Affine map (AffineMap)

AffineMap{N, S<:LazySet{N}, NM, MAT<:AbstractMatrix{NM},
          VN<:AbstractVector{NM}} <: AbstractAffineMap{N, S}

Type that represents an affine transformation $M⋅X ⊕ v$ of a set $X$, i.e., the set

\[Y = \{ y ∈ ℝ^n : y = Mx + v,\qquad x ∈ X \}.\]

If $X$ is $n$-dimensional, then $M$ should be an $m × n$ matrix and $v$ should be an $m$-dimensional vector.

Fields

• M – matrix
• X – set
• v – translation vector

The fields' getter functions are matrix, set and vector, respectively.

Notes

An affine map is the composition of a linear map and a translation. This type is parametric in the coefficients of the linear map, NM, which may be different from the numeric type of the wrapped set, N. However, the numeric type of the translation vector should be NM.

An affine map preserves convexity: if X is convex, then any affine map of X is convex as well.

Examples

For the examples we create a $3×2$ matrix, a two-dimensional unit square, and a three-dimensional vector. Then we combine them in an AffineMap.

julia> A = [1 2; 1 3; 1 4]; X = BallInf([0, 0], 1); b2 = [1, 2]; b3 = [1, 2, 3];

julia> AffineMap(A, X, b3)
AffineMap{Int64, BallInf{Int64, Vector{Int64}}, Int64, Matrix{Int64}, Vector{Int64}}([1 2; 1 3; 1 4], BallInf{Int64, Vector{Int64}}([0, 0], 1), [1, 2, 3])

For convenience, A does not need to be a matrix; we also allow to use UniformScalings resp. scalars (interpreted as a scaling, i.e., a scaled identity matrix). Scaling by $1$ is ignored and simplified to a pure Translation.

julia> using LinearAlgebra

julia> am = AffineMap(2I, X, b2)
AffineMap{Int64, BallInf{Int64, Vector{Int64}}, Int64, Diagonal{Int64, Vector{Int64}}, Vector{Int64}}([2 0; 0 2], BallInf{Int64, Vector{Int64}}([0, 0], 1), [1, 2])

julia> AffineMap(2, X, b2) == am
true

julia> AffineMap(1, X, b2)
Translation{Int64, BallInf{Int64, Vector{Int64}}, Vector{Int64}}(BallInf{Int64, Vector{Int64}}([0, 0], 1), [1, 2])

Applying a linear map to an AffineMap object combines the two maps into a new AffineMap instance. Again we can make use of the conversion for convenience.

julia> B = [2 0; 0 2]; am2 = B * am
AffineMap{Int64, BallInf{Int64, Vector{Int64}}, Int64, Matrix{Int64}, Vector{Int64}}([4 0; 0 4], BallInf{Int64, Vector{Int64}}([0, 0], 1), [2, 4])

julia> 2 * am == am2
true

The application of an AffineMap to a ZeroSet or an EmptySet is simplified automatically.

julia> AffineMap(A, ZeroSet{Int}(2), b3)
Singleton{Int64, Vector{Int64}}([1, 2, 3])

julia> AffineMap(A, EmptySet{Int}(2), b3)
EmptySet{Int64}(2)