[metapost] Re: Intersections of NURBs

[Larry Siebenmann]

Hi Laurence F.

I carelessly wrote on Sun, 30 Jan 2005 22:02:27 -0500:

 > The notion of convex hull in RP^n seems well defined for
 > objects that miss some hyperplane.

I should have written

 | The notion of convex hull in n-dimensional real
 | projective space RP^n seems well defined for
 | objects that miss some hyperplane and are affine convex 
 | in the the complement of that hyperplane.

OR (consequently)

 | The notion of convex hull in RP^n seems well defined for
 | all small objects.

I had just carefully given a counterexample to the quoted
statement!! -- namely, my four point set {A',B',C',D'}. The
modified statements seem to get around the difficulty I used
{A',B',C',D'} to illustrate to Laurence F. concerning
generalization of the de Casteljau convex hull property for
projectively transformed bezier curves.

 LF> At any rate, my sources on 3D graphics all indicate that
 > NURBs are the de facto standard for 3D applications.  So
 > far, I've had to take this on faith.

I lack faith. NURBSs cannot be the choice to end all choices
since the envelope of a circular penstroke along a bezier
cubic is not a NURBS; it is in general an *non*-rational
algebraic curve; hence not a NURBS!  Curve nirvana has not
yet been found.

Cheers

    Laurent S.