clifford (version 1.0-2)

rev: Reverse of a Clifford object

Description

The “reverse” of a term is simply the basis vectors written in reverse order; this changes the sign of the term if the number of basis vectors is 2 or 3 (modulo 4). Taking the reverse is a linear operation.

Both Hestenes and Chisholm use a dagger to denote the reverse of \(A\), as in \(A^\dag\). But both Perwass and Dorst use a tilde, as in \(\tilde{A}\).

$$ \left(A^\dag\right)^\dag=A\qquad \left(AB\right)^\dag=B^\dag A^\dag\qquad \left(A+B\right)^\dag=A^\dag+B^\dag\qquad \left<A^\dag\right>=\left<A\right> $$

where \(\left<A\right>\) is the grade operator; and it is easy to prove that

$$ \left<A^\dag\right>_r=\left<A\right>^\dag_r=(-1)^{r(r-1)/2}\left<A\right>_r $$

We can also show that

$$ \left<AB\right>_r=(-1)^{r(r-1)/2}\left<B^\dag A^\dag\right>_r $$

Usage

# S3 method for clifford
rev(x)

Arguments

x

Clifford object

See Also

grade,Conj

Examples

# NOT RUN {
x <- rcliff()
rev(x)


A <- rblade(g=3)
B <- rblade(g=4)
rev(A %^% B) == rev(B) %^% rev(A)   # should be small
rev(A * B) == rev(B) * rev(A)           # should be small
# }