grade: The grade of a clifford object

In clifford: Arbitrary Dimensional Clifford Algebras

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)

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.

See Also

signature, const

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)}))

clifford documentation built on June 8, 2025, 10:56 a.m.