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 columns1,2 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.