# DomainSets.jl

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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.

## Examples

For more information, see the [documentation](https://JuliaApproximation.github.io/DomainSets.jl/dev).

### Intervals

DomainSets.jl uses [IntervalSets.jl](https://github.com/JuliaMath/IntervalSets.jl) for closed and open intervals. In addition, it defines a few standard intervals.

```julia
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

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

```julia
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)
true

julia> UnitInterval()^3
UnitCube()
```

### 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
julia> using StaticArrays

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

julia> [0.0,1.0,0.0,0.0] in UnitSphere(4)
true
```
`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
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
julia> SVector(1,0) in UnitCircle()
true
```



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

julia> [0.1,0.2,0.3,-0.1] in UnitBall(4)
true
```
By default `N=3`, but `UnitDisk` is a special case in 2D, and so are `ComplexUnitDisk` and `ComplexUnitCircle` in the complex plane:
```julia
julia> SVector(0.1,0.2) in UnitDisk()
true

julia> 0.5+0.2im ∈ ComplexUnitDisk()
true
```

`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
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
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
julia> 1:5 in ProductDomain([0..i for i in 1:5])
true
```
In other cases, the points are concatenated into a tuple and membership is
evaluated element-wise:
```julia
julia> ("a", 0.4) ∈ ProductDomain(["a","b"], 0..1)
true
```

Some arguments are recognized and return a more specialized product domain.
Examples are the unit box and more general hyperrectangles:
```julia
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
julia> d = UnitCircle() ∪ 2UnitCircle();

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

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
false

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


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

julia> [0.5,2] ∈ d
true

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
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
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
true
```

### 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)`.
