section2trans {zonohedra}

compute the intersection of a plane and the 2-transition surface associated with a zonohedron

Description

In general, the 2-transition surface may be highly non-convex, possibly with self-intersections. The intersection of a plane and the 2-transition surface is a union of polygons, possibly with self-intersections and intersecting each other. This function computes one of those polygons. If there are other polygons, it issues a warning and does not try to compute them.

Usage

section2trans( x, normal, beta, invert=FALSE, plot=FALSE, tol=1.e-12, ... )

Arguments

x

a zonohedron object as returned by the constructor zonohedron()

normal

a non-zero numeric 3-vector - the normal of all the planes

beta

a numeric M-vector of plane constants. The equation of the k'th plane k is: <x,normal> = beta[k].

invert

if TRUE, then compute a point in the unit cube that maps to the point on the 2-transition surface, and add it as a column in the returned data.frame

plot

if TRUE, the polygons formed by the the intersection of the planes and the 2-transition surface. added to an existing 3D plot of the zonohedron x, see plot.zonohedron(). The polygons are drawn in red.

tol

a small positive number, used as the tolerance for the plane intersecting the interior of each parallelogram, see Details.

...

not used

Details

The function is designed for the situation when the intersection of a plane and the surface is a single polygon.

Given a plane, the function finds all the parallelograms of the surface whose interiors intersect the plane. Each intersection is a line segment. For each parallelogram it associates one of the endpoints of the segment. The parallelograms are put in polygon order by picking an arbitrary one as the starting point, and then "marching" from one to the next using the canonical parallelogram adjacency relation. After returning to the starting point, if there are other parallelograms remaining, it means that there are other polygons in the section and a warning is issued.

Value

section2trans() returns a list of length M (=length(beta)), and the i'th item in the list is a data frame with these columns:

point

a Px3 matrix with the P points of the i'th polygon in the rows. If the plane does not intersect the 2-transition surface, then P=0 and the matrix has 0 rows. The row names of point are the indexes of the facets that contain the vertices of the polygon; see Details.

gndpair

the 2 indexes from the ground set that generates the parallelogram containing point. See Details for a description of the "marching parallelogram" procedure.

And if invert is TRUE, then this column is added:

pcube

a point in the unit cube that maps to point. This point in the cube always has 2 transitions.

The names of the returned list are readable strings that contain normal and beta[i].

In case of error, the function returns NULL.

Note

The package rgl is required for 3D plotting.

See Also

zonohedron(), plot.zonohedron(), section.zonohedron(), raytrace2trans()