[metapost] Re: Some observations on turning and self intersection

Larry Siebenmann laurent at math.toronto.edu
Tue Feb 8 22:24:59 CET 2005

Hello Dan, and others

You write concerning the global affine classification
of cubic b'ezier curves:

 > This is probably all well known to geometers, but I only
 > discovered it recently and thought it might help with some
 > of the questions discussed here lately.

Yes probably known.  (But maybe badly known:)

There is a more complex problem of classifying
implicitly defined real cubic curves that was first
solved by Isaac Newton via heroic calculations.
(I believe there were some omissions/duplications.)  The
problem is that Newton discussed loci (zeros of a cubic
polynomial in 2 variables) so that it is tiresome for anyone
but an expert to see exactly what Newton's theory says about
bezier curves.

I expect that Knuth and Hobby learned the affine as well as
orthogonal classification to create metafont.  Hobby is an
expert on numerical implicitization of bezier cubics so he
probably knows exactly how his beziers fit into Newton's

There are now dozens of books on computer graphics and any
one of them could/should include the classification.  But our
math library has few -- and *none* of the volumes recommended
by/to Laurence F.

I agree that mp users should be able to think about and
manipulate various classifications of bezier cubics such as
orthogonal, symplectic, and affine.  That will 
require some macros and maybe some improvements in
the mp core.

Laurent S.

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