#583 - Concrete intersection of two HPolygons

[schillic]

Closes #583.

[mforets]

I was expecting that addconstraint! takes care of adding the constraints properly. And it provides different methods such as linear and binary search (and eventually can use a specialized addconstraint! based on a specific concrete AbstractHPolygon, but i don't have an example right now).

This is:

function intersection(...)
    P = HPolygon{N}()
    addconstraint!.(P, constraints_list(P1))
    addconstraint!.(P, constraints_list(P2))
    return P
end

But the repeated halfspaces is something to be fixed in addconstraint!. Or we add a method to remove the redundant constraints. For example:

julia> h = convert(HPolygon, BallInf(ones(2), 1.0));

julia> constraints_list(h)
4-element Array{LazySets.HalfSpace{Float64},1}:
 LazySets.HalfSpace{Float64}([1.0, 0.0], 2.0)   
 LazySets.HalfSpace{Float64}([0.0, 1.0], 2.0)   
 LazySets.HalfSpace{Float64}([-1.0, -0.0], -0.0)
 LazySets.HalfSpace{Float64}([-0.0, -1.0], -0.0)

julia> P = HPolygon{Float64}()
LazySets.HPolygon{Float64}(LazySets.HalfSpace{Float64}[])

julia> addconstraint!.(P, [constraints_list(h); constraints_list(h)]);

julia> constraints_list(P)
8-element Array{LazySets.HalfSpace{Float64},1}:
 LazySets.HalfSpace{Float64}([1.0, 0.0], 2.0)   
 LazySets.HalfSpace{Float64}([1.0, 0.0], 2.0)   
 LazySets.HalfSpace{Float64}([0.0, 1.0], 2.0)   
 LazySets.HalfSpace{Float64}([0.0, 1.0], 2.0)   
 LazySets.HalfSpace{Float64}([-1.0, -0.0], -0.0)
 LazySets.HalfSpace{Float64}([-1.0, -0.0], -0.0)
 LazySets.HalfSpace{Float64}([-0.0, -1.0], -0.0)
 LazySets.HalfSpace{Float64}([-0.0, -1.0], -0.0)

[schillic]

The thing is that adding constraints all at once gives a worst-case quadratic algorithm. Interleaved insertion is worst-case linear.

[mforets]

Maybe extend a bit: "The thing is that adding constraints all at once gives a worst-case quadratic algorithm. Interleaved insertion is worst-case linear."