I have a question concerning the wonderful talk at SIGGRAPH2019 from Charles Gunn and Steven De Keninck.
What are the benefits of an explicit dimension with e0^2=0.
In Geometric Algebra over R^2 (with e1 and e2 as basis), i can define a bivector e1e2 for rotation, furthermore i can ad a vector to the bivector to build a nilpotent element like e0:=e1+ e1e2 with e0^2=0. So i need at least no extra dimension for mapping a vector along a straight line by an exponential function…
I guess the extra dimension simplifies other operations like the meet operation etc. Is there a simple mathematical argument for the extra dimension?
You cannot rotate around an arbitrary point, so you do not have all rotations in the Euclidean motions. For that, and the translations that involves, you will need a bit more. And the extra null dimension provides it precisely.
You can also see the motions as double reflections in general lines. To characterize a line in 2D, you need 3 homogeneous parameters. Anyway, bivector.net contains material elucidating this, see the sections on 2D PGA and 3D PGA.
Thank you very much for answering my question! I know your great book, what an honor!
At least in principle i can rotate around an arbitrary point by composing three exponential function to translate, rotate, translate… it will take more calculation time than using 2D PGA, i guess. Is this a crucial argument for using an additional dimension?
Your hint with double reflection is also an interesting point, i have to play a little bit with a 2x2 matrix representation of Clifford algebra Cl(2,0) the next days…
‘Imaginary’ units that square to -1, 0, and 1 are very much connected to elliptic rotations, shear operations, and hyperbolic rotations respectively. This is in fact a continuous family of types of rotation, and is deeply linked to the symmetric and antisymmetric contributions from which a transformation matrix is built.
Shearing a line one dimension up that passes through the origin and looking at how the intersection with a plane one unit away from the origin is proportionally affected by the shear helps to reason about why the shear transformation is the best form of rotation to use as a tool to model lower dimensional translations.