[metapost] re: a generic class of cubic bezier curves
Larry Siebenmann
laurent at math.toronto.edu
Mon Feb 14 13:48:15 CET 2005
Sorry, there is a serious glitch in the comment (ii) in the
preamble to my last posting (Sun, 13 Feb 2005).
The phrase
(*) <<...insisting that the first derivative path F'(t) describe a
non-degenarate parabola in R^2.>>
should have been
(#) <<...insisting that the first derivative path F'(t) describe a
non-degenarate parabola in R^2 not containing the origin (0,0).>>
Condition (*) alone was *equivalent* to the double nondegeneracy
condition in my 2001 notes.
Cheers
Laurent S.
PS. Incidentally, as some of you know very well, the 'bump' case
(1) with two inflexion points on the curve F(t) occurs when the
origin lies *outside* the parabola F'(t). The two times at which
the line (0,0)--F'(t) is tangent to the parabola described by F'(t)
are the two times of inflexion of F(t). The 'loop' case (2) with a
self-intersection on F(t) occurs when the origin lies *inside* the
parabola.
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