Understanding e-, e+ Vs e0 and einf

[Tarunark_Singh]

I am learning my ropes with GA and specifically CGA. I using Clifford library for python and found this doc page useful Conformal Geometric Algebra — Clifford 1.5.0dev0 documentation

The diagram at the above link shows the additional basis vector e- and e+, and also e0 and einf.

1. am I correct in thinking that these four basis vector all lie in the plane E0? I believe e- and e+ are orthogonal basis vector in addition to Euclidean basis vectors. And so are e0 and einf. If so then is the pair e0 and einf rotated from of e- and e+?

2. why does e0 have a factor of 1/2 and why does einf does not have the factor of 1/2 when constructed with e- and e+?

3. what is the null space for these basis vector? I assume it is only at the 0,0,0,0,0…(32 times?), since these are all orthogonal basis vector. But then what does it mean to say that some objects can pass through e0 as the origin.

Forgive me if I am missing something obvious and I will thank you for removing that haze.

[chakravala]

@Tarunark_Singh

these four basis vector all lie in the plane E0?

Yes, if by E0 you mean the bivector formed by e+ and e- or e0 and einf.

@Tarunark_Singh

why does e0 have a factor of 1/2 and why does einf does not have the factor of 1/2 when constructed with e- and e+?

The factor of 1/2 is a purely cosmetic choice, you can choose a differently scaled basis too. This choice is made to eliminate some factors of 2 in calculations involving this basis, experiment with the calculations to see why people choose to use a factor of 1/2 there.

@Tarunark_Singh

what is the null space for these basis vector? I assume it is only at the 0,0,0,0,0…(32 times?), since these are all orthogonal basis vector. But then what does it mean to say that some objects can pass through e0 as the origin.

This doesn’t parse as a coherent question, too vague.