CliffordNumbers

A simple, statically sized multivector (Clifford number) implementation for Julia using graded representations. This allows for common multivector operations, particularly the various products of geometric algebra, to be easily implemented with extremely high performance (faster than matrix multiplications of matrix representations) without depending on any linear algebra library. Additionally, the multivectors provided by this package can be stored inline in arrays or other data structures.

Clifford numbers

Types

This package exports AbstractCliffordNumber{Q,T} and its subtypes, which describe the behavior of multivectors with algebra Q and scalar type T<:Union{Real,Complex}. This is a subtype of Number, and therefore acts as a scalar for the purpose of broadcasting. For this reason, we provide an nblades function separate from Base.length to count the number of blades represented by a type.

To index an AbstractCliffordNumber, we provide the BitIndex{Q} type, which allows for arbitrary components to be indexed, and the BitIndices{Q,C<:AbstractCliffordNumber{Q}} type, which provides all valid indices of instances of C that are not constrained to be zero. Indexing with ordinary integers is disallowed, but Tuple(::AbstractCliffordNumber) obtains the backing Tuple.

AbstractCliffordNumber{Q,T} includes the following concrete subtypes:

• CliffordNumber{Q,T,L}, which represents the coefficients associated with all basis blades.
• EvenCliffordNumber{Q,T,L} and OddCliffordNumber{Q,T,L}, which represents multivectors with only basis blades of even or odd grade being nonzero. These are especially important when dealing with physically realizable Euclidean transformations (rotations and translations).
• KVector{K,Q,T,L}, which represents multivectors with only basis blades of grade K being nonzero. This is especially useful for representing common vectors, bivectors, pseudovectors, etc.

Promotion and conversion

The type promotion system is heavily leveraged to minimize the memory footprint needed to represent the results of various operations. Promotion can convert the numeric types associated with two AbstractCliffordNumber{Q} instances (for instance, the sum of KVector{1,APS,Int} and KVector{1,APS,Float64} is KVector{1,APS,Float64}), but it can also leverage grade information to promote to smaller types: the sum of KVector{1,APS,Int} and KVector{3,APS,Int} is an OddCliffordNumber{APS,Int}, but the sum of KVector{1,APS,Int} and KVector{2,APS,Int} is a CliffordNumber{APS,Int}.

We provide the scalar_promote and scalar_convert functions to allow for promotion of the scalar types backing an AbstractCliffordNumber without needlessly expanding the represented grades.

Indexing

Although AbstractCliffordNumber instances are scalars, the BitIndex{Q} type can be used to retrieve coefficients associated with specific basis blades. The full set of BitIndex{Q} types for some x::AbstractCliffordNumber can be generated with BitIndices(x), and this is a binary ordered vector of BitIndex{Q} objects.

Mathematical operations are defined generically by working with the BitIndex{Q} objects associated with an AbstractCliffordNumber{Q,T}. Elementwise operations on each element of a BitIndices instance returns a TransformedBitIndices, a wrapper which lazily associates a function with a BitIndices object, and this can be used to implement grade dependent operations, such as (anti)automorphisms.

Operations

The following mathematical operations are supported by this package:

• Addition (+), subtraction and negation (-)
• The geometric product (*)
• Efficient muladd operations involving scalars and multivectors
• Scalar left (/) and right (\) division, including rational division (//)
• The reverse ('), grade involution, and Clifford conjugation
• The modulus and absolute value (with abs2 and abs)
• The wedge product (∧)
• The left (⨼) and right (⨽) contractions
• The dot product and the Hestenes dot product
• The commutator product (×) and anticommutator product (⨰)
• Exponentiation

Some of the names or implementations of various operations may change in the near future.

Note that AbstractCliffordNumber{Q,T} is a scalar type, so dotted operators do not apply any operations to each coefficient individually. However, they can be used to perform elementwise operations on collections of Clifford numbers.

Features to be added or improved

Type system

• A StaticMultivector{[G],Q,T,L} type, allowing for non-static implementations.

Relationships between Clifford algebras

• Determining the even subalgebra associated with some Clifford algebra.
• Converting even multivectors of an algebra to multivectors in the even subalgebra.

Mathematical operations

• Better numerical accuracy and stability for some operations. In particular, properly leveraging sinpi, cospi, hypot, and other methods with better numerical accuracy and stability internally.

Interoperability

• How do we handle the dot product of this package and the dot product of the LinearAlgebra standard library? This package has no dependencies, but the semantics should be compatible as much as possible without requiring a LinearAlgebra dependency.
• Secondarily, StaticArrays.similar_type and CliffordNumbers.similar_type have essentially identical behavior.