# DomainSets.jl

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}()`

.

## Examples

### Intervals

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

```
julia> using DomainSets, StaticArrays
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> SVector(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> SVector(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)`

.