# Clifford algebra

by PeteKH
Tags: basics, clifford algebra
 P: 295 Where a vector's tail sits has to be represented separately from the vector, and I don't think that is necessarily any different when using Clifford Algebra unless you introduce some mechanism for encoding that translation. Are you asking about representations of lines in Clifford algebras? For example, a standard parametric representation for points x on a line with points a and b on the line is: $$x = a + \alpha (b-a)$$ you can rewrite this as $$(x - a) = \alpha (b-a)$$ Then wedge both sides with (b-a): $$(x - a) \wedge (b-a) = 0$$ This produces a bivector representation of the line equivalent to the original parameterized equation. Points x on the line will be colinear with the direction vector (b-a), so the wedge product is zero.
 P: 295 Clifford algebra Hi Pete, Again I think that your representation issue probably has to be thought through independent of any Clifford Algebra context. If you wanted to represent a vector and it's origin throughout space, then a pair of vectors is not really enough information. Suppose you represented the base of a 2D vector as a complex number B and it's length and magnetude as another complex number P. Then is a vector formed out of this pairing: $$V = (B, P)$$ really a good representation? I don't think so (consider addition, and what it does to the B values). You probably want a representation something like the computer graphics matrix representation of a point. For this 2D case you could probably use $$V = \begin{bmatrix} B & 0 \\ P & 1 \end{bmatrix}$$ Points with a representation like this can be added. Rotation and scaling are matrix operations of the form $$\begin{bmatrix} T & 0 \\ 0 & 1 \end{bmatrix}$$ And since you have the 1 value in the corner you can rescale your origin after any arbitrary sequence of operations (there's a name for this sort of representation in CG but its been 10 years since school where I used it so I forget that part). I'd conclude that you need at least five real coordinates for your 2D vector field representation problem (one to encode the scale factor). Once you get as far as picking a representation for this sort of vector field space, if desired I'd imagine that there would be a number of possible multivector representations that you could pick from to encode it, just as you have freedom to pick how you want to do so with a matrix representation (could use three by three matrixes with a 2x2 point representation, and 2x1 origin, plus a 1 and two zeros).