[metapost] cubic b'ezier curves

[Daniel H. Luecking]

On Wed, 9 Feb 2005, Larry Siebenmann wrote:

>
>
> Hello Dan, and others
>
> I once worked through the affine classification of bezier
> cubics (in late 1991 it seems) and so I can check yours.
>
> Modulo my own possible slipups you seem to have just 2
> duplications in:
>
[list snipped]
>
> The duplications:- the three cases
>
>  >   (t,t^3)      y = x^3, ypart pauses at t=0
>  >   (t,t+t^3)    like above, but ypart never pauses
>  >   (t,-t+t^3)   as above, but ypart pauses twice.
>
> are one and the same.  The linear selfmap of R^2 given by (x,y)
> |-> (x,y + ax) sends the first curve to (t,at+t^3). So the next
> two are equivalent to the first.  Thus there are 7 not 9 normal
> forms for degree exactly 3.

Yes, I know. I was actually trying for a certain symmetry (three cubic
ypart cases for each type of xpart. I realized these were essentially
the same only after I sent it to the list.

>
> I notice that I even wrote up (but never posted) an exposition
> of one possible classification procedure for the last two cases.
> It was to be a sequel to my posting on the metafont list of Tue,
> 18 Dec 2001 06:34:16 -0500, see metafont archives at
> <http://www.gutenberg.eu.org/pub/gut>. That posting was called
> "# quadratic bezier curves in metafont" and was the third in a
> row on the relation of cubic and quadratic b'ezier curves; none
> received any response -- and radio silence ensued for 3 weeks!

I'll have to reread that.

Regards,
Dan

-- 
Dan Luecking
Dept. of Mathematical Sciences
University of Arkansas
Fayetteville, AR 72101