[metapost] Re: Intersections of NURBs
Larry Siebenmann
laurent at math.toronto.edu
Thu Feb 3 21:36:50 CET 2005
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.
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