Affine maps (AbstractAffineMap)

AbstractAffineMap{N, S<:LazySet{N}} <: LazySet{N}

Abstract type for affine maps.

Notes

See AffineMap for a standard implementation of this interface.

Every concrete AbstractAffineMap must define the following methods:

• matrix(::AbstractAffineMap) – return the linear map
• vector(::AbstractAffineMap) – return the affine translation vector
• set(::AbstractAffineMap) – return the set that the map is applied to

The subtypes of AbstractAffineMap:

julia> subtypes(AbstractAffineMap)
7-element Vector{Any}:
 AffineMap
 ExponentialMap
 ExponentialProjectionMap
 InverseLinearMap
 LinearMap
 ResetMap
 Translation

matrix(X::AbstractAffineMap)

Return the matrix of an affine map.

Input

• X – affine map

Output

The matrix of X.

vector(X::AbstractAffineMap)

Return the vector of an affine map.

Input

• X – affine map

Output

The vector of X.

set(X::AbstractAffineMap)

Return the set of an affine map.

Input

• X – affine map

Output

The set of X before applying the map.

an_element(X::LazySet)

Return some element of a nonempty set.

Input

• X – set

Output

An element of X unless X is empty.

Extended help

Algorithm

The implementation relies on the an_element method of the wrapped set.

center(X::LazySet)

Compute the center of a centrally symmetric set.

Input

• X – centrally symmetric set

Output

A vector with the center, or midpoint, of X.

Extended help

center(am::AbstractAffineMap)

Algorithm

The implementation relies on the center method of the wrapped set.

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(am::AbstractAffineMap)

Notes

We assume that the underlying set X is polyhedral, i.e., offers a method constraints_list(X).

Algorithm

This implementation uses the constraints_list method to compute the list of constraints of the translation of a lazy LinearMap.

isbounded(X::LazySet)

Check whether a set is bounded.

Input

• X – set

Output

true iff the set is bounded.

Notes

See also isboundedtype(::Type{<:LazySet}).

Extended help

isbounded(am::AbstractAffineMap; cond_tol::Number=DEFAULT_COND_TOL)

Input

• cond_tol – (optional) tolerance of matrix condition (used to check whether the matrix is invertible)

Algorithm

We first check if the matrix is zero or the wrapped set is bounded. If not, we perform a sufficient check whether the matrix is invertible. If the matrix is invertible, then the map being bounded is equivalent to the wrapped set being bounded, and hence the map is unbounded. Otherwise, we check boundedness via _isbounded_unit_dimensions.

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

Check whether a set is empty.

Input

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

Output

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

Extended help

isempty(am::AbstractAffineMap)

Algorithm

The result is true iff the wrapped set is empty.

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(am::AbstractAffineMap)

Algorithm

An affine map is universal iff the wrapped set is universal and the matrix does not map any dimension to zero.

∈(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, am::AbstractAffineMap)

Algorithm

Observe that $x ∈ M⋅S ⊕ v$ iff $M^{-1}⋅(x - v) ∈ S$. This implementation does not explicitly invert the matrix, which is why it also works for non-square matrices.

Examples

julia> am = AffineMap([2.0 0.0; 0.0 1.0], BallInf([1., 1.], 1.), [-1.0, -1.0]);

julia> [5.0, 1.0] ∈ am
false

julia> [3.0, 1.0] ∈ am
true

An example with a non-square matrix:

julia> B = BallInf(zeros(4), 1.);

julia> M = [1. 0 0 0; 0 1 0 0]/2;

julia> [0.5, 0.5] ∈ M*B
true

vertices_list(X::LazySet)

Compute a list of vertices of a polytopic set.

Input

• X – polytopic set

Output

A list of the vertices of X.

Extended help

vertices_list(am::AbstractAffineMap; [apply_convex_hull]::Bool=true)

Input

• apply_convex_hull – (optional, default: true) if true, apply the convex hull operation to the list of vertices transformed by the affine map

Algorithm

This implementation computes all vertices of X, then transforms them through the affine map, i.e., x ↦ M*x + v for each vertex x of X. By default, the convex-hull operation is taken before returning this list. For dimensions three or higher, this operation relies on the functionality through the concrete polyhedra library Polyhedra.jl.

If you are not interested in taking the convex hull of the resulting vertices under the affine map, pass apply_convex_hull=false as a keyword argument.

Note that we assume that the underlying set X is polytopic, either concretely or lazily, i.e., the function vertices_list should be applicable.

volume(X::LazySet)

Compute the volume, or Lebesgue measure, of a set.

Input

• X – set

Output

A real number representing the Lebesgue measure of X.

Notes

The Lebesgue measure has the following common special cases:

• In 1D, it coincides with the length.
• In 2D, it coincides with the area (see also area).
• In 3D, it coincides with the volume.

In higher dimensions, it is also known as the hypervolume or simply volume.

Extended help

volume(am::AbstractAffineMap)

Notes

This implementation requires a dimension-preserving map (i.e., a square matrix).

Algorithm

A square linear map scales the volume of any set by its absolute determinant. A translation does not affect the volume. Thus, the volume of M * X + {v} is |det(M)| * volume(X).