Line

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

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=[-4.0, 2, 4])
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 $p ∈ \mathbb{R}^n$ and $q ∈ \mathbb{R}^n$, the line that passes through these two points is L:{y ∈ \mathbb{R}^n: y = p + λ(q - p), λ ∈ \mathbb{R}}``.

dim(L::Line)

Return the ambient dimension of a line.

Input

• L – line

Output

The ambient dimension of the line.

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.

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.

∈(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 $d$.

an_element(L::Line)

Return some element of a line.

Input

• L – line

Output

An element on the line.

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).

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.

isbounded(L::Line)

Determine whether a line is bounded.

Input

• L – line

Output

false.

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.

isempty(L::Line)

Check whether a line is empty or not.

Input

• L – line

Output

false.

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.

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.

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.

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.

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.

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.