A zonohedron \(Z\) is the image of a linear map \([0,1]^n \to Z \subset \bold{R}^3\), from the n-cube to 3D space. For a point on the boundary of the zonohedron, this function computes a point in the unit cube that maps to it. All coordinates of the point in the cube are 0 or 1, except for two of them. The point is not necessarily unique.
# S3 method for zonohedron
invertboundary( x, point, tol=5.e-14 )invertboundary.zonohedron() returns a data.frame
with M rows and these columns:
the given boundary point
signed distance to the boundary of x;
for successful inversion its absolute value is \(\le\) tol
index of the facet pair that contains the point
sign of the facet pair; either +1 or -1
a point in the unit n-cube that maps to the given boundary point;
all coordinates of pcube are 0 or 1,
except for 2 of them.
the number of transitions in pcube - a non-negative even integer
If a point point cannot be inverted, e.g. because
distance is too large, the other columns are all NA.
If the row names of point 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 global error, the function returns NULL.
a zonohedron object as returned by the constructor zonohedron()
Mx3 matrix with points on the boundary of x in the rows.
Such a matrix typically is returned by
raytrace.zonohedron()
or
section.zonohedron().
point can also be a numeric vector that can be converted
to such a matrix, by row.
points that are not within tol of the boundary are
skipped, see Details
Given the boundary point, the function determines the facet
that contains it.
The pcube coordinates of the base vertex of this facet are
all 0 or 1, and fairly easy to determine.
If the facet is a parallelogram, the other two coordinates are
fairly easy to determine too.
If the facet is a zonogon with K generators, with K>2,
then the unknown K coordinates are calculated
using invert.zonogon().
Because of floating point behaviour, coordinates can be slightly
negative or slightly more than 1.
After the calculation, they are clamped to [0,1].
zonohedron(),
section.zonohedron(),
raytrace.zonohedron(),
invert.zonogon()