renamed infinity and origin indices

README.md

@@ -63,11 +63,11 @@ An application of this is in the `Grasmann` package, where dual indexing is used
 
 ### Additional features for projective geometry generators
 
-Not to be confused with the *dual basis*, to declare *dual numbers* it is possible to specify a degenerate basis element with `ϵ` at the first index, i.e. `V"ϵ+++"`
-Optionally, an additional element can be specified as the *origin* by using `o` subsequently, i.e. `V"o+++"` or `V"ϵo+++"`.
+Not to be confused with the *dual basis*, to declare *dual numbers* it is possible to specify a degenerate basis element with `∞` at the first index, i.e. `V"∞+++"`
+Optionally, an additional element can be specified as the *origin* by using `∅` subsequently, i.e. `V"∅+++"` or `V"∞∅+++"`.
 These two basis elements will be interpreted in the type system such that they propagate under transformations when combining a mixed index sets (provided the signature is compatible).
 
-The index number `N` of the `VectorSpace` corresponds to the total number of generator elements, but even though `V"ϵo+++"` of type `VectorSpace{5,3}` has standard generator elements, it can be internally recognized in the direct sum algebra as being generated by a 3-index `VectorSpace{3,0}` due to the encoding of the degenerate and origin elements with `M` in the `VectorSpace{N,M}` type.
+The index number `N` of the `VectorSpace` corresponds to the total number of generator elements, but even though `V"∞∅+++"` of type `VectorSpace{5,3}` has standard generator elements, it can be internally recognized in the direct sum algebra as being generated by a 3-index `VectorSpace{3,0}` due to the encoding of the degenerate and origin elements with `M` in the `VectorSpace{N,M}` type.
 
 ### Future work
 

src/DirectSum.jl

@@ -3,7 +3,7 @@ module DirectSum
 #   This file is part of DirectSum.jl. It is licensed under the GPL license
 #   Grassmann Copyright (C) 2019 Michael Reed
 
-export VectorSpace, vectorspace, @V_str, Signature, DiagonalForm, ℝ, ⊕, value
+export VectorSpace, vectorspace, @V_str, @D_str, Signature, DiagonalForm, ℝ, ⊕, value
 import Base: getindex, abs, @pure, +, *, ^, ∪, ∩, ⊆, ⊇
 import LinearAlgebra: det
 using StaticArrays
@@ -16,6 +16,8 @@ bit2int(b::BitArray{1}) = parse(Bits,join(reverse([t ? '1' : '0' for t ∈ b])),
 
 @pure doc2m(d,o,c=0) = (1<<(d-1))+(1<<(2*o-1))+(c<0 ? 8 : (1<<(3*c-1)))
 
+const vio = ('∞','∅')
+
 ## VectorSpace{N}
 
 abstract type VectorSpace{Indices,Options,Metrics} end
@@ -38,7 +40,7 @@ end
         length(s) < 4 && (s *= join(zeros(Int,5-length(s))))
         Signature(parse(Int,s[1]),parse(Int,s[2]),parse(Int,s[3]),UInt(parse(Int,s[4:end])))
     else
-        Signature{N,doc2m(Int('ϵ'∈s),Int('o'∈s))}(replace(replace(s,'ϵ'=>'+'),'o'=>'+'))
+        Signature{N,doc2m(Int(vio[1]∈s),Int(vio[2]∈s))}(replace(replace(s,vio[1]=>'+'),vio[2]=>'+'))
     end
 end
 
@@ -58,9 +60,9 @@ Base.length(s::VectorSpace{N}) where N = N
 
 @inline function Base.show(io::IO,s::Signature)
     print(io,'⟨')
-    hasdual(s) && print(io,'ϵ')
-    hasorigin(s) && print(io,'o')
-    print(io,sig.(s[hasdual(s)+hasorigin(s)+1:ndims(s)])...)
+    hasinf(s) && print(io,vio[1])
+    hasorigin(s) && print(io,vio[2])
+    print(io,sig.(s[hasinf(s)+hasorigin(s)+1:ndims(s)])...)
     print(io,'⟩')
     C = dualtype(s)
     C ≠ 0 ? print(io, C < 0 ? '*' : ''') : nothing
@@ -88,16 +90,17 @@ end
 DiagonalForm{N,M}(b::Vector) where {N,M} = DiagonalForm{N,M}(SVector(b...))
 DiagonalForm(b::SVector{N}) where N = DiagonalForm{N,0}(b)
 DiagonalForm(b::Vector) = DiagonalForm{length(b),0}(b)
-DiagonalForm(b...) = DiagonalForm(b)
+DiagonalForm(b::Tuple) = DiagonalForm{length(b),0}(SVector(b))
+DiagonalForm(b::Number...) = DiagonalForm(b)
 
 @inline getindex(s::DiagonalForm{N,M,S} where {N,M},i) where S = diagonalform(s)[i]
 getindex(vs::DiagonalForm{N,M,S} where M,i::Colon) where {N,S} = diagonalform(vs)
 
 @inline function Base.show(io::IO,s::DiagonalForm)
     print(io,'⟨')
-    hasdual(s) && print(io,'ϵ')
-    hasorigin(s) && print(io,'o')
-    for k ∈ hasdual(s)+hasorigin(s)+1:ndims(s)
+    hasinf(s) && print(io,vio[1])
+    hasorigin(s) && print(io,vio[2])
+    for k ∈ hasinf(s)+hasorigin(s)+1:ndims(s)
         print(io,s[k])
         k ≠ ndims(s) && print(io,',')
     end
@@ -106,23 +109,29 @@ getindex(vs::DiagonalForm{N,M,S} where M,i::Colon) where {N,S} = diagonalform(vs
     C ≠ 0 ? print(io, C < 0 ? '*' : ''') : nothing
 end
 
+@pure Signature(V::DiagonalForm{N,M}) where {N,M} = Signature{N,M}(Vector(signbit.(V[:])))
+
 # macro
 
 macro V_str(str)
     vectorspace(str)
 end
 
+macro D_str(str)
+    DiagonalForm(Meta.parse(str).args)
+end
+
 # generic
 
 @pure Base.ndims(::VectorSpace{N}) where N = N
-@pure hasdual(M::Int) = M ∈ (1,3,5,7,9,11)
+@pure hasinf(M::Int) = M ∈ (1,3,5,7,9,11)
 @pure hasorigin(M::Int) = M ∈ (2,3,6,7,10,11)
 @pure dualtype(M::Int) = M ∈ 8:11 ? -1 : Int(M ∈ (4,5,6,7))
-@pure hasdual(::VectorSpace{N,M} where N) where M = hasdual(M)
+@pure hasinf(::VectorSpace{N,M} where N) where M = hasinf(M)
 @pure hasorigin(::VectorSpace{N,M} where N) where M = hasorigin(M)
 @pure dualtype(::VectorSpace{N,M} where N) where M = dualtype(M)
 @pure options(::VectorSpace{N,M} where N) where M = M
-@pure options_list(V::VectorSpace) = hasdual(V),hasorigin(V),dualtype(V)
+@pure options_list(V::VectorSpace) = hasinf(V),hasorigin(V),dualtype(V)
 @pure value(::VectorSpace{N,M,S} where {N,M}) where S = S
 
 det(s::Signature) = isodd(count_ones(value(s))) ? -1 : 1
@@ -140,13 +149,13 @@ dual(V::VectorSpace{N},B,M=Int(N/2)) where N = ((B<<M)&((1<<N)-1))|(B>>M)
 @pure function Base.adjoint(V::Signature{N,M,S}) where {N,M,S}
     C = dualtype(V)
     C < 0 && throw(error("$V is the direct sum of a vector space and its dual space"))
-    Signature{N,doc2m(hasdual(V),hasorigin(V),Int(!Bool(C))),flip_sig(N,S)}()
+    Signature{N,doc2m(hasinf(V),hasorigin(V),Int(!Bool(C))),flip_sig(N,S)}()
 end
 
 @pure function Base.adjoint(V::DiagonalForm{N,M,S}) where {N,M,S}
     C = dualtype(V)
     C < 0 && throw(error("$V is the direct sum of a vector space and its dual space"))
-    DiagonalForm{N,doc2m(hasdual(V),hasorigin(V),Int(!Bool(C))),S}()
+    DiagonalForm{N,doc2m(hasinf(V),hasorigin(V),Int(!Bool(C))),S}()
 end
 
 ## default definitions
@@ -158,4 +167,9 @@ const ℝ = Signature(1)
 include("operations.jl")
 include("indices.jl")
 
+export metric
+
+@pure metric(V::Signature,b::Bits) = isodd(count_ones(value(V)&b)) ? -1 : 1
+@pure metric(V::DiagonalForm,b::Bits) = prod(V[indices(b)])
+
 end # module

src/indices.jl

@@ -11,8 +11,8 @@ const alphanumv = digs*low_case*upp_case #*low_greek*upp_greek
 const alphanumw = digs*upp_case*low_case #*upp_greek*low_greek
 
 const subs = Dict{Int,Char}(
-   -1 => 'ϵ',
-    0 => 'o',
+   -1 => vio[1],
+    0 => vio[2],
     1 => '₁',
     2 => '₂',
     3 => '₃',
@@ -27,8 +27,8 @@ const subs = Dict{Int,Char}(
 )
 
 const sups = Dict{Int,Char}(
-   -1 => 'ϵ',
-    0 => 'o',
+   -1 => vio[1],
+    0 => vio[2],
     1 => '¹',
     2 => '²',
     3 => '³',
@@ -57,8 +57,8 @@ end
 function shift_indices(s::VectorSpace{N,M} where N,set::Vector{Int}) where M
     if !isempty(set)
         k = 1
-        hasdual(s) && set[1] == 1 && (set[1] = -1; k += 1)
-        shift = hasdual(s) + hasorigin(s)
+        hasinf(s) && set[1] == 1 && (set[1] = -1; k += 1)
+        shift = hasinf(s) + hasorigin(s)
         hasorigin(s) && length(set)>=k && set[k]==shift && (set[k]=0;k+=1)
         shift > 0 && (set[k:end] .-= shift)
     end