CGAL users discussion list

[Eric Berberich <>]

Thomas Zangl - Home wrote:
>  
> Am Thu, 3 Jul 2008 22:57:58 +0200, schrieb "Pedro_Machado_Manhães_de_Castro" <>:
>
> Hi!
>
> > typedef typename Kernel::FT                                 FT;
> > typedef typename Root_of_traits< FT >::RootOf_2 Root_of_2;
> >
> > const Root_of_2 mu = make_root_of_2(-b/(2*a), FT(1), (b*b - 4*a*c)/(4*a*a));
> > const Root_of_2 x = x1 + mu*(x2-x1);
> > const Root_of_2 y = y1 + mu*(y2-y1);
> > const Root_of_2 z = z1 + mu*(z2-z1);
> >
> > Note: You can&#39;t store them as coordinates for points.
> > You may want to have a look at CORE::Expr (slow) if you need absolutely to work with CGAL points. 
> > Otherwise, you have to find another way to store them. Just remember that whenever using FT&#39;s like Gmpq or MP_Float, you are using something that do not support Root_of_2 (rationals do not include them), and the correspondent Points will not support as well.
>
> I already suspected something like this. At least, any ideas how I can
> improve accuracy of my intersection code?
>
> The result should be a Point_3 based on Gmpq.
>
> TIA,

gmpq is not able to exactly represent the square-root that naturally 
exists (except for degenerate cases, where the root is a rational) for 
the coordinates of the intersection points of a sphere and a line. 
Either you compute an interval of two gmpq that contain the exact 
coordinates, or you switch to CORE::Expr (or leda::real) or you use the 
Root_of number types. I currently don't see another option.

It is the mathematics that result in this "gmpq-is-not-sufficient" case. 
Believe us, it would be much nicer, if curved geometry would stay in 
rational numbers ;-)

eriC