Internal API Reference

Combine the outputs of in of member domains into a single output of the lazy domain.

Return a suitable tolerance to use for verifying whether a point is close to a domain. Typically, the tolerance is close to the precision limit of the numeric type associated with the domain.

The indicator function of a domain is the function f(x) = x ∈ D.

Return the bounding box of the intersection of two or more bounding boxes.

Is the given combination of point and domain compatible?

parametric_domain(fmap, domain)

The domain that results from mapping the given domain.

Can the domains be promoted without throwing an error?

Promote the given domains to have a common element type.

promote_map_domain_pair(map, domain)

Promote the map and the domain such that the output satisfies codomaintype(map) == domaineltype(domain).

Promote point and domain to compatible types.

Return an interval that is similar to the given interval, but with endpoints a and b instead.

simplifies(domain)

Does the domain simplify?

simplify(domain)

Simplify the given domain to an equal domain.

Convert the given domain to a domain defined in DomainSets.jl.

Return the bounding box of the union of two or more bounding boxes.

Vectorized version of approx_in: apply x ∈ d to all elements of A.

Vectorized version of in: apply x ∈ d to all elements of A.

Inverse of tointernalpoint.

Translate a point of the lazy domain to a point (or points) of the composing domain.

Supertype of domains that are defined by an indicator function.

An indicator function is a function f : S -> [0,1] that indicates membership of x to a domain D with D ⊂ S. The indicator function corresponds exactly to the in function of a domain: f(x) = x ∈ D.

Concrete subtypes of AbstractIndicatorFunction store a representation of this indicator function and implement in using that representation, rather than implementing in directly.

Supertype of level set domains of the form f(x)=C.

A MappedDomain represents the mapping of a domain.

The map of a domain d under the mapping y=f(x) consists of all points f(x) with x ∈ d. The characteristic function of a mapped domain is defined in terms of the inverse map g = inverse(f):

x ∈ m ⟺ g(x) ∈ d

Supertype of sublevel set domains.

Supertype of superlevel set domains.

An AbstractVectorDomain is any domain whose eltype is <:AbstractVector{T}.

An indicator function with a known bounding domain.

A composite lazy domain is defined in terms of multiple domains.

A Cube is a hyperrectangle with equal side lengths in each dimension.

Abstract supertype for domains that wrap another domain.

A DomainPoint is a point which is an element of a domain by construction.

A domain point is just a point, not a domain. This is different from a Point type.

The type is a wrapper: retrieve the underlying point using point(p).

The broadcast style associated with domains

A ball in a fixed N-dimensional Euclidean space.

A cube in a fixed N-dimensional Euclidean space.

A hypersphere in a fixed N-dimensional Euclidean space.

A point on the unit sphere represented by a standard Euclidean vector.

Example of a domain that wraps another domain and thus obtains its own type.

The abstract type FixedInterval is the supertype of intervals with endpoints determined by the type, rather than field values. Examples include UnitInterval and ChebyshevInterval.

The N-fold cartesian product of a fixed interval.

A FunctionLevelSet is a set that derives from the levels of a function.

A GenericBall is a ball with a given radius and center.

A GenericSphere is a sphere with a given radius and center.

A HyperRectangle is the cartesian product of intervals.

Isomorphic <: CanonicalType

An isomorphic canonical domain is a domain that is the same up to an isomorphism.

Supertype of all compositions of a lazy domain. The composition determines how the point x is distributed to the member domains of a lazy domain.

Three compositions implemented in the package are:

• NoComposedMap: the lazy domain encapsulates a single domain and x is passed

through unaltered

• Combination: the lazy domain has several members and x is passed to the in

method of all members

• Product: the lazy domain has several members and the components of x are

passed to the components of the lazy domain

A lazy domain evaluates its membership function on the fly in terms of that of other domains.

The in(x, domain::LazyDomain) applies three types of transformations:

1. Point mapping: y = tointernalpoint(domain, x)
2. Distribution of y over member domains given by components(domain)
3. Combination of the outputs into a single boolean result.

The distribution step is determined by the result of composition(domain), see composition. The combination is performed by combine. Mapping between points of the lazy domain and points of its member domains is described by y = tointernalpoint(domain, x) and x = toexternalpoint(domain, y).

The set of all natural numbers.

NegativeRealLine{T=Float64}()

The open negative halfline (-∞,0).

NonnegativeRealLine{T=Float64}()

The closed positive halfline [0,∞).

NonpositiveRealLine{T=Float64}()

The closed negative halfline (-∞,0].

One()

Representation of the number 1.

Origin()

Representation of the origin.

Parameterization <: CanonicalType

A parametric canonical domain can be parameterized from a simpler domain.

A ParametricDomain stores the forward map of a mapped domain.

PositiveRealLine{T=Float64}()

The open positive halfline (0,∞).

A simple lazy domain is defined in terms of a single domain.

It has no composition and no combination of its in function.

Supertype of an N-dimensional simplex.

A point on the unit sphere.

A point on the unit sphere represented in spherical coordinates.

A ball with vector elements of variable length.

A cube with vector elements of variable length.

A sphere with vector elements of variable length.

A WrappedDomain is a wrapper around an object that implements the domain interface, and that is itself a domain.