Dual quaternions via Clifford algebra

To cite the clifford package in publications please use Hankin (2022). This short document shows how dual quaternions may be implemented using Clifford algebra. The dual quaternions are an interesting and useful mathematical object which finds applications in solid-body mechanics. Consider the following multiplication table:

##     1   i   j   k    ε  εi  εj  εk
##  1  1   i   j   k    ε  εi  εj  εk
##  i  i  -1   k  -j   εi  -ε  εk -εj
##  j  j  -k  -1   i   εj -εk  -ε  εi
##  k  k   j  -i  -1   εk  εj -εi  -ε
##  ε  ε  εi  εj  εk    0   0   0   0
## εi εi  -ε  εk -εj    0   0   0   0
## εj εj -εk  -ε  εi    0   0   0   0
## εk εk  εj -εi   -ε   0   0   0   0

Thus, for example, \(ij=k\) (not \(-k\)). Here, \(i,j,k\) are the unit quaternion basis elements (so \(i^2=j^2=k^2=ijk=-1\)) and \(\varepsilon\) is the dual unit that commutes with \(i,j,k\) and satisfies \(\varepsilon^2=0\). We can implement this in a rough-and-ready way with the onion package (Hankin 2006) by observing that a dual quaternion may be represented as \(A+\varepsilon B\), where \(A\) and \(B\) are quaternions. Then

\[ (A+\varepsilon B)(C+\varepsilon D)== AC + \varepsilon(AD+BC)\]

(note the absence of a \(BD\) term, as \(\varepsilon^2=0\)). Very crude R idiom for this would be to define a DQ object as a list of quaternions, as in:

DQ_example <- list(as.quaternion(c(5,8,-3,3),single=TRUE), as.quaternion(c(-1,2,1,12),single=TRUE))
DQ_example

## [[1]]
##    [1]
## Re   5
## i    8
## j   -3
## k    3
## 
## [[2]]
##    [1]
## Re  -1
## i    2
## j    1
## k   12

Then the product can be implemented as follows:

DQ_prod_DQ <- function(DQ1,DQ2){
  A <- DQ1[[1]] ; B <- DQ1[[2]] ; C <- DQ2[[1]] ; D <- DQ2[[2]]
  list(A*C, A*D+B*C) 
}

We will use follow the coercion used in inst/quaternion_clifford.Rmd (not the alternative mapping, which requires a different signature) to coerce from cliffords to quaternions. We may map the dual quaternions to Clifford objects by additionally identifying \(\varepsilon\) with \(e_4\); to ensure \(e_4^2=0\) we work with \(Cl(3,0)\) which in package idiom is set by executing signature(3,0).

signature(3,0)
e(4)*e(4)

## Element of a Clifford algebra, equal to
## the zero clifford element (0)

\[ \mathbf{i}\leftrightarrow -e_{12}\\ \mathbf{j}\leftrightarrow -e_{13}\\ \mathbf{k}\leftrightarrow -e_{23}\\ \varepsilon\leftrightarrow e_4 \]

Conversion functions would be

`cliff_to_DQ` <- function(C){  # terms such as e_3 and e_123 and e_34 are silently discarded
    quat <- getcoeffs(C,list(numeric(0),c(1,2),c(1,3),c(2,3)))
    quat[-1] <- -quat[-1]

    epsi <- getcoeffs(C,list(4,c(1,2,4),c(1,3,4),c(2,3,4)))
    epsi[-1] <- -epsi[-1]

    return(list(as.quaternion(quat,single=TRUE),as.quaternion(epsi,single=TRUE)))
} 


`DQ_to_cliff` <- function(DQ){ # DQ is a two-element list of quaternions
  jj1 <- c(as.matrix(DQ[[1]]))
  jj1[-1] <- -jj1[-1]
  jj2 <- c(as.matrix(DQ[[2]]))
  jj2[-1] <- -jj2[-1]
  
 clifford(list(numeric(0),c(1,2),c(1,3),c(2,3),4,c(1,2,4),c(1,3,4),c(2,3,4)),c(jj1,jj2))
}

Check that DQ_to_cliff() is indeed a homomorphism:

DQ1 <- list(as.quaternion(c(5,6,2,-7),single=TRUE),as.quaternion(c(-3,1,4,8),single=TRUE))
DQ2 <- list(as.quaternion(c(-1,3,1,4),single=TRUE),as.quaternion(c(1,9,-7,4),single=TRUE))
LHS <- DQ_to_cliff(DQ1) * DQ_to_cliff(DQ2)
RHS <- DQ_to_cliff(DQ_prod_DQ(DQ1,DQ2))
LHS == RHS

## [1] TRUE

Checking that cliff_to_DQ() is a homomorphism follows the same line of reasoning:

C1 <- clifford(list(numeric(0),c(1,3),c(1,2,4),c(1,3,4)),c(3,7,11,13))
C2 <- clifford(list(numeric(0),c(1,2),c(1,3),c(1,3,4)),c(2,5,6,17))

LHS <- cliff_to_DQ(C1*C2)
RHS <- DQ_prod_DQ(cliff_to_DQ(C1),cliff_to_DQ(C2))
identical(LHS,RHS)

## [1] TRUE

The final verification is to check that functions cliff_to_DQ() and DQ_to_cliff() are isomorphisms:

identical(DQ1,cliff_to_DQ(DQ_to_cliff(DQ1)))

## [1] TRUE

identical(C1,DQ_to_cliff(cliff_to_DQ(C1)))

## [1] TRUE

References

Hankin, R. K. S. 2006. “Normed Division Algebras with R: Introducing the ‘Onion‘ Package.” R News 6 (2): 49–52.

———. 2022. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.

Robin K. S. Hankin

References