Package 'clifford' reference manual

Title:	Arbitrary Dimensional Clifford Algebras
Description:	A suite of routines for Clifford algebras, using the 'Map' class of the Standard Template Library. Canonical reference: Hestenes (1987, ISBN 90-277-1673-0, "Clifford algebra to geometric calculus"). Special cases including Lorentz transforms, quaternion multiplication, and Grassmann algebra, are discussed. Vignettes presenting conformal geometric algebra, quaternions and split quaternions, dual numbers, and Lorentz transforms are included. The package follows 'disordR' discipline.
Authors:	Robin K. S. Hankin [aut, cre]
Maintainer:	Robin K. S. Hankin <[email protected]>
License:	GPL (>= 2)
Version:	1.0-9
Built:	2025-02-13 08:28:51 UTC
Source:	https://github.com/robinhankin/clifford

Arbitrary Dimensional Clifford Algebras

Description

A suite of routines for Clifford algebras, using the 'Map' class of the Standard Template Library. Canonical reference: Hestenes (1987, ISBN 90-277-1673-0, "Clifford algebra to geometric calculus"). Special cases including Lorentz transforms, quaternion multiplication, and Grassmann algebra, are discussed. Vignettes presenting conformal geometric algebra, quaternions and split quaternions, dual numbers, and Lorentz transforms are included. The package follows 'disordR' discipline.

Details

The DESCRIPTION file:

Package:	clifford
Type:	Package
Title:	Arbitrary Dimensional Clifford Algebras
Version:	1.0-9
Depends:	R (>= 4.1.0)
Authors@R:	person(given=c("Robin", "K.", "S."), family="Hankin", role = c("aut","cre"), email="[email protected]", comment = c(ORCID = "0000-0001-5982-0415"))
Maintainer:	Robin K. S. Hankin <[email protected]>
Description:	A suite of routines for Clifford algebras, using the 'Map' class of the Standard Template Library. Canonical reference: Hestenes (1987, ISBN 90-277-1673-0, "Clifford algebra to geometric calculus"). Special cases including Lorentz transforms, quaternion multiplication, and Grassmann algebra, are discussed. Vignettes presenting conformal geometric algebra, quaternions and split quaternions, dual numbers, and Lorentz transforms are included. The package follows 'disordR' discipline.
License:	GPL (>= 2)
LazyData:	yes
Suggests:	knitr, rmarkdown, testthat, onion (>= 1.5-3), lorentz (>= 1.1-1), emulator (>= 1.2-24), jordan (>= 1.0-5), permutations (>= 1.1-5), covr, quadform
VignetteBuilder:	knitr
Imports:	Rcpp (>= 0.12.5), disordR (>= 0.9-8-4), magrittr, methods, partitions (>= 1.10-4), freealg (>= 1.0-4)
LinkingTo:	Rcpp,BH
URL:	https://github.com/RobinHankin/clifford, https://robinhankin.github.io/clifford/
BugReports:	https://github.com/RobinHankin/clifford/issues
Config/pak/sysreqs:	libgmp3-dev
Repository:	https://robinhankin.r-universe.dev
RemoteUrl:	https://github.com/robinhankin/clifford
RemoteRef:	HEAD
RemoteSha:	0b41f101fe3ebad82c64f3c5184ab19949f03869
Author:	Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Index of help topics:

Ops.clifford            Arithmetic Ops Group Methods for 'clifford'
                        objects
[.clifford              Extract or Replace Parts of a clifford
allcliff                Clifford object containing all possible terms
antivector              Antivectors or pseudovectors
as.vector               Coerce a clifford vector to a numeric vector
cartan                  Cartan map between clifford algebras
clifford                Create, coerce, and test for 'clifford' objects
clifford-package        Arbitrary Dimensional Clifford Algebras
const                   The constant term of a Clifford object
dot-class               Class "dot"
drop                    Drop redundant information
even                    Even and odd clifford objects
grade                   The grade of a clifford object
homog                   Homogenous Clifford objects
horner                  Horner's method
involution              Clifford involutions
lowlevel                Low-level helper functions for 'clifford'
                        objects
magnitude               Magnitude of a clifford object
minus                   Take the negative of a vector
numeric_to_clifford     Coercion from numeric to Clifford form
print.clifford          Print clifford objects
pseudoscalar            Coercion from numeric to Clifford form
quaternion              Quaternions using Clifford algebras
rcliff                  Random clifford objects
signature               The signature of the Clifford algebra
summary.clifford        Summary methods for clifford objects
term                    Deal with terms
zap                     Zap small values in a clifford object
zero                    The zero Clifford object

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[email protected]>

References

J. Snygg (2012). A new approach to differential geometry using Clifford's geometric Algebra, Birkhauser. ISBN 978-0-8176-8282-8
D. Hestenes (1987). Clifford algebra to geometric calculus, Kluwer. ISBN 90-277-1673-0
C. Perwass (2009). Geometric algebra with applications in engineering, Springer. ISBN 978-3-540-89068-3
D. Hildenbrand (2013). Foundations of geometric algebra computing. Springer, ISBN 978-3-642-31794-1

Examples


as.1vector(1:4)

as.1vector(1:4) * rcliff()

# Following from Ablamowicz and Fauser (see vignette):
x <- clifford(list(1:3,c(1,5,7,8,10)),c(4,-10)) + 2
y <- clifford(list(c(1,2,3,7),c(1,5,6,8),c(1,4,6,7)),c(4,1,-3)) - 1
x*y # signature irrelevant



as.1vector(1:4)

as.1vector(1:4) * rcliff()

# Following from Ablamowicz and Fauser (see vignette):
x <- clifford(list(1:3,c(1,5,7,8,10)),c(4,-10)) + 2
y <- clifford(list(c(1,2,3,7),c(1,5,6,8),c(1,4,6,7)),c(4,1,-3)) - 1
x*y # signature irrelevant

Clifford object containing all possible terms

Description

The Clifford algebra on basis vectors $e_1,e_2,\ldots, e_n$ has $2^n$ independent multivectors. Function allcliff() generates a clifford object with a nonzero coefficient for each multivector.

Usage

allcliff(n,grade)allcliff(n,grade)

Arguments

`n`	Integer specifying dimension of underlying vector space
`grade`	Grade of multivector to be returned. If missing, multivector contains every term of every grade $\leq n$

Author(s)

Robin K. S. Hankin

Examples

allcliff(6)

a <- allcliff(5)
a[] <- rcliff()*100

allcliff(6)

a <- allcliff(5)
a[] <- rcliff()*100

Antivectors or pseudovectors

Description

Antivectors or pseudovectors

Usage

antivector(v, n = length(v))
as.antivector(v)
is.antivector(C, include.pseudoscalar=FALSE)
antivector(v, n = length(v))
as.antivector(v)
is.antivector(C, include.pseudoscalar=FALSE)

Arguments

`v`	Numeric vector
`n`	Integer specifying dimensionality of underlying vector space
`C`	Clifford object
`include.pseudoscalar`	Boolean: should the pseudoscalar be considered an antivector?

Details

An antivector is an $n$ -dimensional Clifford object, all of whose terms are of grade $n-1$ . An antivector has $n$ degrees of freedom. Function antivector(v,n) interprets v[i] as the coefficient of $e_1e_2\ldots e_{i-1}e_{i+1}\ldots e_n$ .

Function as.antivector() is a convenience wrapper, coercing its argument to an antivector of minimal dimension (zero entries are interpreted consistently).

The pseudoscalar is a peculiar edge case. Consider:

  A <- clifford(list(c(1,2,3)))
  B <- A + clifford(list(c(1,2,4)))

> is.antivector(A)
[1] FALSE
> is.antivector(B)
[1] TRUE
> is.antivector(A,include.pseudoscalar=TRUE)
[1] TRUE
> is.antivector(B,include.pseudoscalar=TRUE)
[1] TRUE

One could argue that A should be an antivector as it is a term in B, which is definitely an antivector. Use include.pseudoscalar=TRUE to ensure consistency in this case.

Compare as.1vector(), which returns a clifford object of grade 1.

Note

An antivector is always a blade.

Author(s)

Robin K. S. Hankin

References

Wikipedia contributors. (2018, July 20). “Antivector”. In Wikipedia, The Free Encyclopedia. Retrieved 19:06, January 27, 2020, from https://en.wikipedia.org/w/index.php?title=Antivector&oldid=851094060

Examples

antivector(1:5)

as.1vector(c(1,1,2)) %X% as.1vector(c(3,2,2))
c(1*2-2*2, 2*3-1*2, 1*2-1*3)  # note sign of e_13

antivector(1:5)

as.1vector(c(1,1,2)) %X% as.1vector(c(3,2,2))
c(1*2-2*2, 2*3-1*2, 1*2-1*3)  # note sign of e_13

Coerce a clifford vector to a numeric vector

Description

Given a clifford object with all terms of grade 1, return the corresponding numeric vector

Usage

## S3 method for class 'clifford'
as.vector(x,mode = "any")
## S3 method for class 'clifford'
as.vector(x,mode = "any")

Arguments

`x`	Object of class clifford
`mode`	ignored

Note

The awkward R idiom of this function is because the terms may be stored in any order; see the examples

Author(s)

Robin K. S. Hankin

Examples

x <- clifford(list(6,2,9),1:3)
as.vector(x)

as.1vector(as.vector(x)) == x  # should be TRUE

x <- clifford(list(6,2,9),1:3)
as.vector(x)

as.1vector(as.vector(x)) == x  # should be TRUE

Cartan map between clifford algebras

Description

Cartan's map isomorphisms from $\operatorname{Cl}(p,q)$ to $\operatorname{Cl}(p-4,q+4)$ and $\operatorname{Cl}(p+4,q-4)$

Usage

cartan(C, n = 1)
cartan_inverse(C, n = 1)
cartan(C, n = 1)
cartan_inverse(C, n = 1)

Arguments

`C`	Object of class `clifford`
`n`	Strictly positive integer

Value

Returns an object of class clifford. The default value n=1 maps $\operatorname{Cl}(4,q)$ to $\operatorname{Cl}(0,q+4)$ (cartan()) and $\operatorname{Cl}(0,q)$ to $\operatorname{Cl}(4,q-4)$ .

Author(s)

Robin K. S. Hankin

References

E. Hitzer and S. Sangwine 2017. “Multivector and multivector matrix inverses in real Clifford algebras”, Applied Mathematics and Computation. 311:3755-89

Examples


a <- rcliff(d=7)   # Cl(4,3)
b <- rcliff(d=7)   # Cl(4,3)
signature(4,3)     # e1^2 = e2^2 = e3^2 = e4^2 = +1; e5^2 = e6^2=e7^2 = -1
ab <- a*b          # multiplication in Cl(4,3)

signature(0,7)   # e1^2 = ... = e7^2 = -1
cartan(a)*cartan(b) == cartan(ab) # multiplication in Cl(0,7); should be TRUE

signature(Inf)  # restore default
a <- rcliff(d=7)   # Cl(4,3)
b <- rcliff(d=7)   # Cl(4,3)
signature(4,3)     # e1^2 = e2^2 = e3^2 = e4^2 = +1; e5^2 = e6^2=e7^2 = -1
ab <- a*b          # multiplication in Cl(4,3)

signature(0,7)   # e1^2 = ... = e7^2 = -1
cartan(a)*cartan(b) == cartan(ab) # multiplication in Cl(0,7); should be TRUE

signature(Inf)  # restore default

Create, coerce, and test for `clifford` objects

Description

An object of class clifford is a member of a Clifford algebra. These objects may be added and multiplied, and have various applications in physics and mathematics.

Usage

clifford(terms, coeffs=1)
is_ok_clifford(terms, coeffs)
as.clifford(x)
is.clifford(x)
nbits(x)
nterms(x)
## S3 method for class 'clifford'
dim(x)
clifford(terms, coeffs=1)
is_ok_clifford(terms, coeffs)
as.clifford(x)
is.clifford(x)
nbits(x)
nterms(x)
## S3 method for class 'clifford'
dim(x)

Arguments

`terms`	A list of integer vectors with strictly increasing entries corresponding to the basis vectors of the underlying vector space
`coeffs`	Numeric vector of coefficients
`x`	Object of class `clifford`

Details

Function clifford() is the formal creation mechanism for clifford objects. If coeffs is of length 1, it will be recycled (even if terms is empty, in which case the zero Clifford object is returned). Argument terms is passed through list_modifier(), so a zero entry is interpreted as numeric(0)
Function as.clifford() is much more user-friendly and attempts to coerce a range of input arguments to clifford form
Function nbits() returns the number of bits required in the low-level C routines to store the terms (this is the largest entry in the list of terms). For a scalar, this is zero and for the zero clifford object it (currently) returns zero as well although a case could be made for NULL
Function nterms() returns the number of terms in the expression
Function is_ok_clifford() is a helper function that checks for consistency of its arguments

Author(s)

Robin K. S. Hankin

References

Snygg 2012. “A new approach to differential geometry using Clifford's geometric algebra”. Birkhauser; Springer Science+Business.

Examples


(x <- clifford(list(1,2,1:4),1:3))   # Formal creation method
(y <- as.1vector(4:2))
(z <- rcliff(include.fewer=TRUE))

terms(x+100)
coeffs(z)

## Clifford objects may be added and multiplied:

x + y
x*y

(x <- clifford(list(1,2,1:4),1:3))   # Formal creation method
(y <- as.1vector(4:2))
(z <- rcliff(include.fewer=TRUE))

terms(x+100)
coeffs(z)

## Clifford objects may be added and multiplied:

x + y
x*y

The constant term of a Clifford object

Description

Get and set the constant term of a clifford object.

Usage

const(C,drop=TRUE)
is.real(C)
## S3 replacement method for class 'clifford'
const(x) <- value
const(C,drop=TRUE)
is.real(C)
## S3 replacement method for class 'clifford'
const(x) <- value

Arguments

`C`, `x`	Clifford object
`value`	Replacement value
`drop`	Boolean, with default `TRUE` meaning to return the constant coerced to numeric, and `FALSE` meaning to return a (constant) Clifford object

Details

Extractor method for specific terms. Function const() returns the constant element of a Clifford object. Note that const(C) returns the same as grade(C,0), but is faster. If C is a numeric vector, the first element is returned: any other elements are silently discarded, but this may change in future.

The R idiom in const<-() is slightly awkward:

> body(`const<-.clifford`)
{
  stopifnot(length(value) == 1)
  x <- x - const(x)
  return(x + value)
}

The reason that it is not simply return(x-const(x)+value) or return(x+value-const(x)) is to ensure numerical accuracy; see examples.

Author(s)

Robin K. S. Hankin

Examples


X <- clifford(list(1,1:2,1:3,3:5),6:9)
X
X <- X + 1e300
X

const(X) # should be 1e300

const(X) <- 0.6
const(X) # should be 0.6, no numerical error

# compare naive approach:

X <- clifford(list(1,1:2,1:3,3:5),6:9)+1e300
X+0.6-const(X)  # constant gets lost in the numerics

X <- clifford(list(1,1:2,1:3,3:5),6:9)+1e-300
X-const(X)+0.6  # answer correct by virtue of left-associativity


x <- 2+rcliff(d=3,g=3)
jj <- x*cliffconj(x)
is.real(jj*rev(jj))   # should be TRUE
X <- clifford(list(1,1:2,1:3,3:5),6:9)
X
X <- X + 1e300
X

const(X) # should be 1e300

const(X) <- 0.6
const(X) # should be 0.6, no numerical error

# compare naive approach:

X <- clifford(list(1,1:2,1:3,3:5),6:9)+1e300
X+0.6-const(X)  # constant gets lost in the numerics

X <- clifford(list(1,1:2,1:3,3:5),6:9)+1e-300
X-const(X)+0.6  # answer correct by virtue of left-associativity


x <- 2+rcliff(d=3,g=3)
jj <- x*cliffconj(x)
is.real(jj*rev(jj))   # should be TRUE

Class “dot”

Description

The dot object is defined so that idiom like .[x,y] returns the commutator, that is, (x*y-y*x)/2. The factor of 2 ensures that .[x,y] == x %X% y.

The dot object is generated by running script inst/dot.Rmd, which includes some further discussion and technical documentation, and creates file dot.rda which resides in the data/ directory.

Arguments

`x`	Object of any class
`i`, `j`	elements to commute
`...`	Further arguments to `dot_error()`, currently ignored

Value

Always returns an object of the same class as xy.

Author(s)

Robin K. S. Hankin

Examples

x <- rcliff()
y <- rcliff()
z <- rcliff()

.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]]  # Jacobi identity


x <- rcliff()
y <- rcliff()
z <- rcliff()

.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]]  # Jacobi identity

Drop redundant information

Description

Coerce scalar Clifford objects to numeric

Usage

drop(x)
drop_clifford(x)
drop(x)
drop_clifford(x)

Arguments

`x`	Clifford object

Details

If its argument is a pure scalar clifford object, or the pseudoscalar, coerce to numeric. Scalar or pseudoscalar clifford objects are coerced to an unnamed numeric vector (of length 1). Checking for being the pseudoscalar requires that option maxdim be set.

Function drop() is generic, dispatching to helper function drop_clifford() for clifford objects. The logic of drop_clifford() prevents is.pseudoscalar() being called if maxdim is NULL.

Note

Many functions in the package take drop as an argument which, if TRUE, means that the function returns a dropped value.

Author(s)

Robin K. S. Hankin

Examples

drop(as.clifford(5))

const(rcliff())
const(rcliff(),drop=FALSE)

drop(as.clifford(5))

const(rcliff())
const(rcliff(),drop=FALSE)

Even and odd clifford objects

Description

A clifford object is even if every term has even grade, and odd if every term has odd grade.

Functions is.even() and is.odd() test a clifford object for evenness or oddness.

Functions evenpart() and oddpart() extract the even or odd terms from a clifford object, and we write $A_+$ and $A_-$ respectively; we have $A=A_+ + A_-$

Usage

is.even(C)
is.odd(C)
evenpart(C)
oddpart(C)
is.even(C)
is.odd(C)
evenpart(C)
oddpart(C)

Arguments

`C`	Clifford object

Author(s)

Robin K. S. Hankin

Examples


A <- rcliff()
A == evenpart(A) + oddpart(A) # should be true

A <- rcliff()
A == evenpart(A) + oddpart(A) # should be true

Extract or Replace Parts of a clifford

Description

Extract or replace subsets of cliffords.

Usage

## S3 method for class 'clifford'
C[index, ...,drop=FALSE]
## S3 replacement method for class 'clifford'
C[index, ...] <- value
coeffs(x)
coeffs(x) <- value
list_modifier(B)
getcoeffs(C, B)
## S3 method for class 'clifford'
Im(z)
## S3 method for class 'clifford'
Re(z)
## S3 method for class 'clifford'
C[index, ...,drop=FALSE]
## S3 replacement method for class 'clifford'
C[index, ...] <- value
coeffs(x)
coeffs(x) <- value
list_modifier(B)
getcoeffs(C, B)
## S3 method for class 'clifford'
Im(z)
## S3 method for class 'clifford'
Re(z)

Arguments

`C`, `x`, `z`	A clifford object
`index`	elements to extract or replace
`value`	replacement value
`B`	A list of integer vectors, terms
`drop`	Boolean: should constant clifford objects be coerced to numeric?
`...`	Further arguments

Details

Extraction and replacement methods. The extraction method uses getcoeffs() and the replacement method uses low-level helper function c_overwrite().

In the extraction function a[index], if index is a list, further arguments are ignored; if not, the dots are used. If index is a list, its elements are interpreted as integer vectors indicating which terms to be extracted (even if it is a disord object). If index is a disord object, standard consistency rules are applied. The extraction methods are designed so that idiom such as a[coeffs(a)>3] works.

For replacement methods, the standard use-case is a[i] <- b in which argument i is a list of integer vectors and b a length-one numeric vector; (replacement vectors of length greater than one are currently not implemented, whether or not they violate disordR discipline). Otherwise, to manipulate parts of a clifford object, use coeffs(a) <- value; disord discipline is enforced. Idiom such as a[coeffs(a)<2] <- 0 is implemented experimentally, as syntactic sugar for coeffs(a)[coeffs(a)<2] <- 0. Replacement using a list-valued index, as in A[i] <- value uses an ugly hack if value is zero. Replacement methods are not yet finalised and not yet fully integrated with the disordR package.

Idiom such as a[] <- b follows the spray package. If b is a length-one scalar, then coeffs(a) <- b has the same effect as a[] <- b.

Grade-based replacement methods such as grade(C,n) <- value are impemented and documented at grade.Rd.

Functions terms() [see term.Rd] and coeffs() extract the terms and coefficients from a clifford object. These functions return disord objects but the ordering is consistent between them (an extended discussion of this phenomenon is presented in the mvp package). Note that coeffs() returns numeric(0) on the zero clifford object.

Function coeffs<-() (idiom coeffs(a) <- b) sets all coefficients of a to b. This has the same effect as a[] <- b.

Extracting or replacing a list with a repeated elements is usually a Bad Idea (tm). However, if option warn_on_repeats is set to FALSE, no warning will be given (and the coefficient will be the sum of the coefficients of the term; see the examples).

Function getcoeffs() is a lower-level helper function that lacks the succour offered by [.clifford(). It returns a named numeric vector [not a disord object: the order of the elements is determined by the order of argument B]. Compare standard extraction, eg a[index], which returns a clifford object. The names of the returned vector are determined by function catterm().

Attempting to extract a coefficient of a term that includes a negative index will throw an error. The coefficient of a term not present in the Clifford object (including term with an index larger than indicated by maxyterm()) will return zero.

The index for the constant is formally list(numeric(0)), but this is a pain to type. Square bracket extraction and getcoeffs() have special dispensation for zero entries, which are translated by helper function list_modifier() to numeric(0) and listified if necessary. The upshot is that x[0] and getcoeffs(x,0) work as expected, returning the constant.

Function Im() is a generic, which sets the real component of its argument to zero (as per the onion package). Function Re() is a convenience synonym for const().

Vignette getcoeffs gives a more extended discussion of function getcoeffs().

Examples

A <- clifford(list(1,1:2,1:3),1:3)
B <- clifford(list(1:2,1:6),c(44,45))

A[1,c(1,3,4)]

A[2:3, 4] <- 99
A[] <- B



X <- 5 + 6*e(1) -7*e(1:3) + 3*e(4:5)
X[0]           # special dispensation for zero
X[0,drop=TRUE] # coerce to numeric
X[list(0,1:3)] 

getcoeffs(X,0)
getcoeffs(X,list(1,0,1:3))

# clifford(list(1,1:2,1:2),1:3)  # would give a warning

options("warn_on_repeats" = FALSE)
clifford(list(1,1:2,1:2),1:3)  # works; 1e1 + 5e_12

options("warn_on_repeats" = TRUE) # return to default behaviour.


A <- clifford(list(1,1:2,1:3),1:3)
B <- clifford(list(1:2,1:6),c(44,45))

A[1,c(1,3,4)]

A[2:3, 4] <- 99
A[] <- B



X <- 5 + 6*e(1) -7*e(1:3) + 3*e(4:5)
X[0]           # special dispensation for zero
X[0,drop=TRUE] # coerce to numeric
X[list(0,1:3)] 

getcoeffs(X,0)
getcoeffs(X,list(1,0,1:3))

# clifford(list(1,1:2,1:2),1:3)  # would give a warning

options("warn_on_repeats" = FALSE)
clifford(list(1,1:2,1:2),1:3)  # works; 1e1 + 5e_12

options("warn_on_repeats" = TRUE) # return to default behaviour.

The grade of a clifford object

Description

The grade of a term is the number of basis vectors in it.

Usage

grade(C, n, drop=TRUE)
grade(C,n) <- value
grades(x)
gradesplus(x)
gradesminus(x)
gradeszero(x)
grade(C, n, drop=TRUE)
grade(C,n) <- value
grades(x)
gradesplus(x)
gradesminus(x)
gradeszero(x)

Arguments

`C`, `x`	Clifford object
`n`	Integer vector specifying grades to extract
`value`	Replacement value, a numeric vector
`drop`	Boolean, with default `TRUE` meaning to coerce a constant Clifford object to numeric, and `FALSE` meaning not to

Details

A term is a single expression in a Clifford object. It has a coefficient and is described by the basis vectors it comprises. Thus $4e_{234}$ is a term but $e_3 + e_5$ is not.

The grade of a term is the number of basis vectors in it. Thus the grade of $e_1$ is 1, and the grade of $e_{125}=e_1e_2e_5$ is 3. The grade operator $\left\langle\cdot\right\rangle_r$ is used to extract terms of a particular grade, with

$A=\left\langle A\right\rangle_0 + \left\langle A\right\rangle_1 + \left\langle A\right\rangle_2 + \cdots = \sum_r\left\langle A\right\rangle_r$

for any Clifford object $A$ . Thus $\left\langle A\right\rangle_r$ is said to be homogenous of grade $r$ . Hestenes sometimes writes subscripts that specify grades using an overbar as in $\left\langle A\right\rangle_{\overline{r}}$ . It is conventional to denote the zero-grade object $\left\langle A\right\rangle_0$ as simply $\left\langle A\right\rangle$ .

We have

$\left\langle A+B\right\rangle_r=\left\langle A\right\rangle_r + \left\langle B\right\rangle_r\qquad \left\langle\lambda A\right\rangle_r=\lambda\left\langle A\right\rangle_r\qquad \left\langle\left\langle A\right\rangle_r\right\rangle_s=\left\langle A\right\rangle_r\delta_{rs}.$

Function grades() returns an (unordered) vector specifying the grades of the constituent terms. Function grades<-() allows idiom such as grade(x,1:2) <- 7 to operate as expected [here to set all coefficients of terms with grades 1 or 2 to value 7].

Function gradesplus() returns the same but counting only basis vectors that square to $+1$ , and gradesminus() counts only basis vectors that square to $-1$ . Function signature() controls which basis vectors square to $+1$ and which to $-1$ .

From Perwass, page 57, given a bilinear form

$\left\langle{\mathbf x},{\mathbf x}\right\rangle=x_1^2+x_2^2+\cdots +x_p^2-x_{p+1}^2-\cdots -x_{p+q}^2$

and a basis blade $e_\mathbb{A}$ with $A\subseteq\left\lbrace 1,\ldots,p+q\right\rbrace$ , then

$\mathrm{gr}(e_A) = \left|\left\lbrace a\in A\colon 1\leqslant a\leqslant p+q\right\rbrace\right|$

$\mathrm{gr}_{+}(e_A) = \left|\left\lbrace a\in A\colon 1\leqslant a\leqslant p\right\rbrace\right|$

$\mathrm{gr}_{-}(e_A) = \left|\left\lbrace a\in A\colon p < a\leq p+q\right\rbrace\right|$

Function gradeszero() counts only the basis vectors squaring to zero (I have not seen this anywhere else, but it is a logical suggestion).

If the signature is zero, then the Clifford algebra reduces to a Grassmann algebra and products match the wedge product of exterior calculus. In this case, functions gradesplus() and gradesminus() return NA.

Function grade(C,n) returns a clifford object with just the elements of grade g, where g %in% n.

Idiom like grade(C,r) <- value, where r is a non-negative integer (or vector of non-negative integers) should behave as expected. It has two distinct cases: firstly, where value is a length-one numeric vector; and secondly, where value is a clifford object:

Firstly, grade(C,r) <- value with value a length-one numeric vector. This changes the coefficient of all grade-r terms to value. Note that disordR discipline must be respected, so if value has length exceeding one, a disordR consistency error might be raised.
Secondly, grade(C,r) <- value with value a clifford object. This should operate as expected: it will replace the grade-r components of C with value. If value has any grade component not in r, a “grade mismatch” error will be returned. Thus, only the grade-r components of C may be modified with this construction. It is semi vectorised: if r is a vector, it is interpreted as a set of grades to replace.

The zero grade term, grade(C,0), is given more naturally by const(C).

Function c_grade() is a helper function that is documented at Ops.clifford.Rd.

Note

In the C code, “term” has a slightly different meaning, referring to the vectors without the associated coefficient.

Author(s)

Robin K. S. Hankin

References

C. Perwass 2009. “Geometric algebra with applications in engineering”. Springer.

Examples


a <- clifford(sapply(seq_len(7),seq_len),seq_len(7))
a
grades(a)
grade(a,5)


a <- clifford(list(0,3,7,1:2,2:3,3:4,1:3,1:4),1:8)
b <- clifford(list(4,1:2,2:3),c(101,102,103))

grade(a,1) <- 13*e(6)
grade(a,2) <- grade(b,2)
grade(a,0:2) <- grade(b,0:2)*7


signature(2,2)
x <- rcliff()
drop(gradesplus(x) + gradesminus(x) + gradeszero(x) - grades(x))

a <- rcliff()
a == Reduce(`+`,sapply(unique(grades(a)),function(g){grade(a,g)}))

a <- clifford(sapply(seq_len(7),seq_len),seq_len(7))
a
grades(a)
grade(a,5)


a <- clifford(list(0,3,7,1:2,2:3,3:4,1:3,1:4),1:8)
b <- clifford(list(4,1:2,2:3),c(101,102,103))

grade(a,1) <- 13*e(6)
grade(a,2) <- grade(b,2)
grade(a,0:2) <- grade(b,0:2)*7


signature(2,2)
x <- rcliff()
drop(gradesplus(x) + gradesminus(x) + gradeszero(x) - grades(x))

a <- rcliff()
a == Reduce(`+`,sapply(unique(grades(a)),function(g){grade(a,g)}))

Homogenous Clifford objects

Description

A clifford object is homogenous if all its terms are the same grade. A scalar (including the zero clifford object) is considered to be homogenous. This ensures that is.homog(grade(C,n)) always returns TRUE.

Usage

is.homog(C)
is.homog(C)

Arguments

`C`	Object of class clifford

Note

Nonzero homogenous clifford objects have a multiplicative inverse.

Author(s)

Robin K. S. Hankin

Examples

is.homog(rcliff())
is.homog(rcliff(include.fewer=FALSE))
is.homog(rcliff())
is.homog(rcliff(include.fewer=FALSE))

Horner's method

Description

Horner's method for Clifford objects

Usage

horner(P,v)horner(P,v)

Arguments

`P`	A Clifford object
`v`	Numeric vector of coefficients

Details

Given a polynomial

$p(x) = a_0 +a_1+a_2x^2+\cdots + a_nx^n$

it is possible to express $p(x)$ in the algebraically equivalent form

$p(x) = a_0 + x\left(a_1+x\left(a_2+\cdots + x\left(a_{n-1} +xa_n \right)\cdots\right)\right)$

which is much more efficient for evaluation, as it requires only $n$ multiplications and $n$ additions, and this is optimal. The output of horner() depends on the signature().

Note

Horner's method is not as cool for Clifford objects as it is for (e.g.) multivariate polynomials or freealg objects. This is because powers of Clifford objects don't get more complicated as the power increases.

Author(s)

Robin K. S. Hankin

Examples


horner(1+e(1:3)+e(2:3) , 1:6)

rcliff() |> horner(1:4)

horner(1+e(1:3)+e(2:3) , 1:6)

rcliff() |> horner(1:4)

Clifford involutions

Description

An involution is a function that is its own inverse, or equivalently $f(f(x))=x$ . There are several important involutions on Clifford objects; these commute past the grade operator with $f(\left\langle A\right\rangle_r)=\left\langle f(A)\right\rangle_r$ and are linear: $f(\alpha A+\beta B)=\alpha f(A)+\beta f(B)$ .

The dual is documented here for convenience, even though it is not an involution (applying the dual four times is the identity).

The reverse $A^\sim$ is given by rev() (both Perwass and Dorst use a tilde, as in $\tilde{A}$ or $A^\sim$ . However, both Hestenes and Chisholm use a dagger, as in $A^\dagger$ . This page uses Perwass's notation). The reverse of a term written as a product of basis vectors is simply the product of the same basis vectors but written in reverse order. This changes the sign of the term if the number of basis vectors is 2 or 3 (modulo 4). Thus, for example, $\left(e_1e_2e_3\right)^\sim=e_3e_2e_1=-e_1e_2e_3$ and $\left(e_1e_2e_3e_4\right)^\sim=e_4e_3e_2e_1=+e_1e_2e_3e_4$ . Formally, if $X=e_{i_1}\ldots e_{i_k}$ , then $\tilde{X}=e_{i_k}\ldots e_{i_1}$ .

$\left\langle A^\sim\right\rangle_r=\widetilde{\left\langle A\right\rangle_r}=(-1)^{r(r-1)/2}\left\langle A\right\rangle_r$

Perwass shows that $\left\langle AB\right\rangle_r=(-1)^{r(r-1)/2}\left\langle\tilde{B}\tilde{A}\right\rangle_r$
The Conjugate $A^\dagger$ is given by Conj() (we use Perwass's notation, def 2.9 p59). This depends on the signature of the Clifford algebra; see grade.Rd for notation. Given a basis blade $e_\mathbb{A}$ with $\mathbb{A}\subseteq\left\lbrace 1,\ldots,p+q\right\rbrace$ , then we have $e_\mathbb{A}^\dagger = (-1)^m {e_\mathbb{A}}^\sim$ , where $m=\mathrm{gr}_{-}(\mathbb{A})$ . Alternatively, we might say

$\left(\left\langle A\right\rangle_r\right)^\dagger=(-1)^m(-1)^{r(r-1)/2}\left\langle A\right\rangle_r$

where $m=\mathrm{gr}_{-}(\left\langle A\right\rangle_r)$ [NB I have changed Perwass's notation].
The main (grade) involution or grade involution $\widehat{A}$ is given by gradeinv(). This changes the sign of any term with odd grade:

$\widehat{\left\langle A\right\rangle_r} =(-1)^r\left\langle A\right\rangle_r$

(I don't see this in Perwass or Hestenes; notation follows Hitzer and Sangwine). It is a special case of grade negation.
The grade $r$ -negation $A_{\overline{r}}$ is given by neg(). This changes the sign of the grade $r$ component of $A$ . It is formally defined as $A-2\left\langle A\right\rangle_r$ but function neg() uses a more efficient method. It is possible to negate all terms with specified grades, so for example we might have $\left\langle A\right\rangle_{\overline{\left\lbrace 1,2,5\right\rbrace}} = A-2\left( \left\langle A\right\rangle_1 +\left\langle A\right\rangle_2+\left\langle A\right\rangle_5\right)$ and the R idiom would be neg(A,c(1,2,5)). Note that Hestenes uses “ $A_{\overline{r}}$ ” to mean the same as $\left\langle A\right\rangle_r$ .
The Clifford conjugate $\overline{A}$ is given by cliffconj(). It is distinct from conjugation $A^\dagger$ , and is defined in Hitzer and Sangwine as

$\overline{\left\langle A\right\rangle_r} = (-1)^{r(r+1)/2}\left\langle A\right\rangle_r.$
The dual $C^*$ of a clifford object $C$ is given by dual(C,n); argument n is the dimension of the underlying vector space. Perwass gives $C^*=CI^{-1}$

where $I=e_1e_2\ldots e_n$ is the unit pseudoscalar [note that Hestenes uses $I$ to mean something different]. The dual is sensitive to the signature of the Clifford algebra and the dimension of the underlying vector space.

Usage

## S3 method for class 'clifford'
rev(x)
## S3 method for class 'clifford'
Conj(z)
cliffconj(z)
neg(C,n)
gradeinv(C)
## S3 method for class 'clifford'
rev(x)
## S3 method for class 'clifford'
Conj(z)
cliffconj(z)
neg(C,n)
gradeinv(C)

Arguments

`C`, `x`, `z`	Clifford object
`n`	Integer vector specifying grades to be negated in `neg()`

Author(s)

Robin K. S. Hankin

Examples


x <- rcliff()
x
rev(x)


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

options(maxdim=8)
a <- rcliff(d=8)
dual(dual(dual(dual(a,8),8),8),8) == a # should be TRUE
options(maxdim=NULL) # restore default
x <- rcliff()
x
rev(x)


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

options(maxdim=8)
a <- rcliff(d=8)
dual(dual(dual(dual(a,8),8),8),8) == a # should be TRUE
options(maxdim=NULL) # restore default

Low-level helper functions for `clifford` objects

Description

Helper functions for clifford objects, written in C using the STL map class.

Usage

c_identity(L, p, m)
c_grade(L, c, m, n)
c_add(L1, c1, L2, c2, m)
c_multiply(L1, c1, L2, c2, m, sig)
c_power(L, c, m, p, sig)
c_equal(L1, c1, L2, c2, m)
c_overwrite(L1, c1, L2, c2, m)
c_cartan(L, c, m, n)
c_cartan_inverse(L, c, m, n)
c_identity(L, p, m)
c_grade(L, c, m, n)
c_add(L1, c1, L2, c2, m)
c_multiply(L1, c1, L2, c2, m, sig)
c_power(L, c, m, p, sig)
c_equal(L1, c1, L2, c2, m)
c_overwrite(L1, c1, L2, c2, m)
c_cartan(L, c, m, n)
c_cartan_inverse(L, c, m, n)

Arguments

`L`, `L1`, `L2`	Lists of terms
`c1`, `c2`, `c`	Numeric vectors of coefficients
`m`	Maximum entry of terms
`n`	Grade to extract
`p`	Integer power
`sig`	Two positive integers, $p$ and $q$ , representing the number of $+1$ and $-1$ terms on the main diagonal of quadratic form

Details

The functions documented here are low-level helper functions that wrap the C code. They are called by functions like clifford_plus_clifford(), which are themselves called by the binary operators documented at Ops.clifford.Rd. The functions documented here are not really intended for day-to-day use.

Function c_identity() checks that the list of terms L is the same length as the vector coefficients p; if not, an error is given. Note that R function clifford() will recycle the coefficient vector if of length 1, so that clifford(list(1,1:2),7) works as expected (but c_identity(list(1,1:2),7,2) will throw an error).

Function clifford_inverse() is problematic as nonnull blades always have an inverse; but function is.blade() is not yet implemented. Blades (including null blades) have a pseudoinverse, but this is not implemented yet either.

Value

The high-level functions documented here return an object of class clifford. But don't use the low-level functions.

Author(s)

Robin K. S. Hankin

Magnitude of a clifford object

Description

Following Perwass, the magnitude of a multivector is defined as

$\left|\left|A\right|\right| = \sqrt{A\ast A}$

Where $A\ast A$ denotes the Euclidean scalar product eucprod(). Recall that the Euclidean scalar product is never negative (the function body is sqrt(abs(eucprod(z))); the abs() is needed to avoid numerical roundoff errors in eucprod() giving a negative value).

Usage

## S3 method for class 'clifford'
Mod(z)
## S3 method for class 'clifford'
Mod(z)

Arguments

`z`	Clifford objects

Note

There is a nice example of scalar product at rcliff.Rd.

Author(s)

Robin K. S. Hankin

Examples



Mod(rcliff())


# Perwass, p68, asserts that if A is a k-blade, then (in his notation)
# AA == A*A.

# In package idiom, A*A == A %star% A:

A <- rcliff()          
Mod(A*A - A %star% A)  # meh

A <- rblade()
Mod(A*A - A %star% A)  # should be small



Mod(rcliff())


# Perwass, p68, asserts that if A is a k-blade, then (in his notation)
# AA == A*A.

# In package idiom, A*A == A %star% A:

A <- rcliff()          
Mod(A*A - A %star% A)  # meh

A <- rblade()
Mod(A*A - A %star% A)  # should be small

Take the negative of a vector

Description

Very simple function that takes the negative of a vector, here so that idiom such as

coeffs(z)[gradesminus(z)%%2 != 0] %<>% minus

works as intended (this taken from Conj.clifford()).

Usage

minus(x)minus(x)

Arguments

`x`	Any vector or disord object

Value

Returns a vector or disord

Author(s)

Robin K. S. Hankin

Coercion from numeric to Clifford form

Description

Given a numeric value or vector, return a Clifford algebra element

Usage

numeric_to_clifford(x)
as.1vector(x)
is.1vector(x)
scalar(x=1)
as.scalar(x=1)
is.scalar(C)
basis(n,x=1)
e(n,x=1)
numeric_to_clifford(x)
as.1vector(x)
is.1vector(x)
scalar(x=1)
as.scalar(x=1)
is.scalar(C)
basis(n,x=1)
e(n,x=1)

Arguments

`x`	Numeric vector
`n`	Integer specifying dimensionality of underlying vector space
`C`	Object possibly of class Clifford

Details

Function as.scalar() takes a length-one numeric vector and returns a Clifford scalar of that value (to extract the scalar component of a multivector, use const()).

Function is.scalar() is a synonym for is.real() which is documented at const.Rd.

Function as.1vector() takes a numeric vector and returns the linear sum of length-one blades with coefficients given by x; function is.1vector() returns TRUE if every term is of grade 1.

Function numeric_to_vector() dispatches to either as.scalar() for length-one vectors or as.1vector() if the length is greater than one.

Function basis() returns a wedge product of basis vectors; function e() is a synonym. There is special dispensation for zero, so e(0) returns the Clifford scalar 1.

Function antivector() should arguably be described here but is actually documented at antivector.Rd.

Author(s)

Robin K. S. Hankin

Examples


as.scalar(6)
as.1vector(1:8)

e(5:8)

Reduce(`+`,sapply(seq_len(7),function(n){e(seq_len(n))},simplify=FALSE))


as.scalar(6)
as.1vector(1:8)

e(5:8)

Reduce(`+`,sapply(seq_len(7),function(n){e(seq_len(n))},simplify=FALSE))

Arithmetic Ops Group Methods for `clifford` objects

Description

Different arithmetic operators for clifford objects, including many different types of multiplication.

Usage

## S3 method for class 'clifford'
Ops(e1, e2)
clifford_negative(C)
geoprod(C1,C2)
clifford_times_scalar(C,x)
clifford_plus_clifford(C1,C2)
clifford_eq_clifford(C1,C2)
clifford_inverse(C)
cliffdotprod(C1,C2)
fatdot(C1,C2)
lefttick(C1,C2)
righttick(C1,C2)
wedge(C1,C2)
scalprod(C1,C2=rev(C1),drop=TRUE)
eucprod(C1,C2=C1,drop=TRUE)
maxyterm(C1,C2=as.clifford(0))
C1 %.% C2
C1 %dot% C2
C1 %^% C2
C1 %X% C2
C1 %star% C2
C1 % % C2
C1 %euc% C2
C1 %o% C2
C1 %_|% C2
C1 %|_% C2
## S3 method for class 'clifford'
Ops(e1, e2)
clifford_negative(C)
geoprod(C1,C2)
clifford_times_scalar(C,x)
clifford_plus_clifford(C1,C2)
clifford_eq_clifford(C1,C2)
clifford_inverse(C)
cliffdotprod(C1,C2)
fatdot(C1,C2)
lefttick(C1,C2)
righttick(C1,C2)
wedge(C1,C2)
scalprod(C1,C2=rev(C1),drop=TRUE)
eucprod(C1,C2=C1,drop=TRUE)
maxyterm(C1,C2=as.clifford(0))
C1 %.% C2
C1 %dot% C2
C1 %^% C2
C1 %X% C2
C1 %star% C2
C1 % % C2
C1 %euc% C2
C1 %o% C2
C1 %_|% C2
C1 %|_% C2

Arguments

`e1`, `e2`, `C`, `C1`, `C2`	Objects of class `clifford` or coerced if needed
`x`	Scalar, length one numeric vector
`drop`	Boolean, with default `TRUE` meaning to return the constant coerced to numeric, and `FALSE` meaning to return a (constant) Clifford object

Details

The function Ops.clifford() passes unary and binary arithmetic operators “+”, “-”, “*”, “/” and “^” to the appropriate specialist function. Function maxyterm() returns the maximum index in the terms of its arguments.

The package has several binary operators:


Geometric product	`A*B = geoprod(A,B)`	$\displaystyle AB=\sum_{r,s}\left\langle A\right\rangle_r\left\langle B\right\rangle_s$
Inner product	`A %.% B = cliffdotprod(A,B)`	$\displaystyle A\cdot B=\sum_{r\neq 0\atop s\ne 0}^{\vphantom{s\neq 0}}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{\left\|s-r\right\|}$
Outer product	`A %^% B = wedge(A,B)`	$\displaystyle A\wedge B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{s+r}$
Fat dot product	`A %o% B = fatdot(A,B)`	$\displaystyle A\bullet B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{\left\|s-r\right\|}$
Left contraction	`A %_\|% B = lefttick(A,B)`	$\displaystyle A\rfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{s-r}$
Right contraction	`A %\|_% B = righttick(A,B)`	$\displaystyle A\lfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{r-s}$
Cross product	`A %X% B = cross(A,B)`	$\displaystyle A\times B=\frac{1}{2_{\vphantom{j}}}\left(AB-BA\right)$
Scalar product	`A %star% B = star(A,B)`	$\displaystyle A\ast B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_0$
Euclidean product	`A %euc% B = eucprod(A,B)`	$\displaystyle A\star B= A\ast B^\dagger$

In R idiom, the geometric product geoprod(.,.) has to be indicated with a “*” (as in A*B) and so the binary operator must be %*%: we need a different idiom for scalar product, which is why %star% is used.

Because geometric product is often denoted by juxtaposition, package idiom includes a % % b for geometric product.

Binary operator %dot% is a synonym for %.%, which causes problems for rmarkdown.

Function clifford_inverse() returns an inverse for nonnull Clifford objects $\operatorname{Cl}(p,q)$ for $p+q\leq 5$ , and a few other special cases. The functionality is problematic as nonnull blades always have an inverse; but function is.blade() is not yet implemented. Blades (including null blades) have a pseudoinverse, but this is not implemented yet either.

The scalar product of two clifford objects is defined as the zero-grade component of their geometric product:

$A\ast B=\left\langle AB\right\rangle_0\qquad{\mbox{NB: notation used by both Perwass and Hestenes}}$

In package idiom the scalar product is given by A %star% B or scalprod(A,B). Hestenes and Perwass both use an asterisk for scalar product as in “ $A*B$ ”, but in package idiom, the asterisk is reserved for geometric product.

Note: in the package, A*B is the geometric product.

The Euclidean product (or Euclidean scalar product) of two clifford objects is defined as

$A\star B= A\ast B^\dagger= \left\langle AB^\dagger\right\rangle_0\qquad{\mbox{Perwass}}$

where $B^\dagger$ denotes Conjugate [as in Conj(a)]. In package idiom the Euclidean scalar product is given by eucprod(A,B) or A %euc% B, both of which return A * Conj(B).

Note that the scalar product $A\ast A$ can be positive or negative [that is, A %star% A may be any sign], but the Euclidean product is guaranteed to be non-negative [that is, A %euc% A is always positive or zero].

Dorst defines the left and right contraction (Chisholm calls these the left and right inner product) as $A\rfloor B$ and $A\lfloor B$ . See the vignette for more details.

Division, as in idiom x/y, is defined as x*clifford_inverse(y). Function clifford_inverse() uses the method set out by Hitzer and Sangwine but is limited to $p+q\leq 5$ .

The Lie bracket, $\left[x,y\right]$ is implemented in the package using idiom such as .[x,y], and this is documented at dot.Rd.

Many of the functions documented here use low-level helper functions that wrap C code. For example, fatdot() uses c_fatdotprod(). These are documented at lowlevel.Rd.

Value

The high-level functions documented here return a clifford object. The low-level functions are not really intended for the end-user.

Note

All the different Clifford products have binary operators for convenience including the wedge product %^%. However, as an experimental facility, the caret “^” returns either multiplicative powers [as in A^3=A*A*A], or a wedge product [as in A^B = A %^% B = wedge(A,B)] depending on the class of the second argument. I don't see that “A ^ B” is at all ambiguous but OTOH I might withdraw it if it proves unsatisfactory for some reason.

Compare the stokes package, where multiplicative powers do not really make sense and A^B is interpreted as a wedge product of differential forms $A$ and $B$ . In stokes, the wedge product is the sine qua non for the whole package and needs a terse idiomatic representation (although there A%^%B returns the wedge product too).

Using %.% causes severe and weird difficult-to-debug problems in markdown documents.

Author(s)

Robin K. S. Hankin

References

E. Hitzer and S. Sangwine 2017. “Multivector and multivector matrix inverses in real Clifford algebras”. Applied Mathematics and Computation 311:375-389

Examples


u <- rcliff(5)
v <- rcliff(5)
w <- rcliff(5)

u
v
u*v

u+(v+w) == (u+v)+w            # should be TRUE by associativity of "+"
u*(v*w) == (u*v)*w            # should be TRUE by associativity of "*"
u*(v+w) == u*v + u*w          # should be TRUE by distributivity

# Now if x,y are _vectors_ we have:

x <- as.1vector(sample(5))
y <- as.1vector(sample(5))
x*y == x%.%y + x%^%y
x %^% y == x %^% (y + 3*x)  
x %^% y == (x*y-x*y)/2        # should be TRUE 

#  above are TRUE for x,y vectors (but not for multivectors, in general)


## Inner product "%.%" is not associative:
x <- rcliff(5,g=2)
y <- rcliff(5,g=2)
z <- rcliff(5,g=2)
x %.% (y %.% z) == (x %.% y) %.% z

## Other products should work as expected:

x %|_% y   ## left contraction
x %_|% y   ## right contraction
x %o% y    ## fat dot product
x ^ y        ## Experimental wedge product idiom, plain caret

u <- rcliff(5)
v <- rcliff(5)
w <- rcliff(5)

u
v
u*v

u+(v+w) == (u+v)+w            # should be TRUE by associativity of "+"
u*(v*w) == (u*v)*w            # should be TRUE by associativity of "*"
u*(v+w) == u*v + u*w          # should be TRUE by distributivity

# Now if x,y are _vectors_ we have:

x <- as.1vector(sample(5))
y <- as.1vector(sample(5))
x*y == x%.%y + x%^%y
x %^% y == x %^% (y + 3*x)  
x %^% y == (x*y-x*y)/2        # should be TRUE 

#  above are TRUE for x,y vectors (but not for multivectors, in general)


## Inner product "%.%" is not associative:
x <- rcliff(5,g=2)
y <- rcliff(5,g=2)
z <- rcliff(5,g=2)
x %.% (y %.% z) == (x %.% y) %.% z

## Other products should work as expected:

x %|_% y   ## left contraction
x %_|% y   ## right contraction
x %o% y    ## fat dot product
x ^ y        ## Experimental wedge product idiom, plain caret

Print clifford objects

Description

Print methods for Clifford algebra

Usage

## S3 method for class 'clifford'
print(x,...)
## S3 method for class 'clifford'
as.character(x,...)
catterm(a)
## S3 method for class 'clifford'
print(x,...)
## S3 method for class 'clifford'
as.character(x,...)
catterm(a)

Arguments

`x`	Object of class `clifford` in the print method
`...`	Further arguments, currently ignored
`a`	Integer vector representing a term

Note

The print method does not change the internal representation of a clifford object, which is a two-element list, the first of which is a list of integer vectors representing terms, and the second is a numeric vector of coefficients. The print method has special dispensation for the zero clifford object.

The print method is sensitive to the value of options separate and basissep. If option separate is FALSE (the default), the method prints the basis blades in a compact form, as in “e_134”. The indices of the basis vectors are separated with the value of option basissep which is usually NULL; but if $n>9$ , then setting option basissep to a comma (“,”) might look good as it will print e_10,11,12 instead of e_101112:

options("basissep" = ",")

If option separate is TRUE, the method prints the basis vectors separately, as in e10 e11 e12:

options("separate" = TRUE)

Function catterm() is a low-level helper function, used in the print method, coercion to character, and also in function getcoeffs() to set the names of its output. It takes an integer vector like c(1,5,6) and returns a representation of the corresponding basis blade, in this case “e_156”. Function catterm() is where options basissep and separate are processed. Special dispensation is needed for length-zero vectors, for which the empty string is returned. This is needed to ensure that the constant term (which has a basis blade of numeric(0)) is treated appropriately. See also list_modifier() which deals with this issue.

Experimental bespoke print method print_clifford_quaternion() and print_clifford_pauli() is included. This are executed if option clifford_print_special is quaternion; if NULL, then print_clifford_default() is used. It is straightforward to add further bespoke print methods if needed (modify print.clifford()).

Author(s)

Robin K. S. Hankin

Examples


a <- rclifff(9)
a   # default print method incomprehensible

options("separate" = TRUE)
a    # marginally better


options("separate" = FALSE)
options(basissep=",")
a    #  clearer; YMMV

options(basissep = NULL, maxdim=NULL)  # restore default

options("maxdim" = 3)
signature(3)
a <- clifford(list(0,c(1,2),c(1,3),c(2,3)),6:9)
a

options("clifford_print_special" = "quaternion")
a

options("maxdim" = NULL)
options("clifford_print_special" = NULL)
signature(Inf)
a <- rclifff(9)
a   # default print method incomprehensible

options("separate" = TRUE)
a    # marginally better


options("separate" = FALSE)
options(basissep=",")
a    #  clearer; YMMV

options(basissep = NULL, maxdim=NULL)  # restore default

options("maxdim" = 3)
signature(3)
a <- clifford(list(0,c(1,2),c(1,3),c(2,3)),6:9)
a

options("clifford_print_special" = "quaternion")
a

options("maxdim" = NULL)
options("clifford_print_special" = NULL)
signature(Inf)

Coercion from numeric to Clifford form

Description

Given a numeric value or vector, return a Clifford algebra element

Usage

pseudoscalar()
is.pseudoscalar(C)
pseudoscalar()
is.pseudoscalar(C)

Arguments

`C`	Object possibly of class Clifford

Details

Function pseudoscalar() returns the unit pseudoscalar of dimensionality option("maxdim") and function is.pseudoscalar() checks for a Clifford object being a pseudoscalar. Note that these functions require maxdim to be set; otherwise they are meaningless.

Usually, one will set option(maxdim) at the start of a session, together with the signature. Then one might define I <- pseudoscalar() in the interests of compactness and legibility.

Author(s)

Robin K. S. Hankin

Examples


options(maxdim=6)
I <- pseudoscalar()
is.pseudoscalar(I)
options(maxdim=NULL) # restore default
options(maxdim=6)
I <- pseudoscalar()
is.pseudoscalar(I)
options(maxdim=NULL) # restore default

Quaternions using Clifford algebras

Description

Converting quaternions to and from Clifford objects is not part of the package but functionality and a short discussion is included in inst/quaternion_clifford.Rmd.

Details

Given a quaternion $a+bi+cj+dk$ , one may identify $i$ with $-e_{12}$ , $j$ with $-e_{13}$ , and $k$ with $-e_{23}$ (the constant term is of course $e_0$ ).

Note

A different mapping, from the quaternions to $\operatorname{Cl}(0,2)$ is given at signature.Rd.

Author(s)

Robin K. S. Hankin

Random clifford objects

Description

Random Clifford algebra elements, intended as quick “get you going” examples of clifford objects

Usage

rcliff(n=9, d=6, g=4, include.fewer=TRUE)
rclifff(n=100,d=20,g=10,include.fewer=TRUE)
rblade(d=7, g=3)
rcliff(n=9, d=6, g=4, include.fewer=TRUE)
rclifff(n=100,d=20,g=10,include.fewer=TRUE)
rblade(d=7, g=3)

Arguments

`n`	Number of terms
`d`	Dimensionality of underlying vector space
`g`	Maximum grade of any term
`include.fewer`	Boolean, with `FALSE` meaning to return a clifford object comprising only terms of grade `g`, and default `TRUE` meaning to include terms with grades less than `g` (including a term of grade zero, that is, a scalar)

Details

Function rcliff() gives a quick nontrivial Clifford object, typically with terms having a range of grades (see ‘grade.Rd’); argument include.fewer=FALSE ensures that all terms are of the same grade. Function rclifff() is the same but returns a more complicated object by default.

Function rblade() gives a Clifford object that is a blade (see ‘term.Rd’). It returns the wedge product of a number of 1-vectors, for example $\left(e_1+2e_2\right)\wedge\left(e_1+3e_5\right)$ .

Perwass gives the following lemma:

Given blades $A_{\langle r\rangle}, B_{\langle s\rangle}, C_{\langle t\rangle}$ , then

$\langle A_{\langle r\rangle} B_{\langle s\rangle} C_{\langle t\rangle} \rangle_0 = \langle C_{\langle t\rangle} A_{\langle r\rangle} B_{\langle s\rangle} \rangle_0$

In the proof he notes in an intermediate step that

$\langle A_{\langle r\rangle} B_{\langle s\rangle} \rangle_t * C_{\langle t\rangle} = C_{\langle t\rangle} * \langle A_{\langle r\rangle} B_{\langle s\rangle} \rangle_t = \langle C_{\langle t\rangle} A_{\langle r\rangle} B_{\langle s\rangle} \rangle_0.$

Package idiom is shown in the examples.

Note

If the grade exceeds the dimensionality, $g>d$ , then the result is arguably zero; rcliff() returns an error.

Author(s)

Robin K. S. Hankin

Examples


rcliff()
rcliff(d=3,g=2)
rcliff(3,10,7)
rcliff(3,10,7,include=TRUE)

x1 <- rcliff()
x2 <- rcliff()
x3 <- rcliff()

x1*(x2*x3) == (x1*x2)*x3  # should be TRUE


rblade()

# We can invert blades easily:
a <- rblade()
ainv <- rev(a)/scalprod(a)

zap(a*ainv)  # 1 (to numerical precision)
zap(ainv*a)  # 1 (to numerical precision)

# Perwass 2009, lemma 3.9:


A <- rblade(d=9,g=4)  
B <- rblade(d=9,g=5)  
C <- rblade(d=9,g=6)  

grade(A*B*C,0)-grade(C*A*B,0)   # zero to numerical precision



# Intermediate step

x1 <- grade(A*B,3) %star% C
x2 <- C %star% grade(A*B,3)
x3 <- grade(C*A*B,0)

max(x1,x2,x3) - min(x1,x2,x3)   # zero to numerical precision

rcliff()
rcliff(d=3,g=2)
rcliff(3,10,7)
rcliff(3,10,7,include=TRUE)

x1 <- rcliff()
x2 <- rcliff()
x3 <- rcliff()

x1*(x2*x3) == (x1*x2)*x3  # should be TRUE


rblade()

# We can invert blades easily:
a <- rblade()
ainv <- rev(a)/scalprod(a)

zap(a*ainv)  # 1 (to numerical precision)
zap(ainv*a)  # 1 (to numerical precision)

# Perwass 2009, lemma 3.9:


A <- rblade(d=9,g=4)  
B <- rblade(d=9,g=5)  
C <- rblade(d=9,g=6)  

grade(A*B*C,0)-grade(C*A*B,0)   # zero to numerical precision



# Intermediate step

x1 <- grade(A*B,3) %star% C
x2 <- C %star% grade(A*B,3)
x3 <- grade(C*A*B,0)

max(x1,x2,x3) - min(x1,x2,x3)   # zero to numerical precision

The signature of the Clifford algebra

Description

Getting and setting the signature of the Clifford algebra

Usage

signature(p,q=0)
is_ok_sig(s)
showsig(s)
## S3 method for class 'sigobj'
print(x,...)
signature(p,q=0)
is_ok_sig(s)
showsig(s)
## S3 method for class 'sigobj'
print(x,...)

Arguments

`s`, `p`, `q`	Integers, specifying number of positive elements on the diagonal of the quadratic form, with `s=c(p,q)`
`x`	Object of class `sigobj`
`...`	Further arguments, currently ignored

Details

The signature functionality is modelled on the lorentz package; clifford::signature() operates in the same way as lorentz::sol() which gets and sets the speed of light. The idea is that both the speed of light and the signature of a Clifford algebra are generally set once, at the beginning of an R session, and subsequently change only very infrequently.

Clifford algebras require a bilinear form $\left\langle\cdot,\cdot\right\rangle$ on $\mathbb{R}^n$ . If ${\mathbf x}=\left(x_1,\ldots,x_n\right)$ we define

$\left\langle{\mathbf x},{\mathbf x}\right\rangle=x_1^2+x_2^2+\cdots +x_p^2-x_{p+1}^2-\cdots -x_{p+q}^2$

where $p+q=n$ . With this quadratic form the vector space is denoted $\mathbb{R}^{p,q}$ and we say that $(p,q)$ is the signature of the bilinear form $\left\langle\cdot,\cdot\right\rangle$ . This gives rise to the Clifford algebra $C_{p,q}$ .

If the signature is $(p,q)$ , then we have

$e_i e_i = +1\, (\mbox{if } 1\leq i\leq p), -1\, (\mbox{if } p+1\leq i\leq p+q), 0\, (\mbox{if } i>p+q).$

Note that $(p,0)$ corresponds to a positive-semidefinite quadratic form in which $e_ie_i=+1$ for all $i\leq p$ and $e_ie_i=0$ for all $i > p$ . Similarly, $(0,q)$ corresponds to a negative-semidefinite quadratic form in which $e_ie_i=-1$ for all $i\leq q$ and $e_ie_i=0$ for all $i > q$ .

A strictly positive-definite quadratic form is specified by infinite $p$ [in which case $q$ is irrelevant], and signature(Inf) implements this. For a strictly negative-definite quadratic form we would have $p=0,q=\infty$ which would be signature(0,Inf).

If we specify $e_ie_i=0$ for all $i$ , then the operation reduces to the wedge product of a Grassmann algebra. Package idiom for this is to set $p=q=0$ with signature(0,0), but this is not recommended: use the stokes package for Grassmann algebras, which is much more efficient and uses nicer idiom.

Function signature(p,q) returns the signature invisibly; but setting option show_signature to TRUE makes showsig() [which is called by signature()] change the default prompt so it displays the signature, much like showSOL in the lorentz package. Note that changing the signature changes the prompt immediately (if show_signature is TRUE), but changing option show_signature has no effect until showsig() is called.

Calling signature() [that is, with no arguments] returns an object of class sigobj with elements corresponding to $p$ and $q$ . There is special dispensation for “infinite” $p$ or $q$ : the sigobj class ensures that a near-infinite integer such as .Machine$integer.max will be printed as “Inf” rather than, for example, “2147483647”.

Function is_ok_sig() is a helper function that checks for a proper signature. If we set signature(p,q), then technically $n>p+q$ implies $e_n^2=0$ , but usually we are not interested in $e_n$ when $n>p+q$ and want this to be an error. Option maxdim specifies the maximum value of $n$ , with default NULL corresponding to infinity. If $n$ exceeds maxdim, then is_ok_sig() throws an error. Note that it is sometimes fine to have maxdim > p+q [and indeed this is useful in the context of dual numbers]. This option is intended to be a super-strict safety measure.

> e(6)
Element of a Clifford algebra, equal to
+ 1e_6
> options(maxdim=5)
> e(5)
Element of a Clifford algebra, equal to
+ 1e_5
> e(6)
Error in is_ok_clifford(terms, coeffs) : option maxdim exceeded

Author(s)

Robin K. S. Hankin

Examples



signature()

e(1)^2
e(2)^2

signature(1)
e(1)^2
e(2)^2   # note sign

signature(3,4)
sapply(1:10,function(i){drop(e(i)^2)})


signature(Inf)   # restore default




# Nice mapping from Cl(0,2) to the quaternions (loading clifford and
# onion simultaneously is discouraged):

# library("onion")
# signature(0,2)
# Q1 <- rquat(1)
# Q2 <- rquat(1)
# f <- function(H){Re(H)+i(H)*e(1)+j(H)*e(2)+k(H)*e(1:2)}
# f(Q1)*f(Q2) - f(Q1*Q2) # zero to numerical precision
# signature(Inf)

signature()

e(1)^2
e(2)^2

signature(1)
e(1)^2
e(2)^2   # note sign

signature(3,4)
sapply(1:10,function(i){drop(e(i)^2)})


signature(Inf)   # restore default




# Nice mapping from Cl(0,2) to the quaternions (loading clifford and
# onion simultaneously is discouraged):

# library("onion")
# signature(0,2)
# Q1 <- rquat(1)
# Q2 <- rquat(1)
# f <- function(H){Re(H)+i(H)*e(1)+j(H)*e(2)+k(H)*e(1:2)}
# f(Q1)*f(Q2) - f(Q1*Q2) # zero to numerical precision
# signature(Inf)

Summary methods for clifford objects

Description

Summary method for clifford objects, and a print method for summaries.

Usage

## S3 method for class 'clifford'
summary(object, ...)
## S3 method for class 'summary.clifford'
print(x, ...)
first_n_last(x)
## S3 method for class 'clifford'
summary(object, ...)
## S3 method for class 'summary.clifford'
print(x, ...)
first_n_last(x)

Arguments

`object`, `x`	Object of class clifford
`...`	Further arguments, currently ignored

Details

Summary of a clifford object. Note carefully that the “typical terms” are implementation specific. Function first_n_last() is a helper function.

Author(s)

Robin K. S. Hankin

Examples

summary(rcliff())
summary(rcliff())

Deal with terms

Description

By basis vector, I mean one of the basis vectors of the underlying vector space $R^n$ , that is, an element of the set $\left\lbrace e_1,\ldots,e_n\right\rbrace$ . A term is a wedge product of basis vectors (or a geometric product of linearly independent basis vectors), something like $e_{12}$ or $e_{12569}$ . Sometimes I use the word “term” to mean a wedge product of basis vectors together with its associated coefficient: so $7e_{12}$ would be described as a term.

From Perwass: a blade is the outer product of a number of 1-vectors (or, equivalently, the wedge product of linearly independent 1-vectors). Thus $e_{12}=e_1\wedge e_2$ and $e_{12} + e_{13}=e_1\wedge(e_2+e_3)$ are blades, but $e_{12} + e_{34}$ is not.

Function rblade(), documented at ‘rcliff.Rd’, returns a random blade.

Function is.blade() is not currently implemented: there is no easy way to detect whether a Clifford object is a product of 1-vectors.

Usage

terms(x)
is.blade(x)
is.basisblade(x)
terms(x)
is.blade(x)
is.basisblade(x)

Arguments

`x`	Object of class `clifford`

Details

Functions terms() and coeffs() are the extraction methods. These are unordered vectors but the ordering is consistent between them (an extended discussion of this phenomenon is presented in the mvp package).
Function term() returns a clifford object that comprises a single term with unit coefficient.
Function is.basisterm() returns TRUE if its argument has only a single term, or is a nonzero scalar; the zero clifford object is not considered to be a basis term.

Author(s)

Robin K. S. Hankin

References

C. Perwass. “Geometric algebra with applications in engineering”. Springer, 2009.

Examples


x <- rcliff()
terms(x)

is.basisblade(x)


a <- as.1vector(1:3)
b <- as.1vector(c(0,0,0,12,13))

a %^% b # a blade

x <- rcliff()
terms(x)

is.basisblade(x)


a <- as.1vector(1:3)
b <- as.1vector(c(0,0,0,12,13))

a %^% b # a blade

Zap small values in a clifford object

Description

Generic version of zapsmall()

Usage

zap(x, drop=TRUE, digits = getOption("digits"))
zap(x, drop=TRUE, digits = getOption("digits"))

Arguments

`x`	Clifford object
`drop`	Boolean with default `TRUE` meaning to coerce the output to numeric with `drop()`
`digits`	number of digits to retain

Details

Given a clifford object, coefficients close to zero are ‘zapped’, i.e., replaced by ‘0’ in much the same way as base::zapsmall().

The function should be called zapsmall(), and dispatch to the appropriate base function, but I could not figure out how to do this with S3 (the docs were singularly unhelpful) and gave up.

Note, this function actually changes the numeric value, it is not just a print method.

Author(s)

Robin K. S. Hankin

Examples


a <- clifford(sapply(1:10,seq_len),90^-(1:10))
zap(a)
options(digits=3)
zap(a)


a-zap(a)   # nonzero

B <- rblade(g=3)
mB <- B*rev(B)
zap(mB)
drop(mB)
a <- clifford(sapply(1:10,seq_len),90^-(1:10))
zap(a)
options(digits=3)
zap(a)


a-zap(a)   # nonzero

B <- rblade(g=3)
mB <- B*rev(B)
zap(mB)
drop(mB)

The zero Clifford object

Description

Dealing with the zero Clifford object presents particular challenges. Some of the methods need special dispensation for the zero object.

Usage

is.zero(x)
is.zero(x)

Arguments

`x`	Clifford object

Details

To test for a Clifford object's being zero, use is.zero(). Idiom such as x==0 will work irregardless, but sometimes one might prefer the functional form for stylistic reasons.

To create the zero object ab initio, use

clifford(list(),numeric(0))

although note that scalar(0) will work too.

Note

The coefficient of the zero clifford object, as in coeff(scalar(0)), is numeric(0) (but note that 1 + NULL also returns numeric(0)).

Function is.zero() is problematic if another package which also has an is.zero() generic is loaded, for this will mask clifford::is.zero(). Specifically, the jordan package includes jordan::is.zero() and the two do not play nicely together.

Author(s)

Robin K. S. Hankin

Examples


is.zero(rcliff())
is.zero(rcliff())

Package 'clifford'

Help Index

Arbitrary Dimensional Clifford Algebras

Description

Details

Author(s)

References

See Also

Examples

Clifford object containing all possible terms

Description

Usage

Arguments

Author(s)

Examples

Antivectors or pseudovectors

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Coerce a clifford vector to a numeric vector

Description

Usage

Arguments

Note

Author(s)

See Also

Examples

Cartan map between clifford algebras

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Create, coerce, and test for clifford objects

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

The constant term of a Clifford object

Description

Usage

Arguments

Details

Author(s)

See Also

Examples

Class “dot”

Description

Arguments

Value

Author(s)

Examples

Drop redundant information

Description

Usage

Arguments

Details

Note

Author(s)

See Also

Examples

Even and odd clifford objects

Description

Usage

Arguments

Author(s)

See Also

Create, coerce, and test for `clifford` objects

Low-level helper functions for `clifford` objects