join {RConics} | R Documentation |
The join and meet of two points and the parallel
Description
The join operation of two points is the cross-product of these two points
and represents the line passing through them. The meet operation of two lines
is the cross-product of these two lines and represents their intersection.
The line parallel to a line l
and passing through the point p
corresponds to the join of p
with the meet of l
and the line
at infinity.
Usage
join(p, q)
meet(l, m)
parallel(p, l)
Arguments
p |
|
q |
|
l |
|
m |
|
Value
A (3 \times 1)
vector of either the homogeneous coordinates of
the meet of two lines (a point), the homogeneous representation of the
join of two points (line), or the homogeneous representation of the
parallel line. The vector has the form (x,y,1)
.
Source
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
p <- c(3,1,1)
q <- c(0,2,1)
l <- c(0.75,0.25,1)
# m is the line passin through p and q
m <- join(p,q)
# intersection point of m and l
ml <- meet(l,m)
# line parallel to l and through p
lp <- parallel(p,l)
# plot
plot(rbind(p,q),xlim=c(-5,5),ylim=c(-5,5))
abline(h=0,v=0,col="grey",lty=3)
addLine(l,col="red")
addLine(m,col="blue")
points(t(ml),cex=1.5,pch=20,col="blue")
addLine(lp,col="green")