Line

Line{N, VN<:AbstractVector{N}} <: AbstractPolyhedron{N}

Type that represents a line of the form

\[ \{y ∈ ℝ^n: y = p + λd, λ ∈ ℝ\}\]

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

There are three constructors. The optional keyword argument normalize (default: false) can be used to normalize the direction of the resulting line to have norm 1 (w.r.t. the Euclidean norm).

1. The default constructor takes the fields p and d:

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

julia> Line([-1.0, 2, 3], [3.0, 0, -1])
Line{Float64, Vector{Float64}}([-1.0, 2.0, 3.0], [3.0, 0.0, -1.0])

julia> Line([-1.0, 2, 3], [3.0, 0, -1]; normalize=true)
Line{Float64, Vector{Float64}}([-1.0, 2.0, 3.0], [0.9486832980505138, 0.0, -0.31622776601683794])

1. The second constructor takes two points, from and to, as keyword

arguments, and returns the line through them. See the algorithm section for details.

julia> Line(from=[-1.0, 2, 3], to=[2.0, 2, 2])
Line{Float64, Vector{Float64}}([-1.0, 2.0, 3.0], [3.0, 0.0, -1.0])

1. The third constructor resembles Line2D and only works for two-dimensional

lines. It takes two inputs, a and b, and constructs the line such that $a ⋅ x = b$.

julia> Line([2.0, 0], 1.)
Line{Float64, Vector{Float64}}([0.5, 0.0], [0.0, 1.0])

Algorithm

Given two points $p ∈ ℝ^n$ and $q ∈ ℝ^n$, the line that passes through these two points is L:{y ∈ ℝ^n: y = p + λ(q - p), λ ∈ ℝ}``.

constraints_list(X::LazySet)

Compute a list of linear constraints of a polyhedral set.

Input

• X – polyhedral set

Output

A list of the linear constraints of X.

Extended help

constraints_list(L::Line)

Output

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

direction(L::Line)

Return the direction of the line.

Input

• L – line

Output

The direction of the line.

Notes

The direction is not necessarily normalized. See normalize(::Line, ::Real) / normalize!(::Line, ::Real).

isuniversal(X::LazySet, witness::Bool=false)

Check whether a set is universal.

Input

• X – set
• witness – (optional, default: false) compute a witness if activated

Output

• If the witness option is deactivated: true iff $X = ℝ^n$
• If the witness option is activated:
  • (true, []) iff $X = ℝ^n$
  • (false, v) iff $X ≠ ℝ^n$ for some $v ∉ X$

Extended help

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

Algorithm

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

and false otherwise.

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

one, and (false, v) where $v ∉ P$ otherwise.

normalize(L::Line{N}, p::Real=N(2)) where {N}

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.

normalize!(L::Line{N}, p::Real=N(2)) where {N}

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.

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

Algorithm

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

∈(x::AbstractVector, X::LazySet)

Check whether a point lies in a set.

Input

• x – point/vector
• X – set

Output

true iff $x ∈ X$.

Extended help

∈(x::AbstractVector, L::Line)

Algorithm

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

linear_map(M::AbstractMatrix, X::LazySet)

Compute the linear map $M · X$.

Input

• M – matrix
• X – set

Output

A set representing the linear map $M · X$.

Extended help

linear_map(M::AbstractMatrix, 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.