The grade of a term is the number of basis vectors in it.
grade(C, n, drop=TRUE)
grade(C,n) <- value
grades(x)
gradesplus(x)
gradesminus(x)
gradeszero(x)
Clifford object
Integer vector specifying grades to extract
Replacement value, a numeric vector
Boolean, with default TRUE meaning to coerce a
constant Clifford object to numeric, and FALSE meaning not
to
Robin K. S. Hankin
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_2344e_123 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_1e1 is 1, and the grade of e_125=e_1e_2e_5e_125=e1 e2 e5 is 3. The grade operator _r<.>_r is used to extract terms of a particular grade, with
A= A_0 + A_1 + A_2 + = _r A_r
A = <A>_0 + <A>_1 + <A>_2 +... = sum <A>_r
for any Clifford object \(A\). Thus A_r<A>_r is said to be homogenous of grade \(r\). Hestenes sometimes writes subscripts that specify grades using an overbar as in A_romitted. It is conventional to denote the zero-grade object A_0<A>_0 as simply A<A>.
We have
A+B_r= A_r + B_r A_r= A_r A_r_s= A_r_rs.
omitted; see PDF
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
x, x=x_1^2+x_2^2+ +x_p^2-x_p+1^2- -x_p+q^2
<x,x>=x_1^2+...+x_p^2-x_p+1^2-...-x_p+q^2
and a basis blade e_Ae_Ae_A with A 1,...,p+qA 1,...,p+qomitted, then
gr(e_A) = | a A 1 a p+q|
gr(e_A) = | aA 1 a p+q|
omitted
gr_+(e_A) = | a A 1 a p|
gr_+(e_A) = | aA 1 a p|
omitted
gr_-(e_A) = | a A p < a p+q|
gr_-(e_A) = | aA p < a p+q|
omitted
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
Grassman 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.
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.
C. Perwass 2009. “Geometric algebra with applications in engineering”. Springer.
signature, const
a <- clifford(sapply(seq_len(7),seq_len),seq_len(7))
a
grades(a)
grade(a,5)
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)}))
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