The join operation of two points is the crossproduct of these two points and represents the line passing through them. The meet operation of two lines is the crossproduct 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.
1 2 3 
p, q 
(3 \times 1) vectors of the homogeneous coordinates of a point. 
l, m 
(3 \times 1) vectors of the homogeneous representation of a line. 
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).
RichterGebert, Jürgen (2011). Perspectives on Projective Geometry  A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 9783642172854
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  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")

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