intersection of Interval and HalfSpace/Hyperplane

docs/src/lib/binary_functions.md

@@ -56,6 +56,7 @@ intersection(::AbstractHyperrectangle{N}, ::AbstractHyperrectangle{N}) where {N<
 intersection(::Interval{N}, ::Interval{N}) where {N<:Real}
 intersection(::Interval{N}, ::HalfSpace{N}) where {N<:Real}
 intersection(::Interval{N}, ::Hyperplane{N}) where {N<:Real}
+intersection(::Interval{N}, ::LazySet{N}) where {N<:Real}
 intersection(::AbstractHPolygon{N}, ::AbstractHPolygon{N}, ::Bool=true) where {N<:Real}
 intersection(::AbstractPolyhedron{N}, ::AbstractPolyhedron{N}) where {N<:Real}
 intersection(::Union{VPolytope{N}, VPolygon{N}}, ::Union{VPolytope{N}, VPolygon{N}}) where {N<:Real}

src/concrete_intersection.jl

@@ -151,7 +151,7 @@ end
 
 """
     intersection(x::Interval{N}, y::Interval{N}
-                 )::Union{Interval{N}, EmptySet{N}} where {N<:Real}
+                )::Union{Interval{N}, EmptySet{N}} where {N<:Real}
 
 Return the intersection of two intervals.
 
@@ -293,6 +293,60 @@ function intersection(hp::Hyperplane{N}, X::Interval{N}
     return intersection(X, hp)
 end
 
+"""
+    intersection(X::Interval{N}, Y::LazySet{N}
+                )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+
+Compute the intersection of an interval and a convex set.
+
+### Input
+
+- `X` -- interval
+- `Y` -- convex set
+
+### Output
+
+If the sets do not intersect, the result is the empty set.
+Otherwise the result is the interval that describes the intersection, which may
+be of type `Singleton` if the intersection is very small.
+"""
+function intersection(X::Interval{N}, Y::LazySet{N}
+                     )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+    lower = max(min(X), -ρ(N[-1], Y))
+    upper = min(max(X), ρ(N[1], Y))
+    if _isapprox(lower, upper)
+        return Singleton([lower])
+    elseif lower < upper
+        return Interval(lower, upper)
+    else
+        return EmptySet{N}()
+    end
+end
+
+# symmetric method
+function intersection(Y::LazySet{N}, X::Interval{N}
+                     )::Union{Singleton{N}, EmptySet{N}} where {N<:Real}
+    return intersection(X, Y)
+end
+
+# disambiguation
+function intersection(X::Interval{N}, H::AbstractHyperrectangle{N}
+                     )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{Interval{N}, LazySet{N}}, X, H)
+end
+function intersection(H::AbstractHyperrectangle{N}, X::Interval{N}
+                     )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{Interval{N}, LazySet{N}}, X, H)
+end
+function intersection(X::Interval{N}, S::AbstractSingleton{N}
+                     )::Union{Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{typeof(S), LazySet{N}}, S, X)
+end
+function intersection(S::AbstractSingleton{N}, X::Interval{N}
+                     )::Union{Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{typeof(S), LazySet{N}}, S, X)
+end
+
 """
     intersection(P1::AbstractHPolygon{N},
                  P2::AbstractHPolygon{N}
@@ -500,6 +554,14 @@ function intersection(P::AbstractPolyhedron{N},
                      )::Union{Singleton{N}, EmptySet{N}} where {N<:Real}
     return invoke(intersection, Tuple{typeof(S), LazySet{N}}, S, P)
 end
+function intersection(X::Interval{N}, P::AbstractPolyhedron{N}
+                     )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{Interval{N}, LazySet{N}}, X, P)
+end
+function intersection(P::AbstractPolyhedron{N}, X::Interval{N}
+                     )::Union{Interval{N}, Singleton{N}, EmptySet{N}} where {N<:Real}
+    return invoke(intersection, Tuple{Interval{N}, LazySet{N}}, X, P)
+end
 
 """
     intersection(P1::Union{VPolytope{N}, VPolygon{N}},
@@ -670,6 +732,12 @@ end
 function intersection(S::AbstractSingleton{N}, U::Universe{N}) where {N<:Real}
     return S
 end
+function intersection(X::Interval{N}, U::Universe{N}) where {N<:Real}
+    return X
+end
+function intersection(U::Universe{N}, X::Interval{N}) where {N<:Real}
+    return X
+end
 
 """
     intersection(P::AbstractPolyhedron{N}, rm::ResetMap{N}) where {N<:Real}

test/unit_Interval.jl

@@ -105,6 +105,11 @@ for N in [Float64, Float32, Rational{Int}]
     @test intersection(i, hp) == Singleton(N[1.5])
     hp = Hyperplane(N[-1], N(-3))
     @test intersection(i, hp) == EmptySet{N}()
+    # other intersections
+    Y = Ball1(N[2], N(0.5))
+    @test intersection(i, Y) == Interval(N(1.5), N(2))
+    Y = ConvexHull(Singleton(N[-5]), Singleton(N[-1]))
+    @test intersection(i, Y) == EmptySet{N}()
 
     # disjointness check
     @test !isdisjoint(A, B)