Ops.clifford: Arithmetic Ops Group Methods for 'clifford' objects

Description Usage Arguments Details Value Author(s) See Also Examples

Description

Allows arithmetic operators to be used for multivariate polynomials such as addition, multiplication, integer powers, etc.

Usage

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## 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 %^% 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

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.

Functions c_foo() are low-level helper functions that wrap the C code; 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) see PDF
Inner product A %.% B = cliffdotprod(A,B) see PDF
Outer productA %^% B = wedge(A,B) see PDF
Fat dot productA %o% B = fatdot(A,B) see PDF
Left contractionA %_|% B = lefttick(A,B) see PDF
Right contractionA %|_% B = righttick(A,B) see PDF
Cross productA %X% B = cross(A,B) see PDF
Scalar productA %star% B = star(A,B) see PDF
Euclidean productA %euc% B = eucprod(A,B) see PDF

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.

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.

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

omitted; see PDF

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

omitted: see PDF

where ? 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 see PDF and A\lfloor B. See the vignette for more details.

Value

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

Author(s)

Robin K. S. Hankin

See Also

scalprod

Examples

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

u*v

u^3


u+(v+w) == (u+v)+w              # should be TRUE 
u*(v*w) == (u*v)*w              # should be TRUE 
u %^% v == (u*v-v*u)/2        # should be TRUE 

# Now if x,y,z are _vectors_ we would have:

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

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

## Geometric product *is* associative:
x * (y * z)
(x * y) * z

clifford documentation built on March 13, 2020, 3:19 a.m.