Universe

Universe{N} <: AbstractPolyhedron{N}

Type that represents the universal set, i.e., the set of all elements.

Fields

• dim – the ambient dimension of the set

an_element(X::LazySet)

Return some element of a nonempty set.

Input

• X – set

Output

An element of X unless X is empty.

Extended help

an_element(U::Universe)

Algorithm

The output is the origin.

constrained_dimensions(U::Universe)

Return the indices in which a universe is constrained.

Input

• U – universe

Output

The empty vector, as the universe is unconstrained in every dimension.

rand(T::Type{<:LazySet}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing
    )

Create a random set of the given set type.

Input

• T – set type
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A random set of the given set type.

Extended help

rand(::Type{Universe}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)

Output

The (only) universe of the given numeric type and dimension.

ρ(d::AbstractVector, X::LazySet)

Evaluate the support function of a set in a given direction.

Input

• d – direction
• X – set

Output

The evaluation of the support function of X in direction d.

Notes

A convenience alias support_function is also available.

We have the following identity based on the support vector $σ$:

\[ ρ(d, X) = d ⋅ σ(d, X)\]

Extended help

ρ(d::AbstractVector, U::Universe)

Algorithm

If the direction is all zero, the result is zero. Otherwise, the result is Inf.

σ(d::AbstractVector, X::LazySet)

Compute a support vector of a set in a given direction.

Input

• d – direction
• X – set

Output

A support vector of X in direction d.

Notes

A convenience alias support_vector is also available.

Extended help

σ(d::AbstractVector, U::Universe)

Output

A vector with infinity values, except in dimensions where the direction is zero.