Convex hull

Binary convex hull (ConvexHull)

LazySets.ConvexHull — Type.

ConvexHull{N<:Real, S1<:LazySet{N}, S2<:LazySet{N}} <: LazySet{N}

Type that represents the convex hull of the union of two convex sets.

Fields

X – convex set
Y – convex set

Notes

The EmptySet is the neutral element for ConvexHull.

Examples

Convex hull of two 100-dimensional Euclidean balls:

julia> b1, b2 = Ball2(zeros(100), 0.1), Ball2(4*ones(100), 0.2);

julia> c = ConvexHull(b1, b2);

julia> typeof(c)
ConvexHull{Float64,Ball2{Float64,Array{Float64,1}},Ball2{Float64,Array{Float64,1}}}

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LazySets.CH — Type.

CH

Alias for ConvexHull.

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LazySets.swap — Method.

swap(ch::ConvexHull)

Return a new ConvexHull object with the arguments swapped.

Input

ch – convex hull of two convex sets

Output

A new ConvexHull object with the arguments swapped.

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LazySets.dim — Method.

dim(ch::ConvexHull)

Return the dimension of a convex hull of two convex sets.

Input

ch – convex hull of two convex sets

Output

The ambient dimension of the convex hull of two convex sets.

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LazySets.ρ — Method.

ρ(d::AbstractVector{N}, ch::ConvexHull{N}) where {N<:Real}

Return the support function of a convex hull of two convex sets in a given direction.

Input

d – direction
ch – convex hull of two convex sets

Output

The support function of the convex hull in the given direction.

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LazySets.σ — Method.

σ(d::AbstractVector{N}, ch::ConvexHull{N}) where {N<:Real}

Return the support vector of a convex hull of two convex sets in a given direction.

Input

d – direction
ch – convex hull of two convex sets

Output

The support vector of the convex hull in the given direction.

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LazySets.isbounded — Method.

isbounded(ch::ConvexHull)

Determine whether a convex hull of two convex sets is bounded.

Input

ch – convex hull of two convex sets

Output

true iff both wrapped sets are bounded.

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Base.isempty — Method.

isempty(ch::ConvexHull)

Return if a convex hull of two convex sets is empty or not.

Input

ch – convex hull

Output

true iff both wrapped sets are empty.

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Inherited from LazySet:

$n$-ary convex hull (ConvexHullArray)

LazySets.ConvexHullArray — Type.

ConvexHullArray{N<:Real, S<:LazySet{N}} <: LazySet{N}

Type that represents the symbolic convex hull of a finite number of convex sets.

Fields

array – array of sets

Notes

The EmptySet is the neutral element for ConvexHullArray.

Constructors:

ConvexHullArray(array::Vector{<:LazySet}) – default constructor
ConvexHullArray([n]::Int=0, [N]::Type=Float64) – constructor for an empty hull with optional size hint and numeric type

Examples

Convex hull of 100 two-dimensional balls whose centers follows a sinusoidal:

julia> b = [Ball2([2*pi*i/100, sin(2*pi*i/100)], 0.05) for i in 1:100];

julia> c = ConvexHullArray(b);

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LazySets.CHArray — Type.

CHArray

Alias for ConvexHullArray.

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LazySets.dim — Method.

dim(cha::ConvexHullArray)

Return the dimension of the convex hull of a finite number of convex sets.

Input

cha – convex hull array

Output

The ambient dimension of the convex hull of a finite number of convex sets.

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LazySets.ρ — Method.

ρ(d::AbstractVector{N}, cha::ConvexHullArray{N}) where {N<:Real}

Return the support function of a convex hull array in a given direction.

Input

d – direction
cha – convex hull array

Output

The support function of the convex hull array in the given direction.

Algorithm

This algorihm calculates the maximum over all $ρ(d, X_i)$ where the $X_1, …, X_k$ are the sets in the array cha.

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LazySets.σ — Method.

σ(d::AbstractVector{N}, cha::ConvexHullArray{N}) where {N<:Real}

Return the support vector of a convex hull array in a given direction.

Input

d – direction
cha – convex hull array

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LazySets.isbounded — Method.

isbounded(cha::ConvexHullArray)

Determine whether a convex hull of a finite number of convex sets is bounded.

Input

cha – convex hull of a finite number of convex sets

Output

true iff all wrapped sets are bounded.

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LazySets.array — Method.

array(cpa::CartesianProductArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a Cartesian product of a finite number of convex sets.

Input

cpa – Cartesian product array

Output

The array of a Cartesian product of a finite number of convex sets.

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array(cha::ConvexHullArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a convex hull of a finite number of convex sets.

Input

cha – convex hull array

Output

The array of a convex hull of a finite number of convex sets.

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array(ia::IntersectionArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of an intersection of a finite number of convex sets.

Input

ia – intersection of a finite number of convex sets

Output

The array of an intersection of a finite number of convex sets.

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array(msa::MinkowskiSumArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a Minkowski sum of a finite number of convex sets.

Input

msa – Minkowski sum array

Output

The array of a Minkowski sum of a finite number of convex sets.

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array(cms::CachedMinkowskiSumArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a caching Minkowski sum.

Input

cms – caching Minkowski sum

Output

The array of a caching Minkowski sum.

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array(cup::UnionSetArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a union of a finite number of convex sets.

Input

cup – union of a finite number of convex sets

Output

The array that holds the union of a finite number of convex sets.

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Base.isempty — Method.

isempty(cha::ConvexHullArray)

Return if a convex hull array is empty or not.

Input

cha – convex hull array

Output

true iff all wrapped sets are empty.

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LazySets.vertices_list — Method.

vertices_list(X::ConvexHullArray{N, Singleton{N, VT}}) where {N, VT}

Return the list of vertices of the convex hull array of singletons.

Input

X – convex hull array of singletons

Output

The list of elements in the array that defines X.

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Inherited from LazySet: