zonogon-raytrace {zonohedra}

compute the intersection of a ray, based in the interior of a zonogon, and the boundary of that zonogon

Description

The open ray with basepoint \bold{b} and non-zero direction \bold{d} is the set of the form \bold{b} + t \bold{d} where t > 0.

This function computes the intersection of an open ray and the boundary of a zonogon Z. The basepoint is normally required to be in the interior of Z, but an exception is made if the basepoint is 0, and on the boundary of Z, and the direction points into the interior of Z. In these two cases the intersection of the open ray and the boundary of Z is unique. In the second case, the basepoint is also allowed to be the sum of all the generators - the so-called white point of Z.

Usage

## S3 method for class 'zonogon'
raytrace( x, base, direction, plot=FALSE, ... )

Arguments

x

a zonogon object as returned by the constructor zonogon()

base

a numeric 2-vector - the basepoint of all the rays. base must either be in the interior of x, or 0 or the white point and on the boundary of x.

direction

a numeric Mx2 matrix with M non-zero directions in the rows. The basepoint and these directions define M rays.
direction can also be a numeric vector that can be converted to such a matrix, by row.

plot

if TRUE, the computed rays, up to the boundary, are added to an existing plot of the zonogon x, see plot.zonogon(). The segments are drawn in the color red. If there is no open plot, a warning is issued.

...

not used

Value

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

base

the given basepoint - this is the same in every row

direction

the given direction

facetidx

the index of the facet (an edge) where ray exits the zonogon

sign

of the facet, either +1 or -1

tmax

ray parameter of the intersection with the exit facet, always positive

point

the point on the boundary; the intersection of the ray and the facet

timetrace

the computation time, in seconds

If base and direction in a row cannot be processed, the rest of the row is NA.

If the row names of direction are unique, they are copied to the row names of the output.

In case of error, the function returns NULL.

See Also

zonogon(), plot.zonogon(), section.zonohedron()

Examples

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

#   make 4 random directions
set.seed(0)
dir = matrix(rnorm(4*2),4,2)

#  use basepoint in the interior of the zonogon
raytrace( pz20, c(0.5,0.5), dir )
#   base.1 base.2 direction.1 direction.2 facetidx sign      tmax boundary.1 boundary.2    timetrace
# 1    0.5    0.5   1.2629543   0.4146414        4   -1 2.0503073 3.08944438 1.35014236 7.680000e-05
# 2    0.5    0.5  -0.3262334  -1.5399500        1   -1 0.3246859 0.39407664 0.00000000 4.649995e-05
# 3    0.5    0.5   1.3297993  -0.9285670        2   -1 0.4868719 1.14744192 0.04790678 4.310103e-05
# 4    0.5    0.5   1.2724293  -0.2947204        2   -1 0.9354693 1.69031851 0.22429808 4.149997e-05


#  use basepoint at 0 - on the boundary of the zonogon
raytrace( pz20, c(0,0), dir )
#   base.1 base.2 direction.1 direction.2 facetidx sign     tmax boundary.1 boundary.2 timetrace
# 1      0      0   1.2629543   0.4146414        4   -1 2.192481  2.7690037  0.9090936 0.0001216
# 2      0      0  -0.3262334  -1.5399500       NA   NA       NA         NA         NA        NA
# 3      0      0   1.3297993  -0.9285670       NA   NA       NA         NA         NA        NA
# 4      0      0   1.2724293  -0.2947204       NA   NA       NA         NA         NA        NA