Interval Hulls

In this section we illustrate the interval hull operators as well as several plotting functionalities.

• Interval Hulls
• Balls and Singletons
• Ballinf approximation
• Interval hull approximation
• Symmetric interval hull
• Norm, radius and diameter

Balls and Singletons

Consider a ball in the 2-norm. By default, the coefficients of this set are 64-bit floating point numbers. Other numeric types (such as lower precision floating point, or rational) can be defined with the proper argument types in the Ball2 constructor.

using Plots, LazySets

X = Ball2(ones(2), 0.5)

Ball2{Float64,Array{Float64,1}}([1.0, 1.0], 0.5)

To plot a lazy set, we use the plot function. By design, lazy sets plots overapproximate with box directions only. To have a sharp definition of the borders, use the accuracy as a second argument.

plot(X, 1e-3, aspectratio=1)

To add plots to the same pair of axes we use plot!. Let's add some points of the set which are farthest in some given directions. Single points can be plotted using the Singleton type. In the third line of code we plot the center of the ball picking a custom cross marker.

plot!(Singleton(σ([1., 0], X)))
plot!(Singleton(σ([1., 1], X)))
plot!(Singleton(X.center), markershape=:x)

For the remainder of this section we define another ball in the 2-norm and its convex hull with X.

Y = Ball2([-3,-.5], 0.8)
Z = CH(X, Y)

plot(X, 1e-3, aspectratio=1)
plot!(Y, 1e-3)
plot!(Z, 1e-3, alpha=0.2)

Ballinf approximation

A simple overapproximation with a BallInf is obtained with the ballinf_approximation function, from the Approximations module. It overapproximates a convex set by a tight ball in the infinity norm by evaluating the support vector in the canonical directions.

import LazySets.Approximations.ballinf_approximation

plot(X, 1e-3, aspectratio=1)
plot!(Y, 1e-3)
plot!(Z, 1e-3, alpha=0.2)

Bapprox = ballinf_approximation(Z)

plot!(Bapprox, alpha=0.1)
plot!(Singleton(Bapprox.center), markershape=:x)

Bapprox.center, Bapprox.radius

([-1.15, 0.09999999999999998], 2.65)

Interval hull approximation

If we want to have different lengths for each dimension, instead of the ballinf_approximation, we can use the approximation with a hyperrectangle through the interval_hull function.

import LazySets.Approximations.interval_hull

plot(X, 1e-3, aspectratio=1)
plot!(Y, 1e-3)
plot!(Z, 1e-3, alpha=0.2)

Happrox = interval_hull(Z)

plot!(Happrox, alpha=0.1)
plot!(Singleton(Happrox.center), markershape=:x)

Happrox.center, Happrox.radius

([-1.15, 0.09999999999999998], [2.65, 1.4])

Symmetric interval hull

Contrary to the previous approximations, the symmetric interval hull is centered around the origin. It is defined in the Approximations module as well.

import LazySets.Approximations.symmetric_interval_hull
using SparseArrays

plot(X, 1e-3, aspectratio=1)
plot!(Y, 1e-3)
plot!(Z, 1e-3, alpha=0.2)

S = symmetric_interval_hull(Z)
plot!(S, alpha=0.2)
plot!(Singleton(S.center), markershape=:x)

S.center, S.radius

([0.0, 0.0], [3.8, 1.5])

We can get the list of vertices using the vertices_list function:

vertices_list(S)

4-element Array{Array{Float64,1},1}:
 [3.8, 1.5]
 [-3.8, 1.5]
 [3.8, -1.5]
 [-3.8, -1.5]

For instance, compute the support vector in the south-east direction:

σ([1., -1.], S)

2-element Array{Float64,1}:
  3.8
 -1.5

It is also possible to pass a sparse vector as direction, and the result is a sparse vector:

σ(sparsevec([1., -1.]), S)

2-element SparseArrays.SparseVector{Float64,Int64} with 2 stored entries:
  [1]  =  3.8
  [2]  =  -1.5

Norm, radius and diameter

In this part we illustrate some functions to obtain metric properties of sets, applied to the sets X, Y and Z defined previously, in the infinity norm. These functions apply generally to any LazySet. For some types, specialized methods are triggered automatically through multiple-dispatch.

The norm of a convex set is the norm of the enclosing ball (of the given norm) of minimal volume. For instance:

import LazySets.Approximations: norm, radius, diameter

norm(X), norm(Y), norm(Z)

(1.5, 3.8, 3.8)

The radius of a convex set. It is the radius of the enclosing ball (of the given norm) of minimal volume with the same center. In the previous example,

radius(X), radius(Y), radius(Z)

(0.5, 0.8, 2.65)

Finally, it is sometimes convenient to ask directly the diameter of the set, defined as twice the radius:

diameter(X), diameter(Y), diameter(Z)

(1.0, 1.6, 5.3)