DomainSets.jl is a package designed to represent simple infinite sets, that can be used to encode domains of functions. For example, the domain of the function log(x::Float64) is the infinite open interval, which is represented by the type HalfLine{Float64}().

DomainSets.jl uses IntervalSets.jl for Closed and open intervals.

Rectangles can be constructed as a product of intervals, where the elements of the domain are SVector{2}:

julia> using DomainSets, StaticArrays; import DomainSets: ×

julia> SVector(1,2) in (-1..1) × (0..3)
true

A UnitHyperSphere{N,T} contains x::SVector{N,T} if norm(x) == one(T). UnitCircle and UnitSphere are two important cases:

julia> SVector(1,0) in UnitCircle()
true

julia> SVector(1,0,0) in UnitSphere()
true

A UnitHyperBall{N,T} contains x::SVector{N,T} if norm(x) ≤ one(T). UnitDisk and UnitHyperBall are two important cases:

julia> SVector(0.1,0.2) in UnitDisk()
true

julia> SVector(0.1,0.2,0.3) in UnitBall()
true

Domains can be unioned and intersected together:

julia> d = UnitCircle() ∪ 2UnitCircle();

julia> in.([SVector(1,0),SVector(0,2), SVector(1.5,1.5)], Ref(d))
3-element BitArray{1}:
  true
  true
 false

julia> d = UnitCircle() ∩ (2UnitCircle() + SVector(1.0,0.0))
the intersection of 2 domains:
	1.	: UnitHyperSphere{2,Float64}()
	2.	: A mapped domain based on UnitHyperSphere{2,Float64}()

julia> SVector(1,0) in d
false

julia> SVector(-1,0) in d
true

A domain is any type that implements the functions eltype and in. If d is an instance of a type that implements the domain interface, then the domain consists of all x that is an eltype(d) such that x in d returns true.

Domains often represent continuous mathematical domains, for example, a domain d representing the interval [0,1] would have eltype(d) == Int but still have 0.2 in d return true.

DomainSets.jl contains an abstract type Domain{T}. All subtypes of Domain{T} must implement the domain interface, and in addition support convert(Domain{T}, d).