.9230336 1.0885446 0.00000000 0.00000000 0.39548689 1.00000000 0.04948838
# 4 2.5242122 1.7395069 0.16540765 1.00000000 1.00000000 1.00000000 0.03542132
# 5 2.2598725 1.0601592 0.38111324 1.00000000 1.00000000 0.20192029 0.00000000
# 6 1.1387813 1.2636700 0.00000000 0.00000000 0.65250505 1.00000000 0.07478012
# 7 1.6315341 1.0777737 0.00000000 0.64210923 1.00000000 0.36039509 0.00000000

#   hyper.1 hyper.2 hyperidx
# 1       2       5        7
# 2       2       5        7
# 3       3       5        9
# 4       1       5        4
# 5       1       4        3
# 6       3       5        9
# 7       2       4        6