# [metapost] envelope proposal 4

Larry Siebenmann laurent at math.toronto.edu
Fri Mar 18 00:24:06 CET 2005

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Hi Jacko,

I have been talking too fast about envellopes, sliding over
several difficulties.  Here are a couple of
of my sins.

Sin (1).

<< On the other hand the (possibly disconnected)
"perimeter" ("frontier") of the inked region does vary
continuously with the path, and that is all that meets
the eye of the reader. >>

I should have said "semicontinuously" as a subset of the
plane. Only the area of the inked  region clearly varies
continuously in the ordinary sense. Also one should forget
parametrization here. "Semicontinuous" means roughly that
although things can *vanish* instantaneously, they cannot
get big instantanously. Consider the inner envellope of a
varying rectangle for a fixed pen. It can disaappear
instanty, in case at a given path deformation parameter it is a
mere segment.

Sin (2).

<< ...define continous envelope paths for proposal 4 in all
cases SO AS TO VARY CONTINUOUSLY AS THE PATH CONTROL POINTS
VARY CONTINUOUSLY. >>

This notion of continuous variation only makes sense if
we reparametrize proportional to arc length, keeping the
same interval of definition.  Then it seems to work
except for a finite list of exceptional phenomena such
as cusps. Inflexions that run parallel to a polygonal
pen edge are another exceptional phenomenon.

Fortunately, most of the horrors one meets are ultimately
drowned in ink and forgotten.  Still, I reckon that the
polygonal pen business was not an easy wicket for those who
first worked it out.  There is surely a literature on it;
has anyone seen a good reference?

I am not keen to get seriously involved with polygonal
pens...

LS

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