A Julia package for describing domains in Euclidean space
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May 2017


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DomainSets.jl is a package designed to represent simple infinite sets. The package makes it easy to represent sets, verify membership of the set, compare sets and construct new sets from existing ones. Domains are considered equivalent if they describe the same set, regardless of their type.



DomainSets.jl uses IntervalSets.jl for closed and open intervals. In addition, it defines a few standard intervals.

julia> using DomainSets

julia> UnitInterval()
0.0..1.0 (Unit)

julia> ChebyshevInterval()
-1.0..1.0 (Chebyshev)

julia> HalfLine()
0.0..Inf (closed–open) (HalfLine)


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

julia> using DomainSets: ×

julia> (-1..1) × (0..3) × (4.0..5.0)
(-1.0..1.0) × (0.0..3.0) × (4.0..5.0)

julia> [1,2] in (-1..1) × (0..3)

julia> UnitInterval()^3

Circles and Spheres

A UnitSphere contains x if norm(x) == 1. The unit sphere is N-dimensional, and its dimension is specified with the constructor. The element types are SVector{N,T} when the dimension is specified as Val(3), and they are Vector{T} when the dimension is specified by an integer value instead:

julia> using StaticArrays

julia> SA[0,0,1.0] in UnitSphere(Val(3))

julia> [0.0,1.0,0.0,0.0] in UnitSphere(4)

UnitSphere itself is an abstract type, hence the examples above return concrete types <:UnitSphere. The intended element type can also be explicitly specified with the UnitSphere{T} constructor:

julia> typeof(UnitSphere{SVector{3,BigFloat}}())
EuclideanUnitSphere{3, BigFloat} (alias for StaticUnitSphere{SArray{Tuple{3}, BigFloat, 1, 3}})

julia> typeof(UnitSphere{Vector{Float32}}(6))
VectorUnitSphere{Float32} (alias for DynamicUnitSphere{Array{Float32, 1}})

Without arguments, UnitSphere() defaults to a 3D domain with SVector{3,Float64} elements. Similarly, there is a special case UnitCircle in 2D:

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

Disks and Balls

A UnitBall contains x if norm(x) ≤ 1. As with UnitSphere, the dimension is specified via the constructor by type or by value:

julia> SVector(0.1,0.2,0.3) in UnitBall(Val(3))

julia> [0.1,0.2,0.3,-0.1] in UnitBall(4)

By default N=3, but UnitDisk is a special case in 2D, and so are ComplexUnitDisk and ComplexUnitCircle in the complex plane:

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

julia> 0.5+0.2im  ComplexUnitDisk()

UnitBall itself is an abstract type, hence the examples above return concrete types <:UnitBall. The types are similar to those associated with UnitSphere. Like intervals, balls can also be open or closed:

julia> EuclideanUnitBall{3,Float64,:open}()
the 3-dimensional open unit ball

Product domains

The cartesian product of domains is constructed with the ProductDomain or ProductDomain{T} constructor. This abstract constructor returns concrete types best adapted to the arguments given.

If T is not given, ProductDomain makes a suitable choice based on the arguments. If all arguments are Euclidean, i.e., their element types are numbers or static vectors, then the product is a Euclidean domain as well:

julia> ProductDomain(0..2, UnitCircle())
0.0..2.0 x the unit circle

julia> eltype(ans)
SVector{3, Float64} (alias for SArray{Tuple{3}, Float64, 1, 3})

The elements of the interval and the unit circle are flattened into a single vector, much like the vcat function. The result is a VcatDomain.

If a Vector of domains is given, the element type is a Vector as well:

julia> 1:5 in ProductDomain([0..i for i in 1:5])

In other cases, the points are concatenated into a tuple and membership is evaluated element-wise:

julia> ("a", 0.4)  ProductDomain(["a","b"], 0..1)

Some arguments are recognized and return a more specialized product domain. Examples are the unit box and more general hyperrectangles:

julia> ProductDomain(UnitInterval(), UnitInterval())
0.0..1.0 (Unit) x 0.0..1.0 (Unit)

julia> ProductDomain(0..2, 4..5, 6..7.0)
0.0..2.0 x 4.0..5.0 x 6.0..7.0

julia> typeof(ans)
Rectangle{SVector{3, Float64}}

Union, intersection, and setdiff of domains

Domains can be unioned and intersected together:

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

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

julia> d = UnitCircle()  (2UnitCircle() .+ SVector(1.0,0.0))
the intersection of 2 domains:
	1.	: the unit circle
	2.	: A mapped domain based on the unit circle

julia> SVector(1,0) in d

julia> SVector(-1,0) in d

Level sets

A domain can be defined by the level sets of a function. The domains of all points [x,y] for which x*y = 1 or x*y >= 1 are represented as follows:

julia> d = LevelSet{SVector{2,Float64}}(prod, 1.0)
level set f(x) = 1.0 with f = prod

julia> [0.5,2]  d

julia> SuperlevelSet{SVector{2,Float64}}(prod, 1.0)
superlevel set f(x) >= 1.0 with f = prod

There is also SublevelSet, and there are the special cases ZeroSet, SubzeroSet and SuperzeroSet.

Indicator functions

A domain can be defined by an indicator function or a characteristic function. This is a function f(x) which evaluates to true or false, depending on whether or not the point x belongs to the domain.

julia> d = IndicatorFunction{Float64}( t ->  cos(t) > 0)
indicator domain defined by function f = #5

julia> 0.5  d, 3.1  d
(true, false)

This enables generator syntax to define domains:

julia> d = Domain(x>0 for x in -1..1)
indicator function bounded by: -1..1

julia> 0.5  d, -0.5  d
(true, false)

julia> d = Domain( x*y > 0 for (x,y) in UnitDisk())
indicator function bounded by: the 2-dimensional closed unit ball

julia> [0.2, 0.3]  d, [0.2, -0.3]  d
(true, false)

julia> d = Domain( x+y+z > 0 for (x,y,z) in ProductDomain(UnitDisk(), 0..1))
indicator function bounded by: the 2-dimensional closed unit ball x 0..1

julia> [0.3,0.2,0.5]  d

The domain interface

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.

The Domain type

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).