implemented dispatch of UniformScaling as universal pseudoscalar

README.md

@@ -131,8 +131,11 @@ v₂₃
 ```
 Effort of computation depends on the sparsity. Then it is possible to assign the **quaternion** generators `i,j,k` with
 ```Julia
-julia> i,j,k = complementright.((-Λ(3).v1,-Λ(3).v2,-Λ(3).v3))
-(-1v₂₃, 1v₁₃, -1v₁₂)
+julia> i,j,k = hyperplanes(ℝ^3)
+3-element Array{SValue{⟨+++⟩,2,B,Int64} where B,1}:
+ -1v₂₃
+ 1v₁₃
+ -1v₁₂
 
 julia> @btime i^2, j^2, k^2, i*j*k
   158.925 ns (5 allocations: 112 bytes)

src/Grassmann.jl

@@ -17,6 +17,12 @@ include("forms.jl")
 include("symbolic.jl")
 include("generators.jl")
 
+## fundamentals
+
+export hyperplanes
+
+hyperplanes(V::VectorSpace{N}) where N = map(n->I*getbasis(V,1<<n),0:N-1)
+
 abstract type SubAlgebra{V} <: TensorAlgebra{V} end
 
 adjoint(G::A) where A<:SubAlgebra{V} where V = Λ(dual(V))

src/algebra.jl

@@ -137,6 +137,19 @@ end
 #*(a::Basis{V},b::MultiGrade{V}) where V = MultiGrade{V}(b.v,a*basis(b))
 #*(a::MultiGrade{V},b::MultiGrade{V}) where V = MultiGrade{V}(a.v*b.v,basis(a)*basis(b))
 
+for Value ∈ MSV
+    @eval begin
+        *(a::UniformScaling,b::$Value{V}) where V = V(a)*b
+        *(a::$Value{V},b::UniformScaling) where V = a*V(b)
+    end
+end
+for Blade ∈ MSB
+    @eval begin
+        *(a::UniformScaling,b::$Blade{T,V} where T) where V = V(a)*b
+        *(a::$Blade{T,V} where T,b::UniformScaling) where V = a*V(b)
+    end
+end
+
 ## exterior product
 
 export ∧, ∨
@@ -159,7 +172,10 @@ function ∧(a::X,b::Y) where {X<:TensorTerm{V},Y<:TensorTerm{V}} where V
     return SValue{V}(parity(x,y) ? -v : v,Basis{V}(A⊻B))
 end
 
-∧(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∧,a,b)
+@inline ∧(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∧,a,b)
+@inline ∧(a::TensorAlgebra{V},b::UniformScaling{T}) where {V,T<:Field} = a∧V(b)
+@inline ∧(a::UniformScaling{T},b::TensorAlgebra{V}) where {V,T<:Field} = V(a)∧b
+
 
 @inline exterior_product!(V::VectorSpace,out,α,β,γ) = (count_ones(α&β)==0) && joinaddmulti!(V,out,α,β,γ)
 
@@ -215,6 +231,9 @@ for r ∈ (:reverse,:involve,:conj)
     end
 end
 
+reverse(a::UniformScaling{Bool}) = UniformScaling(!a.λ)
+reverse(a::UniformScaling{T}) where T<:Field = UniformScaling(-a.λ)
+
 ## inner product: a ∨ ⋆(b)
 
 import LinearAlgebra: dot, ⋅
@@ -256,7 +275,10 @@ function ∨(a::X,b::Y) where {X<:TensorTerm{V},Y<:TensorTerm{V}} where V
     return SValue{V}(p ? -v : v,Basis{V}(C))
 end
 
-∨(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∨,a,b)
+@inline ∨(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∨,a,b)
+@inline ∨(a::TensorAlgebra{V},b::UniformScaling{T}) where {V,T<:Field} = a∨V(b)
+@inline ∨(a::UniformScaling{T},b::TensorAlgebra{V}) where {V,T<:Field} = V(a)∨b
+
 
 @pure function regressive_calc(N,M,S,A,B)
     α,β = complement(N,A),complement(N,B)
@@ -270,6 +292,13 @@ end
     end
 end
 
+## cross product
+
+import LinearAlgebra: cross
+export ×
+
+cross(a::TensorAlgebra{V},b::TensorAlgebra{V}) where V = I*(a∧b)
+
 ### parity cache
 
 @eval begin

src/multivectors.jl

@@ -11,6 +11,12 @@ abstract type TensorMixed{T,V} <: TensorAlgebra{V} end
 
 import DirectSum: shift_indices, printindex, printindices, VTI
 
+## pseudoscalar
+
+using LinearAlgebra
+import LinearAlgebra: I
+export UniformScaling, I
+
 ## MultiBasis{N}
 
 struct Basis{V,G,B} <: TensorTerm{V,G}
@@ -382,6 +388,8 @@ end
 
 ## conversions
 
+@inline (V::VectorSpace)(s::UniformScaling{T}) where T = SValue{V}(T<:Bool ? (s.λ ? -(one(T)) : one(T)) : s.λ,getbasis(V,(one(Bits)<<ndims(V))-1))
+
 @pure function (W::VectorSpace)(b::Basis{V}) where V
     V==W && (return b)
     !(V⊆W) && throw(error("cannot convert from $(V) to $(W)"))