CGAL users discussion list

[Iosif Pinelis <>]

 wrote:

> Le Mar 6 mars 2007 22:53, 
>
>  a écrit :
>  
>
> > The definition of convexity seems to be missing from the manual. However,
> > there again, it is the usual one:
> >
> > non-simple polygons are not convex, a simple polygon is convex if the
> > union of its bounded side and its edges is convex.
> >    
>
> I am wrong here:
> a flat polygon (hence non-simple) is considered convex if each point of
> the polygon which is not a vertex belongs to exactly two edges.
> In nicer terms, if P is the set of polygons, and CSP is the set of convex
> simple polygons, the set of convex polygons CP is the topological closure
> of CSP in P for the topology induced by the pseudo-distance
>
> d(P1, P2) = min_{all parameterizations f_P1 f_P2} d(f_P1, f_P2)
>  
Thank you again. This may address one of my 3 remaining questions. I am 
not quite sure, though, what exactly you mean here by:

  1. a polygon (a closed piecewise-linear curve?)
  2. a flat polygon (a polygon for which some of the determinants are
     zero?)
  3. a vertex (where the determinant is nonzero?)

Given the definition of a polygon as a closed piecewise-linear curve, 
your nicer, topological definition would be clear. But the "biggest" 
question remains: is there a rigorous proof  that the CGAL convexity 
test exactly corresponds to the definition that you gave?

Iosif Pinelis