Support complex domains (e.g., circle in the complex plane)

[dlfivefifty]

No description provided.

[daanhb]

The original idea was to support this by identifying R^2 with C. This way, any 2D domain can be used as a subset of the complex plane. As a matter of fact, this is currently working, but it is not user-friendly yet and it does result in a very strange object (a unit disk with an affine map to set the radius, and an embedding map for identifying with the plane).

Of course, for popular domains such as circles and, say, Bernstein ellipses, there could be a special-purpose type as well with a straightforward implementation.

[dlfivefifty]

I think that would be confusing, and one should only support conversion:

in(d::Domain{T}, x) where T = in(d, convert(T, x))

This would throw an error if T is SVector{2,Float64} and x isa Complex128.

I would rewrite the circle code to support non-SVector types via:

struct UnitSphere{N,T} <: Domain{T}
end

const Circle{T} = UnitSphere{2,T}
const Sphere{T} = UnitBall{3,T}
circle(::Type{T} = Float64) where {T} = Circle{SVector{2,T}}()

Then a complex-valued unit circle could be represented as Circle{Complex128}(), or a Vector{Float64} unit circle could be represented as Circle{Vector{Float64}}().

[dlfivefifty]

Sorry, calling convert doesn't work: one wants to actually do the following:

function in(x, d::Domain) 
   T = promote_type(typeof(x), eltype(d))
   T == Any && return false
   in(convert(T, x), convert(Domain{T}, d))
end

so that we can do in(1+im , Interval()) without throwing an error.

[dlfivefifty]

PS I'm going to just make the changes I want to make in the branch approxfun-support, and we can discuss whether they are good ideas in a pull request

[dlfivefifty]

I think this makes sense: change it to

struct UnitHyperSphere{T} <: Domain{T} end

const UnitCircle{T} = UnitHyperSphere{SVector{2,T}}
const UnitSphere{T} = UnitHyperSphere{SVector{3,T}}
const ComplexUnitCircle{T} = UnitHyperSphere{Complex{T}}

Here the definition of UnitHyperSphere{T} is x in T such that norm(x) = 1.

[daanhb]

Old issue but still relevant. With the current generic definitions of balls and spheres, I'm adding:

const ComplexUnitCircle{T} = FixedUnitSphere{Complex{T}}
const ComplexUnitDisk{T,C} = FixedUnitBall{Complex{T},C}

ComplexUnitCircle() = ComplexUnitCircle{Float64}()
ComplexUnitDisk() = ComplexUnitDisk{Float64}()
ComplexUnitDisk{Float64}() = ComplexUnitDisk{Float64,:closed}()

The open complex unit disk can be made, if one is so inclined, using ComplexUnitDisk{Float64,:open}().