# Line

LazySets.LineType
Line{N, VN<:AbstractVector{N}} <: AbstractPolyhedron{N}

Type that represents a line of the form

$$$\{y ∈ \mathbb{R}^n: y = p + λd, λ ∈ \mathbb{R}\}$$$

where $p$ is a point on the line and $d$ is its direction vector (not necessarily normalized).

Fields

• p – point on the line
• d – direction

Examples

The line passing through the point $[-1, 2, 3]$ and parallel to the vector $[3, 0, -1]$:

julia> Line([-1, 2, 3.], [3, 0, -1.])
Line{Float64, Vector{Float64}}([-1.0, 2.0, 3.0], [3.0, 0.0, -1.0])
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LazySets.dimMethod
dim(L::Line)

Return the ambient dimension of a line.

Input

• L – line

Output

The ambient dimension of the line.

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LazySets.ρMethod
ρ(d::AbstractVector, L::Line)

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

Input

• d – direction
• L – line

Output

Evaluation of the support function in the given direction.

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LazySets.σMethod
σ(d::AbstractVector, L::Line)

Return a support vector of a line in a given direction.

Input

• d – direction
• L – line

Output

A support vector in the given direction.

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Base.:∈Method
∈(x::AbstractVector, L::Line)

Check whether a given point is contained in a line.

Input

• x – point/vector
• L – line

Output

true iff x ∈ L.

Algorithm

The point $x$ belongs to the line $L : p + λd$ if and only if $x - p$ is proportional to the direction $d$.

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

Create a random line.

Input

• Line – type for dispatch
• 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 line.

Algorithm

All numbers are normally distributed with mean 0 and standard deviation 1.

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LazySets.isuniversalMethod
isuniversal(L::Line; [witness::Bool]=false)

Check whether a line is universal.

Input

• P – line
• witness – (optional, default: false) compute a witness if activated

Output

• If witness is false: true if the ambient dimension is one, false

otherwise.

• If witness is true: (true, []) if the ambient dimension is one,

(false, v) where $v ∉ P$ otherwise.

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Base.isemptyMethod
isempty(L::Line)

Check whether a line is empty or not.

Input

• L – line

Output

false.

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LazySets.constraints_listMethod
constraints_list(L::Line)

Return the list of constraints of a line.

Input

• L – line

Output

A list containing 2n-2 half-spaces whose intersection is L, where n is the ambient dimension of L.

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LazySets.translateMethod
translate(L::Line, v::AbstractVector)

Translate (i.e., shift) a line by a given vector.

Input

• L – line
• v – translation vector

Output

A translated line.

Notes

See also translate! for the in-place version.

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LazySets.translate!Method
translate!(L::Line, v::AbstractVector)

Translate (i.e., shift) a line by a given vector, in-place.

Input

• L – line
• v – translation vector

Output

The line L translated by v.

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LinearAlgebra.normalizeFunction
normalize(L::Line, p::Real=2.0)

Normalize the direction of a line.

Input

• L – line
• p – (optional, default: 2.0) vector p-norm used in the normalization

Output

A line whose direction has unit norm w.r.t. the given p-norm.

Notes

See also normalize!(::Line, ::Real) for the in-place version.

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LinearAlgebra.normalize!Function
normalize!(L::Line, p::Real=2.0)

Normalize the direction of a line storing the result in L.

Input

• L – line
• p – (optional, default: 2.0) vector p-norm used in the normalization

Output

A line whose direction has unit norm w.r.t. the given p-norm.

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ReachabilityBase.Arrays.distanceMethod
distance(x::AbstractVector, L::Line; [p]::Real=2.0)

Compute the distance between point x and the line with respect to the given p-norm.

Input

• x – point/vector
• L – line
• p – (optional, default: 2.0) the p-norm used; p = 2.0 corresponds to the usual Euclidean norm

Output

A scalar representing the distance between x and the line L.

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LazySets.linear_mapMethod
linear_map(M::AbstractMatrix, L::Line)

Concrete linear map of a line.

Input

• M – matrix
• L – line

Output

The line obtained by applying the linear map, if that still results in a line, or a Singleton otherwise.

Algorithm

We apply the linear map to the point and direction of L. If the resulting direction is zero, the result is a singleton.

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Inherited from LazySet: