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[TL3SHI2 <>]

Hi, Olivier,
I have a question for using duality to solve this problem.
Considering there are two convex polyhedrons which are not intersect with
each other any all.
However, using duality, I can always map the faces of both of the
polyhedrons to
vertices, and the convex hull of the mapped vertices is always exists.
Is it saying that the intersection of the original polyhedrons exists?
I am confused about this problem.


Olivier Devillers-2 wrote
>> Olivier is right.
>> Note that for the dualization, you also need exact constructions as 
>> the dualization is a construction. 
>> Exact_predicates_exact_constructions can be used for that.
> 
> You do not necessarily need exact construction. Exact computation has 
> two effects: ensuring robustness of the algorithms
> and a guarantee of the exactness of the result. If you allow 
> approximation during dualization, you still
> have robustness during the convex hull using epick.
> 
> if you have no vertical faces in your polyhedron, you can use dualization
> by
> plane z=ax+by+c <---->   point (a,b,c)
> since you do not make computation during dualization and thus do not 
> need exact construction here.
> 
> The remaining point is the computation of the representation of input 
> polyhedra has intersection of half spaces
> (it uses computations if the real input is the vertices of the polyhedra).
> 
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