Re: [help-3dldf] Re: all intersections between two paths *

[Martijn van Manen]

I think those Bezier curves are of degree 3, aren't they?
Bezout's theorem tells you that as long as you're in 
PC^2 the number of intersections of two of them is 
3 times 3 =9, multiplicity counted. SO when they're simply
tangent, that counts as two. 
Now you're neither counting complex intersection points, nor
checking what happens at infinity. The maximum remains 9.
Numerical problems are ill-posed in this situation.
To find intersection points you have to take the implicit
equation for the one F(x,y)=0 and the parameterization for
the other (g1(t),g2(t))
You end up with a polynomial of degree 9, as predicted by
Bezout. The zeroes of those can only be determined numerically
generally speaking, that is, you can only approximate.
But there are many friendly looking polynomials with the
following satanic property.  Add some
tiny number to the coefficients and the zeroes change
completely.
Happy new year all of you. I've begun working on the 
next installment of my pondering on 3DLDF.


----- Original Message -----
From: "Laurence Finston" <address@hidden>
To: "Larry Siebenmann" <address@hidden>
Subject: [help-3dldf] Re: all intersections between two paths *
Date: Tue, 11 Jan 2005 12:47:21 +0100 (MET)

>
> On Tue, 11 Jan 2005, Larry Siebenmann wrote:
>
> >
> >  > How does one arrive at the value 9 for the maximum number of 
> intersections
> >  > of other `paths' of length 1?  Is there a (relatively) simple proof, or
> >  > can I look this up somewhere?
> >
> > Given a generic bezier degree n planar path t |--> b(t), there
> > is an "implicitization" process (that goes back to Euler they
> > say) [...]
>
>
> Thank you for your explanation.
>
> >
> >  > Given the manipulations possible with connectors, I think it may be
> >  > difficult to filter out `paths' with infinitely many intersections.
> >
> > I am optimistic that it can be done in some practical sense. Have
> > you a specific challenge?
>
> My specific challenge is implementing routines for finding
> intersections of arbitrary three-dimensional Metafont-like
> `paths' in GNU 3DLDF.  Currently, this is not possible.  I
> plan to reimplement them as NURBs because of the property of
> projective invariance, which the latter possess.
> I am not, however, bound by the arithmetical limitations
> built into MF, since I haven't tried to implement whole
> number arithmetic and just use `floats' or `doubles',
> depending on the value of a preprocessor macro.
>
> >
> > I assume you mean by connectors the short paths that metafont seems
> > to insert to link paths that almost but not quite chain together.
> > (?)
>
> No, that's not what I meant, but I expressed myself poorly.
> I should have said "given the manipulations possible with
> control points."  What I meant by "connectors" was actually
> "path joins", e.g.,  ".. tension a and b .."  and "direction
> specifiers", which, when present, are ultimately used to
> find control points.
>
> >
> > By pairs of `paths' with infinitely many intersections I imagine
> > you include paths that remain very near to one another for a
> > 'noticeable' stretch?
>
> Yes.  I also think that under certain circustances, the
> dimensions of the `pens' used for drawing the `paths' should
> be taken into account, since the drawings of two `paths' might
> intersect or become tangent although the `paths' themselves
> do not.  This opens another can of worms, though.
>
> In discussions about intersections in GNU 3DLDF,
> Martijn van Manen has pointed out that the case of objects
> getting very close without actually becoming tangent or
> intersecting are also problematical.
>
> Laurence
>
>
> _______________________________________________
> help-3dldf mailing list
> address@hidden
> http://lists.gnu.org/mailman/listinfo/help-3dldf



Mieulx est de ris que de larmes escripre,
Pour ce que rire est le propre de l'homme.

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