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update metric signature interface docs
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β€Ždocs/src/design.mdβ€Ž

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@@ -68,32 +68,31 @@ As well as defining the geometric algebra, the signature is used to specify basi
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| Required methods | Description |
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|:-----------------|:------------|
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| `dimension(sig)` | The dimension of the underlying vector space, or number of basis vectors.
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| `basis_vector_square(sig, i)` | The scalar square of the `i`th basis vector. |
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| `canonical_signature(sig)` | Canonical representation as a tuple (e.g., `Cl(1,3)` becomes `(1,-1,-1,-1)`).
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| Optional methods | Description |
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|:-----------------|:------------|
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| `show_signature(io, sig)` | Show the metric signature in a compact human-readable form.
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| `show_basis_blade(io, sig, indices)` | Print a basis blade with the given indices (e.g., `v12` or `π’†β‚βˆ§π’†β‚‚`).
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| `bits_to_indices(sig, bits)` | Define display order of indices for a basis blade (must also implement `basis_blade_parity(sig, bits)` consistently).
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| `componentstype(sig, N)` | Preferred array type for `Multivector{sig}` components. (E.g., `Vector`, `MVector`, `SparseVector`, etc.)
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| `use_symbolic_optim(sig)` | Whether to use symbolic code generation to optimise multivector products. (Default is true for low dimensions.)
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| Optional methods | Description | Default
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|:-----------------|:------------|:-------
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| `dimension(sig)` | The dimension of the underlying vector space | `length(canonical_signature(sig))`
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| `basis_vector_square(sig, i)` | The scalar square of the `i`th basis vector | `canonical_signature(sig)[i]`
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| `show_signature(io, sig)` | Print the metric signature in a compact human-readable form | `show(io, sig)`
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| `show_basis_blade(io, sig, indices)` | Print a basis blade with the given indices (e.g., `v12` or `π’†β‚βˆ§π’†β‚‚`) | prints like `v12`
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| `bits_to_indices(sig, bits)` | Define the order of indices for a basis blade (must also implement `basis_blade_parity(sig, bits)` consistently) | always increasing like `v123`
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| `componentstype(sig, N)` | Preferred array type for `Multivector{sig}` components. (E.g., `Vector`, `MVector`, `SparseVector`, etc.) | `SVector` if small, `Vector` if large
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| `use_symbolic_optim(sig)` | Whether to use symbolic code generation to optimise multivector products. | True for low dimensions
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Below is an example of how one might define a β€œprojectivised” signature which adds a projective dimension ``𝐯_0`` squaring to ``-1`` to any signature:
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```@example ga
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import GeometricAlgebra: dimension, basis_vector_square, show_signature, show_basis_blade
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import GeometricAlgebra: canonical_signature, show_signature, show_basis_blade
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struct β„™{Sig} end
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β„™(sig) = β„™{sig}()
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dimension(::β„™{Sig}) where Sig = dimension(Sig) + 1
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basis_vector_square(::β„™{Sig}, i) where Sig = i == 1 ? -1 : basis_vector_square(Sig, i - 1)
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canonical_signature(::β„™{Sig}) where Sig = (-1, canonical_signature(Sig)...)
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show_signature(io::IO, ::β„™{Sig}) where Sig = print(io, "β„™($Sig)")
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show_basis_blade(io::IO, ::β„™, indices::Vector) = print(io, "v", join(indices .- 1))
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basis(β„™(3)) |> sum
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basis(β„™(3))
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```
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