Lines in 3-space are related to many interesting problems like
visibility, radiosity simulation, etc.
Lines in 2-space are well-understood since the introduction of
primal-dual transformation.
The dual transform is a powerful tool in geometric problem.
The problem dealing with points in 2-space is equivalent to the
transformed problem dealing with lines in 2-space.
The primal-dual idea, when extended into 3-space, is connecting the
problems with points in 3-space and planes in 3-space.
Unfortunately, there is no convenient representation of lines in 3-space
which is both concise in views of representation and computation.
In this talk, I'd like to introduction a representation of lines in
3-space called Plucker coordinates which maps a line in
3-space onto a point in homogeneous 6-projective space.
References
1.
Sommerville, D.Y.M Analytic geometry of three dimensions.
Cambridge, 1934.
2.
Hodge, W.V.D and D. Pedoe Methods of algebraic geometry.
Cambridge, 1947.
3.
Chazelle, Edelsbrunner, Guibas, Sharir,and Stofi,
``Lines in space: Combinatorics and algorithms''.
Algorithmica (1996) 15: 428--447.