Lines in space: Part 1: The 4D cross product [Jim Blinn´s Corner]

Computer graphicists see the world as a big pile of polynomials. Piles of linear polynomials (also known as vector and matrix products) represent flat things and straight things. To get curvy things you need higher-order polynomials. In my last few columns^1,2 I´ve played with such higher-order polynomials and their geometric interpretations in 1D and 2D projective spaces. Before trying this in 3D space it´s a good idea to make sure we understand the simple linear case. So in the next couple of columns I´m going to look at homogeneous linear polynomials and their interpretation in projective 3D space. Geometrically, this means that I´ll discuss 3D points, lines, and planes and their intersection and incidence relations. These columns will basically update the ideas from an old Siggraph paper³ with the tensor diagram notation described in past issues of IEEE Computer Graphics and Applications.^1,4 I´ll start by reviewing the algebraic machinery and its geometric interpretation for the lower dimensional spaces. We´ll begin in two dimensions, drop down briefly to one dimension, and then bound off to three dimensions. Along the way, I´ll also share my thoughts about notational conventions for elements of vectors.