# DomainSets.jl

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

### Intervals

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

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)
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> 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> 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()
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> 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> 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> 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])
true
```

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)
true
```

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}:
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> 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> 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
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.

`Domain`

type

The 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)`

.