Why does the Caley table for 2D PGA use e20 instead of e02?
Moreover a translation motor T in 2D apparently must be defined as 1 - x*e20 + y*e01, otherwise the sandwich product T*x*reverse(T) does not give the expected translation. Compared to the 3D case, where a translation motor is simply 1 + x*e01 + y*e02 + z*e03, there is a negative sign for the x component and the order of x and y are swapped. What is the reason for this inconsistency with the sign?
After some more meditation on this question I understand now that the translator derives from the point/vector definition as 1 + orthogonal(vector(d)/2), and the point/vector definition is dual(E0 + x*E1 + y*E2). The coefficients follow from that, and there is not really a point to expect them to be one form or another.
However the question remains why the definition of PGA 2D on the website uses E20 instead of E02. I tried an implementation which uses E02 and it seems to work fine so far.
I suspect it probably has something to do with keeping the ordering of the indices in some canonical or preferred cyclical order (e.g. 0->1->2->0->…). Other than that, there may not be a compelling reason (as is probably true with many conventions). So long as the signs are consistent with the index ordering and the definitions of the dual operations and geometric products, then the math should work out the same.
I bet that is a trick to optimize products, since e_{02}=-e_{20}, probably the code runs faster with that sign.
regarding e20, here’s what i think:
if you multiply the vectors with the pseudoscaler, you get the bivectors:
e0 e012 = e12
e1 e012 = -e02 = e20 (1 swap get the ones together, one to get rid of the minus)
e2 e012 = -e0212 = e01 (2 swaps to get the twos together)
The reason for the inconsitency is that degenerate metrics do NOT actually model dual numbers.
I have known this for many years, but most of the morons promoting geometric algebra are too stupid to realize that dual numbers cannot be expressed with degenerate metrics due to inconsistencies.
This is part of the reason why I call people like Dorst stupid, because they are not going to be able to confront the fact that their papers and books are inconsistent and wrong.
Fortunately, I have a solution to this inconsistency, and if you are very smart and intelligent, you might figure out that I already gave the correct alternative definitions to express dual numbers with geometric algebra. I wont express it here on this website, because Dorst is not ready to lose his professorship yet, but Grassmann.jl has had an alternative way of expressing dual numbers since 2019 already. If you are very smart, you might have noticed, but most geometric algebra enthusiasts aren’t all that smart.
I have said for years that what these “ga enthusiasts” are promoting is inconsistent mathematics, but they cant handle the truth. I have the correct alternatives which provide a consistent alternative, but these “enthusiasts” dont deserve my help, after what they have done.
I asked for Dorst’s consent on a post here before, if he is ready to confront the inconsistencies and errors in his papers and books. He did not respond. Whenever these people are ready to learn from their mistakes, they can let me know.
But these are people who hate me and hate to see that their papers and reputation is damaged, so they will likely never be willing to learn from mistakes.
It’s the typical problem of an academic ego, their egos and reputation are so inflated that it’s impossible to admit to errors or mistakes in their entire research writings, because it would deflate their ego and reputation. As a result, science suffers.