https://github.com/cran/clifford
Tip revision: b9c5906e0220582174d0c19b7d8c7533235aac08 authored by Robin K. S. Hankin on 13 August 2022, 21:00:03 UTC
version 1.0-8
version 1.0-8
Tip revision: b9c5906
dual_quaternion_clifford.Rmd
---
title: "Dual Quaternions via Clifford algebra"
output: html_document
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library("clifford")
library("onion")
library("disordR")
```
<p style="text-align: right;">
![](`r system.file("help/figures/clifford.png", package = "clifford")`){width=10%}
</p>
Consider the following multiplication table:
```{r,echo=FALSE}
M <- noquote(matrix(c(
" 1", " i", " j", " k", " ε", " εi", " εj", " εk",
" i", " -1", " k", " -j", " εi", " -ε", " εk", "-εj",
" j", " -k", " -1", " i", " εj", "-εk", " -ε", " εi",
" k", " j", " -i", " -1", " εk", " εj", "-εi", " -ε",
" ε", " εi", " εj", " εk", " 0", " 0", " 0", " 0",
"εi", " -ε", " εk", "-εj", " 0", " 0", " 0", " 0",
"εj", "-εk", " -ε", " εi", " 0", " 0", " 0", " 0",
"εk", " εj", "-εi", " -ε"," 0", " 0", " 0", " 0"
),8,8,byrow=TRUE))
rownames(M) <- c(" 1"," i"," j"," k"," ε","εi","εj","εk")
colnames(M) <- c(" 1"," i"," j"," k"," ε"," εi"," εj"," εk")
```
```{r,echo=FALSE,print=TRUE}
M
```
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 by observing that a dual quaternion may be
represented as $A+\varepsilon B$. Thus
\[
(A+\varepsilon B)(C+\varepsilon D)== AC + \varepsilon(AD+BC)\]
(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:
```{r}
DQ_example <- list(as.quaternion(c(5,8,-3,3),single=TRUE), as.quaternion(c(-1,2,1,12),single=TRUE))
DQ_example
```
Then the product can be implemented as follows:
```{r}
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)`.
```{r}
signature(3,0)
e(4)*e(4)
```
\[
\mathbf{i}\leftrightarrow -e_{12}\\
\mathbf{j}\leftrightarrow -e_{13}\\
\mathbf{k}\leftrightarrow -e_{23}\\
\varepsilon\leftrightarrow e_4
\]
Conversion functions would be
```{r definecoercion}
`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:
```{r}
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
```
Checking that `cliff_to_DQ()` is a homomorphism follows the same line
of reasoning:
```{r}
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)
```
The final verification is to check that functions `cliff_to_DQ()` and
`DQ_to_cliff()` are isomorphisms:
```{r}
identical(DQ1,cliff_to_DQ(DQ_to_cliff(DQ1)))
identical(C1,DQ_to_cliff(cliff_to_DQ(C1)))
```