# 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.

### Rectangles

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

:

```
julia> using DomainSets, StaticArrays; using DomainSets: ×
julia> SVector(1,2) in (-1..1) × (0..3)
true
```

### Circles and Spheres

A `EuclideanUnitSphere{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
```

### Disks and Balls

A `EuclideanUnitBall{N,T}`

contains `x::SVector{N,T}`

if `norm(x) ≤ one(T)`

. `UnitDisk`

and `UnitBall`

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 with variable length Vector elements

Domains with `Vector`

elements may have an arbitrary dimension. Several of the
examples above have analogues for `Vector`

elements:

```
julia> [0.1, 0.2, 0.3, 0.2, 0.1] in VectorUnitBall(5)
true
julia> [1,0,0,0,0,0,0,0,0,0] in VectorUnitSphere(10)
true
```

Product domains with elements of type `Vector`

, rather than `SVector{N}`

, may
be created by invoking `ProductDomain`

with a vector of domains:

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

### 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)], Ref(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
```

### 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)`

.