CGAL users discussion list

Le Mer 7 mars 2007 1:06, 

 a écrit :
> - in the case of non flat polygons, we can assume that three successive
> vertices are never aligned (by removing unnecessary vertices), and we have
> a non-flat locally convex curve P. If it has no self intersection, it is
> convex, and it has exactly 2 changes of lexicographic order. If it has
> self intersections, we have to show that there are at least 3 changes of
> lexicographic order. Let us call X some intersection point. We can split P
> at X into two polygons, P1 and P2. As any polygon, P1 and P2 have at least
> 2 changes of order each. P is the union of P1 and P2. As such, it has at
> least 2 + 2 changes, maybe minus one, in case X is a point with an order
> change for both P1 and P2. Hence, we always have at least 3 changes in
> case there is an auto intersection. This concludes the proof.

Sorry, slight mistake: "maybe minus one, in case X is a point with an
order change for both P1 and P2" should be replaced by "maybe minus one,
in case X is a point with an order change for P1 or for P2".
-- 
Camille Wormser